Linear

📁 Source: Mathlib/Analysis/Analytic/Linear.lean

Statistics

Metric	Count
Definitions fpowerSeriesBilinear, uncurryBilinear	2
Theorems analyticAt, analyticOn, analyticOnNhd, analyticWithinAt, analyticAt, analyticAt_bilinear, analyticOn, analyticOnNhd, analyticOnNhd_bilinear, analyticOn_bilinear, analyticWithinAt, analyticWithinAt_bilinear, cpolynomialAt, cpolynomialOn, fpowerSeriesBilinear_apply_add_three, fpowerSeriesBilinear_apply_one, fpowerSeriesBilinear_apply_two, fpowerSeriesBilinear_apply_zero, fpowerSeriesBilinear_radius, fpowerSeries_radius, hasFPowerSeriesAt, hasFPowerSeriesAt_bilinear, hasFPowerSeriesOnBall, hasFPowerSeriesOnBall_bilinear, hasFiniteFPowerSeriesOnBall, uncurryBilinear_apply, analyticAt, analyticOn, analyticOnNhd, analyticWithinAt, analyticAt_fst, analyticAt_id, analyticAt_snd, analyticOnNhd_fst, analyticOnNhd_id, analyticOnNhd_snd, analyticOn_fst, analyticOn_id, analyticOn_snd, analyticWithinAt_fst, analyticWithinAt_id, analyticWithinAt_snd	42
Total	44

ContinuousLinearEquiv

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
analyticAt 📖	mathematical	—	AnalyticAt DFunLike.coe ContinuousLinearEquiv DivisionSemiring.toSemiring Semifield.toDivisionSemiring Field.toSemifield NormedField.toField NontriviallyNormedField.toNormedField RingHom.id Semiring.toNonAssocSemiring RingHomInvPair.ids UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddCommGroup.toPseudoMetricSpace NormedAddCommGroup.toSeminormedAddCommGroup ESeminormedAddCommMonoid.toAddCommMonoid ENormedAddCommMonoid.toESeminormedAddCommMonoid NormedAddCommGroup.toENormedAddCommMonoid NormedSpace.toModule EquivLike.toFunLike equivLike	—	RingHomInvPair.ids HasFPowerSeriesAt.analyticAt ContinuousLinearMap.hasFPowerSeriesAt
analyticOn 📖	mathematical	—	AnalyticOn DFunLike.coe ContinuousLinearEquiv DivisionSemiring.toSemiring Semifield.toDivisionSemiring Field.toSemifield NormedField.toField NontriviallyNormedField.toNormedField RingHom.id Semiring.toNonAssocSemiring RingHomInvPair.ids UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddCommGroup.toPseudoMetricSpace NormedAddCommGroup.toSeminormedAddCommGroup ESeminormedAddCommMonoid.toAddCommMonoid ENormedAddCommMonoid.toESeminormedAddCommMonoid NormedAddCommGroup.toENormedAddCommMonoid NormedSpace.toModule EquivLike.toFunLike equivLike	—	RingHomInvPair.ids analyticWithinAt
analyticOnNhd 📖	mathematical	—	AnalyticOnNhd DFunLike.coe ContinuousLinearEquiv DivisionSemiring.toSemiring Semifield.toDivisionSemiring Field.toSemifield NormedField.toField NontriviallyNormedField.toNormedField RingHom.id Semiring.toNonAssocSemiring RingHomInvPair.ids UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddCommGroup.toPseudoMetricSpace NormedAddCommGroup.toSeminormedAddCommGroup ESeminormedAddCommMonoid.toAddCommMonoid ENormedAddCommMonoid.toESeminormedAddCommMonoid NormedAddCommGroup.toENormedAddCommMonoid NormedSpace.toModule EquivLike.toFunLike equivLike	—	RingHomInvPair.ids analyticAt
analyticWithinAt 📖	mathematical	—	AnalyticWithinAt DFunLike.coe ContinuousLinearEquiv DivisionSemiring.toSemiring Semifield.toDivisionSemiring Field.toSemifield NormedField.toField NontriviallyNormedField.toNormedField RingHom.id Semiring.toNonAssocSemiring RingHomInvPair.ids UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddCommGroup.toPseudoMetricSpace NormedAddCommGroup.toSeminormedAddCommGroup ESeminormedAddCommMonoid.toAddCommMonoid ENormedAddCommMonoid.toESeminormedAddCommMonoid NormedAddCommGroup.toENormedAddCommMonoid NormedSpace.toModule EquivLike.toFunLike equivLike	—	RingHomInvPair.ids AnalyticAt.analyticWithinAt analyticAt

ContinuousLinearMap

Definitions

Name	Category	Theorems
fpowerSeriesBilinear 📖	CompOp	7 math math: fpowerSeriesBilinear_apply_zero, fpowerSeriesBilinear_apply_add_three, hasFPowerSeriesOnBall_bilinear, hasFPowerSeriesAt_bilinear, fpowerSeriesBilinear_radius, fpowerSeriesBilinear_apply_two, fpowerSeriesBilinear_apply_one
uncurryBilinear 📖	CompOp	2 math math: uncurryBilinear_apply, fpowerSeriesBilinear_apply_two

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
analyticAt 📖	mathematical	—	AnalyticAt DFunLike.coe ContinuousLinearMap DivisionSemiring.toSemiring Semifield.toDivisionSemiring Field.toSemifield NormedField.toField NontriviallyNormedField.toNormedField RingHom.id Semiring.toNonAssocSemiring UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddCommGroup.toPseudoMetricSpace NormedAddCommGroup.toSeminormedAddCommGroup ESeminormedAddCommMonoid.toAddCommMonoid ENormedAddCommMonoid.toESeminormedAddCommMonoid NormedAddCommGroup.toENormedAddCommMonoid NormedSpace.toModule funLike	—	HasFPowerSeriesAt.analyticAt hasFPowerSeriesAt
analyticAt_bilinear 📖	mathematical	—	AnalyticAt Prod.normedAddCommGroup Prod.normedSpace NontriviallyNormedField.toNormedField NormedAddCommGroup.toSeminormedAddCommGroup DFunLike.coe ContinuousLinearMap DivisionSemiring.toSemiring Semifield.toDivisionSemiring Field.toSemifield NormedField.toField RingHom.id Semiring.toNonAssocSemiring UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddCommGroup.toPseudoMetricSpace ESeminormedAddCommMonoid.toAddCommMonoid ENormedAddCommMonoid.toESeminormedAddCommMonoid NormedAddCommGroup.toENormedAddCommMonoid NormedSpace.toModule funLike topologicalSpace SeminormedAddCommGroup.toAddCommGroup SeminormedAddCommGroup.toIsTopologicalAddGroup addCommMonoid IsTopologicalAddGroup.toContinuousAdd NormedAddGroup.toAddGroup NormedAddCommGroup.toNormedAddGroup module smulCommClass_self CommRing.toCommMonoid Field.toCommRing DistribMulAction.toMulAction CommMonoid.toMonoid AddCommMonoid.toAddMonoid Module.toDistribMulAction Ring.toSemiring CommRing.toRing UniformContinuousConstSMul.to_continuousConstSMul SMulZeroClass.toSMul AddZero.toZero AddZeroClass.toAddZero AddMonoid.toAddZeroClass DistribSMul.toSMulZeroClass DistribMulAction.toDistribSMul MonoidWithZero.toMonoid Semiring.toMonoidWithZero IsBoundedSMul.toUniformContinuousConstSMul SeminormedRing.toPseudoMetricSpace SeminormedCommRing.toSeminormedRing NormedCommRing.toSeminormedCommRing NormedField.toNormedCommRing MulZeroClass.toZero NonUnitalNonAssocSemiring.toMulZeroClass NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing NonUnitalCommRing.toNonUnitalNonAssocCommRing NonUnitalNormedCommRing.toNonUnitalCommRing NormedCommRing.toNonUnitalNormedCommRing NegZeroClass.toZero SubNegZeroMonoid.toNegZeroClass SubtractionMonoid.toSubNegZeroMonoid SubtractionCommMonoid.toSubtractionMonoid AddCommGroup.toDivisionAddCommMonoid NormedAddCommGroup.toAddCommGroup NormedSpace.toIsBoundedSMul	—	SeminormedAddCommGroup.toIsTopologicalAddGroup IsTopologicalAddGroup.toContinuousAdd smulCommClass_self UniformContinuousConstSMul.to_continuousConstSMul IsBoundedSMul.toUniformContinuousConstSMul NormedSpace.toIsBoundedSMul HasFPowerSeriesAt.analyticAt hasFPowerSeriesAt_bilinear
analyticOn 📖	mathematical	—	AnalyticOn DFunLike.coe ContinuousLinearMap DivisionSemiring.toSemiring Semifield.toDivisionSemiring Field.toSemifield NormedField.toField NontriviallyNormedField.toNormedField RingHom.id Semiring.toNonAssocSemiring UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddCommGroup.toPseudoMetricSpace NormedAddCommGroup.toSeminormedAddCommGroup ESeminormedAddCommMonoid.toAddCommMonoid ENormedAddCommMonoid.toESeminormedAddCommMonoid NormedAddCommGroup.toENormedAddCommMonoid NormedSpace.toModule funLike	—	analyticWithinAt
analyticOnNhd 📖	mathematical	—	AnalyticOnNhd DFunLike.coe ContinuousLinearMap DivisionSemiring.toSemiring Semifield.toDivisionSemiring Field.toSemifield NormedField.toField NontriviallyNormedField.toNormedField RingHom.id Semiring.toNonAssocSemiring UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddCommGroup.toPseudoMetricSpace NormedAddCommGroup.toSeminormedAddCommGroup ESeminormedAddCommMonoid.toAddCommMonoid ENormedAddCommMonoid.toESeminormedAddCommMonoid NormedAddCommGroup.toENormedAddCommMonoid NormedSpace.toModule funLike	—	analyticAt
analyticOnNhd_bilinear 📖	mathematical	—	AnalyticOnNhd Prod.normedAddCommGroup Prod.normedSpace NontriviallyNormedField.toNormedField NormedAddCommGroup.toSeminormedAddCommGroup DFunLike.coe ContinuousLinearMap DivisionSemiring.toSemiring Semifield.toDivisionSemiring Field.toSemifield NormedField.toField RingHom.id Semiring.toNonAssocSemiring UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddCommGroup.toPseudoMetricSpace ESeminormedAddCommMonoid.toAddCommMonoid ENormedAddCommMonoid.toESeminormedAddCommMonoid NormedAddCommGroup.toENormedAddCommMonoid NormedSpace.toModule funLike topologicalSpace SeminormedAddCommGroup.toAddCommGroup SeminormedAddCommGroup.toIsTopologicalAddGroup addCommMonoid IsTopologicalAddGroup.toContinuousAdd NormedAddGroup.toAddGroup NormedAddCommGroup.toNormedAddGroup module smulCommClass_self CommRing.toCommMonoid Field.toCommRing DistribMulAction.toMulAction CommMonoid.toMonoid AddCommMonoid.toAddMonoid Module.toDistribMulAction Ring.toSemiring CommRing.toRing UniformContinuousConstSMul.to_continuousConstSMul SMulZeroClass.toSMul AddZero.toZero AddZeroClass.toAddZero AddMonoid.toAddZeroClass DistribSMul.toSMulZeroClass DistribMulAction.toDistribSMul MonoidWithZero.toMonoid Semiring.toMonoidWithZero IsBoundedSMul.toUniformContinuousConstSMul SeminormedRing.toPseudoMetricSpace SeminormedCommRing.toSeminormedRing NormedCommRing.toSeminormedCommRing NormedField.toNormedCommRing MulZeroClass.toZero NonUnitalNonAssocSemiring.toMulZeroClass NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing NonUnitalCommRing.toNonUnitalNonAssocCommRing NonUnitalNormedCommRing.toNonUnitalCommRing NormedCommRing.toNonUnitalNormedCommRing NegZeroClass.toZero SubNegZeroMonoid.toNegZeroClass SubtractionMonoid.toSubNegZeroMonoid SubtractionCommMonoid.toSubtractionMonoid AddCommGroup.toDivisionAddCommMonoid NormedAddCommGroup.toAddCommGroup NormedSpace.toIsBoundedSMul	—	SeminormedAddCommGroup.toIsTopologicalAddGroup IsTopologicalAddGroup.toContinuousAdd smulCommClass_self UniformContinuousConstSMul.to_continuousConstSMul IsBoundedSMul.toUniformContinuousConstSMul NormedSpace.toIsBoundedSMul analyticAt_bilinear
analyticOn_bilinear 📖	mathematical	—	AnalyticOn Prod.normedAddCommGroup Prod.normedSpace NontriviallyNormedField.toNormedField NormedAddCommGroup.toSeminormedAddCommGroup DFunLike.coe ContinuousLinearMap DivisionSemiring.toSemiring Semifield.toDivisionSemiring Field.toSemifield NormedField.toField RingHom.id Semiring.toNonAssocSemiring UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddCommGroup.toPseudoMetricSpace ESeminormedAddCommMonoid.toAddCommMonoid ENormedAddCommMonoid.toESeminormedAddCommMonoid NormedAddCommGroup.toENormedAddCommMonoid NormedSpace.toModule funLike topologicalSpace SeminormedAddCommGroup.toAddCommGroup SeminormedAddCommGroup.toIsTopologicalAddGroup addCommMonoid IsTopologicalAddGroup.toContinuousAdd NormedAddGroup.toAddGroup NormedAddCommGroup.toNormedAddGroup module smulCommClass_self CommRing.toCommMonoid Field.toCommRing DistribMulAction.toMulAction CommMonoid.toMonoid AddCommMonoid.toAddMonoid Module.toDistribMulAction Ring.toSemiring CommRing.toRing UniformContinuousConstSMul.to_continuousConstSMul SMulZeroClass.toSMul AddZero.toZero AddZeroClass.toAddZero AddMonoid.toAddZeroClass DistribSMul.toSMulZeroClass DistribMulAction.toDistribSMul MonoidWithZero.toMonoid Semiring.toMonoidWithZero IsBoundedSMul.toUniformContinuousConstSMul SeminormedRing.toPseudoMetricSpace SeminormedCommRing.toSeminormedRing NormedCommRing.toSeminormedCommRing NormedField.toNormedCommRing MulZeroClass.toZero NonUnitalNonAssocSemiring.toMulZeroClass NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing NonUnitalCommRing.toNonUnitalNonAssocCommRing NonUnitalNormedCommRing.toNonUnitalCommRing NormedCommRing.toNonUnitalNormedCommRing NegZeroClass.toZero SubNegZeroMonoid.toNegZeroClass SubtractionMonoid.toSubNegZeroMonoid SubtractionCommMonoid.toSubtractionMonoid AddCommGroup.toDivisionAddCommMonoid NormedAddCommGroup.toAddCommGroup NormedSpace.toIsBoundedSMul	—	SeminormedAddCommGroup.toIsTopologicalAddGroup IsTopologicalAddGroup.toContinuousAdd smulCommClass_self UniformContinuousConstSMul.to_continuousConstSMul IsBoundedSMul.toUniformContinuousConstSMul NormedSpace.toIsBoundedSMul AnalyticOnNhd.analyticOn analyticOnNhd_bilinear
analyticWithinAt 📖	mathematical	—	AnalyticWithinAt DFunLike.coe ContinuousLinearMap DivisionSemiring.toSemiring Semifield.toDivisionSemiring Field.toSemifield NormedField.toField NontriviallyNormedField.toNormedField RingHom.id Semiring.toNonAssocSemiring UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddCommGroup.toPseudoMetricSpace NormedAddCommGroup.toSeminormedAddCommGroup ESeminormedAddCommMonoid.toAddCommMonoid ENormedAddCommMonoid.toESeminormedAddCommMonoid NormedAddCommGroup.toENormedAddCommMonoid NormedSpace.toModule funLike	—	AnalyticAt.analyticWithinAt analyticAt
analyticWithinAt_bilinear 📖	mathematical	—	AnalyticWithinAt Prod.normedAddCommGroup Prod.normedSpace NontriviallyNormedField.toNormedField NormedAddCommGroup.toSeminormedAddCommGroup DFunLike.coe ContinuousLinearMap DivisionSemiring.toSemiring Semifield.toDivisionSemiring Field.toSemifield NormedField.toField RingHom.id Semiring.toNonAssocSemiring UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddCommGroup.toPseudoMetricSpace ESeminormedAddCommMonoid.toAddCommMonoid ENormedAddCommMonoid.toESeminormedAddCommMonoid NormedAddCommGroup.toENormedAddCommMonoid NormedSpace.toModule funLike topologicalSpace SeminormedAddCommGroup.toAddCommGroup SeminormedAddCommGroup.toIsTopologicalAddGroup addCommMonoid IsTopologicalAddGroup.toContinuousAdd NormedAddGroup.toAddGroup NormedAddCommGroup.toNormedAddGroup module smulCommClass_self CommRing.toCommMonoid Field.toCommRing DistribMulAction.toMulAction CommMonoid.toMonoid AddCommMonoid.toAddMonoid Module.toDistribMulAction Ring.toSemiring CommRing.toRing UniformContinuousConstSMul.to_continuousConstSMul SMulZeroClass.toSMul AddZero.toZero AddZeroClass.toAddZero AddMonoid.toAddZeroClass DistribSMul.toSMulZeroClass DistribMulAction.toDistribSMul MonoidWithZero.toMonoid Semiring.toMonoidWithZero IsBoundedSMul.toUniformContinuousConstSMul SeminormedRing.toPseudoMetricSpace SeminormedCommRing.toSeminormedRing NormedCommRing.toSeminormedCommRing NormedField.toNormedCommRing MulZeroClass.toZero NonUnitalNonAssocSemiring.toMulZeroClass NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing NonUnitalCommRing.toNonUnitalNonAssocCommRing NonUnitalNormedCommRing.toNonUnitalCommRing NormedCommRing.toNonUnitalNormedCommRing NegZeroClass.toZero SubNegZeroMonoid.toNegZeroClass SubtractionMonoid.toSubNegZeroMonoid SubtractionCommMonoid.toSubtractionMonoid AddCommGroup.toDivisionAddCommMonoid NormedAddCommGroup.toAddCommGroup NormedSpace.toIsBoundedSMul	—	SeminormedAddCommGroup.toIsTopologicalAddGroup IsTopologicalAddGroup.toContinuousAdd smulCommClass_self UniformContinuousConstSMul.to_continuousConstSMul IsBoundedSMul.toUniformContinuousConstSMul NormedSpace.toIsBoundedSMul AnalyticAt.analyticWithinAt analyticAt_bilinear
cpolynomialAt 📖	mathematical	—	CPolynomialAt DFunLike.coe ContinuousLinearMap DivisionSemiring.toSemiring Semifield.toDivisionSemiring Field.toSemifield NormedField.toField NontriviallyNormedField.toNormedField RingHom.id Semiring.toNonAssocSemiring UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddCommGroup.toPseudoMetricSpace NormedAddCommGroup.toSeminormedAddCommGroup ESeminormedAddCommMonoid.toAddCommMonoid ENormedAddCommMonoid.toESeminormedAddCommMonoid NormedAddCommGroup.toENormedAddCommMonoid NormedSpace.toModule funLike	—	HasFiniteFPowerSeriesOnBall.cpolynomialAt hasFiniteFPowerSeriesOnBall
cpolynomialOn 📖	mathematical	—	CPolynomialOn DFunLike.coe ContinuousLinearMap DivisionSemiring.toSemiring Semifield.toDivisionSemiring Field.toSemifield NormedField.toField NontriviallyNormedField.toNormedField RingHom.id Semiring.toNonAssocSemiring UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddCommGroup.toPseudoMetricSpace NormedAddCommGroup.toSeminormedAddCommGroup ESeminormedAddCommMonoid.toAddCommMonoid ENormedAddCommMonoid.toESeminormedAddCommMonoid NormedAddCommGroup.toENormedAddCommMonoid NormedSpace.toModule funLike	—	cpolynomialAt
fpowerSeriesBilinear_apply_add_three 📖	mathematical	—	fpowerSeriesBilinear ContinuousMultilinearMap DivisionSemiring.toSemiring Semifield.toDivisionSemiring Field.toSemifield NormedField.toField NontriviallyNormedField.toNormedField Prod.instAddCommMonoid ESeminormedAddCommMonoid.toAddCommMonoid UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddCommGroup.toPseudoMetricSpace NormedAddCommGroup.toSeminormedAddCommGroup ENormedAddCommMonoid.toESeminormedAddCommMonoid NormedAddCommGroup.toENormedAddCommMonoid Prod.instModule NormedSpace.toModule instTopologicalSpaceProd ContinuousMultilinearMap.instZero	—	SeminormedAddCommGroup.toIsTopologicalAddGroup IsTopologicalAddGroup.toContinuousAdd smulCommClass_self UniformContinuousConstSMul.to_continuousConstSMul IsBoundedSMul.toUniformContinuousConstSMul NormedSpace.toIsBoundedSMul
fpowerSeriesBilinear_apply_one 📖	mathematical	—	fpowerSeriesBilinear DFunLike.coe LinearIsometryEquiv DivisionSemiring.toSemiring Semifield.toDivisionSemiring Field.toSemifield NormedField.toField NontriviallyNormedField.toNormedField RingHom.id Semiring.toNonAssocSemiring RingHomInvPair.ids ContinuousLinearMap UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddCommGroup.toPseudoMetricSpace NormedAddCommGroup.toSeminormedAddCommGroup Prod.normedAddCommGroup ESeminormedAddCommMonoid.toAddCommMonoid ENormedAddCommMonoid.toESeminormedAddCommMonoid NormedAddCommGroup.toENormedAddCommMonoid NormedSpace.toModule Prod.normedSpace ContinuousMultilinearMap toSeminormedAddCommGroup RingHomIsometric.ids SeminormedCommRing.toSeminormedRing NormedCommRing.toSeminormedCommRing NormedField.toNormedCommRing ContinuousMultilinearMap.seminormedAddCommGroup' Fin.fintype module smulCommClass_self CommRing.toCommMonoid Field.toCommRing DistribMulAction.toMulAction CommMonoid.toMonoid AddCommMonoid.toAddMonoid Module.toDistribMulAction Ring.toSemiring CommRing.toRing UniformContinuousConstSMul.to_continuousConstSMul SMulZeroClass.toSMul AddZero.toZero AddZeroClass.toAddZero AddMonoid.toAddZeroClass DistribSMul.toSMulZeroClass DistribMulAction.toDistribSMul MonoidWithZero.toMonoid Semiring.toMonoidWithZero IsBoundedSMul.toUniformContinuousConstSMul SeminormedRing.toPseudoMetricSpace MulZeroClass.toZero NonUnitalNonAssocSemiring.toMulZeroClass NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing NonUnitalCommRing.toNonUnitalNonAssocCommRing NonUnitalNormedCommRing.toNonUnitalCommRing NormedCommRing.toNonUnitalNormedCommRing NegZeroClass.toZero SubNegZeroMonoid.toNegZeroClass SubtractionMonoid.toSubNegZeroMonoid SubtractionCommMonoid.toSubtractionMonoid AddCommGroup.toDivisionAddCommMonoid NormedAddCommGroup.toAddCommGroup NormedSpace.toIsBoundedSMul SeminormedAddCommGroup.toAddCommGroup SeminormedAddCommGroup.toIsTopologicalAddGroup ContinuousMultilinearMap.instModule EquivLike.toFunLike LinearIsometryEquiv.instEquivLike LinearIsometryEquiv.symm continuousMultilinearCurryFin1 instTopologicalSpaceProd Prod.instAddCommMonoid AddCommGroup.toAddCommMonoid Prod.instModule topologicalSpace Prod.instAddCommGroup addCommMonoid IsTopologicalAddGroup.toContinuousAdd SeminormedAddGroup.toAddGroup SeminormedAddCommGroup.toSeminormedAddGroup funLike deriv₂	—	SeminormedAddCommGroup.toIsTopologicalAddGroup IsTopologicalAddGroup.toContinuousAdd smulCommClass_self UniformContinuousConstSMul.to_continuousConstSMul IsBoundedSMul.toUniformContinuousConstSMul NormedSpace.toIsBoundedSMul
fpowerSeriesBilinear_apply_two 📖	mathematical	—	fpowerSeriesBilinear uncurryBilinear	—	SeminormedAddCommGroup.toIsTopologicalAddGroup IsTopologicalAddGroup.toContinuousAdd smulCommClass_self UniformContinuousConstSMul.to_continuousConstSMul IsBoundedSMul.toUniformContinuousConstSMul NormedSpace.toIsBoundedSMul
fpowerSeriesBilinear_apply_zero 📖	mathematical	—	fpowerSeriesBilinear ContinuousMultilinearMap.uncurry0 Prod.normedAddCommGroup Prod.normedSpace NontriviallyNormedField.toNormedField NormedAddCommGroup.toSeminormedAddCommGroup DFunLike.coe ContinuousLinearMap DivisionSemiring.toSemiring Semifield.toDivisionSemiring Field.toSemifield NormedField.toField RingHom.id Semiring.toNonAssocSemiring UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddCommGroup.toPseudoMetricSpace ESeminormedAddCommMonoid.toAddCommMonoid ENormedAddCommMonoid.toESeminormedAddCommMonoid NormedAddCommGroup.toENormedAddCommMonoid NormedSpace.toModule funLike topologicalSpace SeminormedAddCommGroup.toAddCommGroup SeminormedAddCommGroup.toIsTopologicalAddGroup addCommMonoid IsTopologicalAddGroup.toContinuousAdd NormedAddGroup.toAddGroup NormedAddCommGroup.toNormedAddGroup module smulCommClass_self CommRing.toCommMonoid Field.toCommRing DistribMulAction.toMulAction CommMonoid.toMonoid AddCommMonoid.toAddMonoid Module.toDistribMulAction Ring.toSemiring CommRing.toRing UniformContinuousConstSMul.to_continuousConstSMul SMulZeroClass.toSMul AddZero.toZero AddZeroClass.toAddZero AddMonoid.toAddZeroClass DistribSMul.toSMulZeroClass DistribMulAction.toDistribSMul MonoidWithZero.toMonoid Semiring.toMonoidWithZero IsBoundedSMul.toUniformContinuousConstSMul SeminormedRing.toPseudoMetricSpace SeminormedCommRing.toSeminormedRing NormedCommRing.toSeminormedCommRing NormedField.toNormedCommRing MulZeroClass.toZero NonUnitalNonAssocSemiring.toMulZeroClass NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing NonUnitalCommRing.toNonUnitalNonAssocCommRing NonUnitalNormedCommRing.toNonUnitalCommRing NormedCommRing.toNonUnitalNormedCommRing NegZeroClass.toZero SubNegZeroMonoid.toNegZeroClass SubtractionMonoid.toSubNegZeroMonoid SubtractionCommMonoid.toSubtractionMonoid AddCommGroup.toDivisionAddCommMonoid NormedAddCommGroup.toAddCommGroup NormedSpace.toIsBoundedSMul	—	SeminormedAddCommGroup.toIsTopologicalAddGroup IsTopologicalAddGroup.toContinuousAdd smulCommClass_self UniformContinuousConstSMul.to_continuousConstSMul IsBoundedSMul.toUniformContinuousConstSMul NormedSpace.toIsBoundedSMul
fpowerSeriesBilinear_radius 📖	mathematical	—	FormalMultilinearSeries.radius Prod.normedAddCommGroup Prod.normedSpace NontriviallyNormedField.toNormedField NormedAddCommGroup.toSeminormedAddCommGroup fpowerSeriesBilinear Top.top ENNReal instTopENNReal	—	SeminormedAddCommGroup.toIsTopologicalAddGroup IsTopologicalAddGroup.toContinuousAdd smulCommClass_self UniformContinuousConstSMul.to_continuousConstSMul IsBoundedSMul.toUniformContinuousConstSMul NormedSpace.toIsBoundedSMul FormalMultilinearSeries.radius_eq_top_of_forall_image_add_eq_zero
fpowerSeries_radius 📖	mathematical	—	FormalMultilinearSeries.radius fpowerSeries Top.top ENNReal instTopENNReal	—	FormalMultilinearSeries.radius_eq_top_of_forall_image_add_eq_zero
hasFPowerSeriesAt 📖	mathematical	—	HasFPowerSeriesAt DFunLike.coe ContinuousLinearMap DivisionSemiring.toSemiring Semifield.toDivisionSemiring Field.toSemifield NormedField.toField NontriviallyNormedField.toNormedField RingHom.id Semiring.toNonAssocSemiring UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddCommGroup.toPseudoMetricSpace NormedAddCommGroup.toSeminormedAddCommGroup ESeminormedAddCommMonoid.toAddCommMonoid ENormedAddCommMonoid.toESeminormedAddCommMonoid NormedAddCommGroup.toENormedAddCommMonoid NormedSpace.toModule funLike fpowerSeries	—	hasFPowerSeriesOnBall
hasFPowerSeriesAt_bilinear 📖	mathematical	—	HasFPowerSeriesAt Prod.normedAddCommGroup Prod.normedSpace NontriviallyNormedField.toNormedField NormedAddCommGroup.toSeminormedAddCommGroup DFunLike.coe ContinuousLinearMap DivisionSemiring.toSemiring Semifield.toDivisionSemiring Field.toSemifield NormedField.toField RingHom.id Semiring.toNonAssocSemiring UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddCommGroup.toPseudoMetricSpace ESeminormedAddCommMonoid.toAddCommMonoid ENormedAddCommMonoid.toESeminormedAddCommMonoid NormedAddCommGroup.toENormedAddCommMonoid NormedSpace.toModule funLike topologicalSpace SeminormedAddCommGroup.toAddCommGroup SeminormedAddCommGroup.toIsTopologicalAddGroup addCommMonoid IsTopologicalAddGroup.toContinuousAdd NormedAddGroup.toAddGroup NormedAddCommGroup.toNormedAddGroup module smulCommClass_self CommRing.toCommMonoid Field.toCommRing DistribMulAction.toMulAction CommMonoid.toMonoid AddCommMonoid.toAddMonoid Module.toDistribMulAction Ring.toSemiring CommRing.toRing UniformContinuousConstSMul.to_continuousConstSMul SMulZeroClass.toSMul AddZero.toZero AddZeroClass.toAddZero AddMonoid.toAddZeroClass DistribSMul.toSMulZeroClass DistribMulAction.toDistribSMul MonoidWithZero.toMonoid Semiring.toMonoidWithZero IsBoundedSMul.toUniformContinuousConstSMul SeminormedRing.toPseudoMetricSpace SeminormedCommRing.toSeminormedRing NormedCommRing.toSeminormedCommRing NormedField.toNormedCommRing MulZeroClass.toZero NonUnitalNonAssocSemiring.toMulZeroClass NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing NonUnitalCommRing.toNonUnitalNonAssocCommRing NonUnitalNormedCommRing.toNonUnitalCommRing NormedCommRing.toNonUnitalNormedCommRing NegZeroClass.toZero SubNegZeroMonoid.toNegZeroClass SubtractionMonoid.toSubNegZeroMonoid SubtractionCommMonoid.toSubtractionMonoid AddCommGroup.toDivisionAddCommMonoid NormedAddCommGroup.toAddCommGroup NormedSpace.toIsBoundedSMul fpowerSeriesBilinear	—	SeminormedAddCommGroup.toIsTopologicalAddGroup IsTopologicalAddGroup.toContinuousAdd smulCommClass_self UniformContinuousConstSMul.to_continuousConstSMul IsBoundedSMul.toUniformContinuousConstSMul NormedSpace.toIsBoundedSMul hasFPowerSeriesOnBall_bilinear
hasFPowerSeriesOnBall 📖	mathematical	—	HasFPowerSeriesOnBall DFunLike.coe ContinuousLinearMap DivisionSemiring.toSemiring Semifield.toDivisionSemiring Field.toSemifield NormedField.toField NontriviallyNormedField.toNormedField RingHom.id Semiring.toNonAssocSemiring UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddCommGroup.toPseudoMetricSpace NormedAddCommGroup.toSeminormedAddCommGroup ESeminormedAddCommMonoid.toAddCommMonoid ENormedAddCommMonoid.toESeminormedAddCommMonoid NormedAddCommGroup.toENormedAddCommMonoid NormedSpace.toModule funLike fpowerSeries Top.top ENNReal instTopENNReal	—	HasFiniteFPowerSeriesOnBall.toHasFPowerSeriesOnBall hasFiniteFPowerSeriesOnBall
hasFPowerSeriesOnBall_bilinear 📖	mathematical	—	HasFPowerSeriesOnBall Prod.normedAddCommGroup Prod.normedSpace NontriviallyNormedField.toNormedField NormedAddCommGroup.toSeminormedAddCommGroup DFunLike.coe ContinuousLinearMap DivisionSemiring.toSemiring Semifield.toDivisionSemiring Field.toSemifield NormedField.toField RingHom.id Semiring.toNonAssocSemiring UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddCommGroup.toPseudoMetricSpace ESeminormedAddCommMonoid.toAddCommMonoid ENormedAddCommMonoid.toESeminormedAddCommMonoid NormedAddCommGroup.toENormedAddCommMonoid NormedSpace.toModule funLike topologicalSpace SeminormedAddCommGroup.toAddCommGroup SeminormedAddCommGroup.toIsTopologicalAddGroup addCommMonoid IsTopologicalAddGroup.toContinuousAdd NormedAddGroup.toAddGroup NormedAddCommGroup.toNormedAddGroup module smulCommClass_self CommRing.toCommMonoid Field.toCommRing DistribMulAction.toMulAction CommMonoid.toMonoid AddCommMonoid.toAddMonoid Module.toDistribMulAction Ring.toSemiring CommRing.toRing UniformContinuousConstSMul.to_continuousConstSMul SMulZeroClass.toSMul AddZero.toZero AddZeroClass.toAddZero AddMonoid.toAddZeroClass DistribSMul.toSMulZeroClass DistribMulAction.toDistribSMul MonoidWithZero.toMonoid Semiring.toMonoidWithZero IsBoundedSMul.toUniformContinuousConstSMul SeminormedRing.toPseudoMetricSpace SeminormedCommRing.toSeminormedRing NormedCommRing.toSeminormedCommRing NormedField.toNormedCommRing MulZeroClass.toZero NonUnitalNonAssocSemiring.toMulZeroClass NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing NonUnitalCommRing.toNonUnitalNonAssocCommRing NonUnitalNormedCommRing.toNonUnitalCommRing NormedCommRing.toNonUnitalNormedCommRing NegZeroClass.toZero SubNegZeroMonoid.toNegZeroClass SubtractionMonoid.toSubNegZeroMonoid SubtractionCommMonoid.toSubtractionMonoid AddCommGroup.toDivisionAddCommMonoid NormedAddCommGroup.toAddCommGroup NormedSpace.toIsBoundedSMul fpowerSeriesBilinear Top.top ENNReal instTopENNReal	—	SeminormedAddCommGroup.toIsTopologicalAddGroup IsTopologicalAddGroup.toContinuousAdd smulCommClass_self UniformContinuousConstSMul.to_continuousConstSMul IsBoundedSMul.toUniformContinuousConstSMul NormedSpace.toIsBoundedSMul fpowerSeriesBilinear_radius ENNReal.coe_lt_top hasSum_nat_add_iff' map_add_add Finset.sum_range_succ Finset.sum_congr zero_add sub_self
hasFiniteFPowerSeriesOnBall 📖	mathematical	—	HasFiniteFPowerSeriesOnBall DFunLike.coe ContinuousLinearMap DivisionSemiring.toSemiring Semifield.toDivisionSemiring Field.toSemifield NormedField.toField NontriviallyNormedField.toNormedField RingHom.id Semiring.toNonAssocSemiring UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddCommGroup.toPseudoMetricSpace NormedAddCommGroup.toSeminormedAddCommGroup ESeminormedAddCommMonoid.toAddCommMonoid ENormedAddCommMonoid.toESeminormedAddCommMonoid NormedAddCommGroup.toENormedAddCommMonoid NormedSpace.toModule funLike fpowerSeries Top.top ENNReal instTopENNReal	—	fpowerSeries_radius ENNReal.coe_lt_top hasSum_nat_add_iff' SeminormedAddCommGroup.toIsTopologicalAddGroup map_add SemilinearMapClass.toAddHomClass ContinuousSemilinearMapClass.toSemilinearMapClass continuousSemilinearMapClass Finset.sum_congr Finset.sum_range_succ Finset.sum_singleton sub_self Mathlib.Tactic.Linarith.lt_irrefl Nat.instAtLeastTwoHAddOfNat Mathlib.Tactic.Ring.of_eq Mathlib.Tactic.Ring.add_congr Mathlib.Tactic.Ring.mul_congr Mathlib.Tactic.Ring.cast_pos Mathlib.Meta.NormNum.isNat_ofNat Mathlib.Tactic.Ring.neg_congr Mathlib.Tactic.Ring.neg_add Mathlib.Tactic.Ring.neg_one_mul Mathlib.Meta.NormNum.IsInt.to_raw_eq Mathlib.Meta.NormNum.isInt_mul Mathlib.Meta.NormNum.IsInt.of_raw Mathlib.Meta.NormNum.IsNat.to_isInt Mathlib.Meta.NormNum.IsNat.of_raw Mathlib.Tactic.Ring.neg_zero Mathlib.Tactic.Ring.add_mul Mathlib.Tactic.Ring.mul_add Mathlib.Tactic.Ring.mul_zero Mathlib.Tactic.Ring.add_pf_add_zero Mathlib.Tactic.Ring.zero_mul Mathlib.Tactic.Ring.sub_congr Mathlib.Tactic.Ring.cast_zero Mathlib.Tactic.Ring.sub_pf Mathlib.Tactic.Ring.add_pf_add_overlap_zero Mathlib.Meta.NormNum.IsInt.to_isNat Mathlib.Meta.NormNum.isInt_add Mathlib.Tactic.Ring.add_pf_zero_add Mathlib.Tactic.Linarith.add_lt_of_neg_of_le Int.instIsStrictOrderedRing Mathlib.Tactic.Linarith.mul_neg neg_neg_of_pos Int.instIsOrderedAddMonoid Mathlib.Tactic.Linarith.zero_lt_one Mathlib.Meta.NormNum.isNat_lt_true IsStrictOrderedRing.toIsOrderedRing Int.instCharZero Mathlib.Tactic.Linarith.sub_nonpos_of_le Nat.cast_zero Mathlib.Tactic.Ring.add_pf_add_overlap Mathlib.Meta.NormNum.IsNat.to_raw_eq Nat.cast_one
uncurryBilinear_apply 📖	mathematical	—	DFunLike.coe ContinuousMultilinearMap DivisionSemiring.toSemiring Semifield.toDivisionSemiring Field.toSemifield NormedField.toField NontriviallyNormedField.toNormedField Prod.instAddCommMonoid ESeminormedAddCommMonoid.toAddCommMonoid UniformSpace.toTopologicalSpace PseudoMetricSpace.toUniformSpace SeminormedAddCommGroup.toPseudoMetricSpace NormedAddCommGroup.toSeminormedAddCommGroup ENormedAddCommMonoid.toESeminormedAddCommMonoid NormedAddCommGroup.toENormedAddCommMonoid Prod.instModule NormedSpace.toModule instTopologicalSpaceProd ContinuousMultilinearMap.funLike uncurryBilinear ContinuousLinearMap RingHom.id Semiring.toNonAssocSemiring funLike topologicalSpace SeminormedAddCommGroup.toAddCommGroup SeminormedAddCommGroup.toIsTopologicalAddGroup addCommMonoid IsTopologicalAddGroup.toContinuousAdd NormedAddGroup.toAddGroup NormedAddCommGroup.toNormedAddGroup module smulCommClass_self CommRing.toCommMonoid Field.toCommRing DistribMulAction.toMulAction CommMonoid.toMonoid AddCommMonoid.toAddMonoid Module.toDistribMulAction Ring.toSemiring CommRing.toRing UniformContinuousConstSMul.to_continuousConstSMul SMulZeroClass.toSMul AddZero.toZero AddZeroClass.toAddZero AddMonoid.toAddZeroClass DistribSMul.toSMulZeroClass DistribMulAction.toDistribSMul MonoidWithZero.toMonoid Semiring.toMonoidWithZero IsBoundedSMul.toUniformContinuousConstSMul SeminormedRing.toPseudoMetricSpace SeminormedCommRing.toSeminormedRing NormedCommRing.toSeminormedCommRing NormedField.toNormedCommRing MulZeroClass.toZero NonUnitalNonAssocSemiring.toMulZeroClass NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing NonUnitalCommRing.toNonUnitalNonAssocCommRing NonUnitalNormedCommRing.toNonUnitalCommRing NormedCommRing.toNonUnitalNormedCommRing NegZeroClass.toZero SubNegZeroMonoid.toNegZeroClass SubtractionMonoid.toSubNegZeroMonoid SubtractionCommMonoid.toSubtractionMonoid AddCommGroup.toDivisionAddCommMonoid NormedAddCommGroup.toAddCommGroup NormedSpace.toIsBoundedSMul	—	SeminormedAddCommGroup.toIsTopologicalAddGroup IsTopologicalAddGroup.toContinuousAdd smulCommClass_self UniformContinuousConstSMul.to_continuousConstSMul IsBoundedSMul.toUniformContinuousConstSMul NormedSpace.toIsBoundedSMul

LinearIsometryEquiv

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
analyticAt 📖	mathematical	—	AnalyticAt DFunLike.coe LinearIsometryEquiv DivisionSemiring.toSemiring Semifield.toDivisionSemiring Field.toSemifield NormedField.toField NontriviallyNormedField.toNormedField RingHom.id Semiring.toNonAssocSemiring RingHomInvPair.ids NormedAddCommGroup.toSeminormedAddCommGroup NormedSpace.toModule EquivLike.toFunLike instEquivLike	—	RingHomInvPair.ids HasFPowerSeriesAt.analyticAt ContinuousLinearMap.hasFPowerSeriesAt
analyticOn 📖	mathematical	—	AnalyticOn DFunLike.coe LinearIsometryEquiv DivisionSemiring.toSemiring Semifield.toDivisionSemiring Field.toSemifield NormedField.toField NontriviallyNormedField.toNormedField RingHom.id Semiring.toNonAssocSemiring RingHomInvPair.ids NormedAddCommGroup.toSeminormedAddCommGroup NormedSpace.toModule EquivLike.toFunLike instEquivLike	—	RingHomInvPair.ids analyticWithinAt
analyticOnNhd 📖	mathematical	—	AnalyticOnNhd DFunLike.coe LinearIsometryEquiv DivisionSemiring.toSemiring Semifield.toDivisionSemiring Field.toSemifield NormedField.toField NontriviallyNormedField.toNormedField RingHom.id Semiring.toNonAssocSemiring RingHomInvPair.ids NormedAddCommGroup.toSeminormedAddCommGroup NormedSpace.toModule EquivLike.toFunLike instEquivLike	—	RingHomInvPair.ids analyticAt
analyticWithinAt 📖	mathematical	—	AnalyticWithinAt DFunLike.coe LinearIsometryEquiv DivisionSemiring.toSemiring Semifield.toDivisionSemiring Field.toSemifield NormedField.toField NontriviallyNormedField.toNormedField RingHom.id Semiring.toNonAssocSemiring RingHomInvPair.ids NormedAddCommGroup.toSeminormedAddCommGroup NormedSpace.toModule EquivLike.toFunLike instEquivLike	—	RingHomInvPair.ids AnalyticAt.analyticWithinAt analyticAt

(root)

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
analyticAt_fst 📖	mathematical	—	AnalyticAt Prod.normedAddCommGroup Prod.normedSpace NontriviallyNormedField.toNormedField NormedAddCommGroup.toSeminormedAddCommGroup	—	ContinuousLinearMap.analyticAt
analyticAt_id 📖	mathematical	—	AnalyticAt	—	ContinuousLinearMap.analyticAt
analyticAt_snd 📖	mathematical	—	AnalyticAt Prod.normedAddCommGroup Prod.normedSpace NontriviallyNormedField.toNormedField NormedAddCommGroup.toSeminormedAddCommGroup	—	ContinuousLinearMap.analyticAt
analyticOnNhd_fst 📖	mathematical	—	AnalyticOnNhd Prod.normedAddCommGroup Prod.normedSpace NontriviallyNormedField.toNormedField NormedAddCommGroup.toSeminormedAddCommGroup	—	analyticAt_fst
analyticOnNhd_id 📖	mathematical	—	AnalyticOnNhd	—	analyticAt_id
analyticOnNhd_snd 📖	mathematical	—	AnalyticOnNhd Prod.normedAddCommGroup Prod.normedSpace NontriviallyNormedField.toNormedField NormedAddCommGroup.toSeminormedAddCommGroup	—	analyticAt_snd
analyticOn_fst 📖	mathematical	—	AnalyticOn Prod.normedAddCommGroup Prod.normedSpace NontriviallyNormedField.toNormedField NormedAddCommGroup.toSeminormedAddCommGroup	—	analyticWithinAt_fst
analyticOn_id 📖	mathematical	—	AnalyticOn	—	analyticWithinAt_id
analyticOn_snd 📖	mathematical	—	AnalyticOn Prod.normedAddCommGroup Prod.normedSpace NontriviallyNormedField.toNormedField NormedAddCommGroup.toSeminormedAddCommGroup	—	analyticWithinAt_snd
analyticWithinAt_fst 📖	mathematical	—	AnalyticWithinAt Prod.normedAddCommGroup Prod.normedSpace NontriviallyNormedField.toNormedField NormedAddCommGroup.toSeminormedAddCommGroup	—	AnalyticAt.analyticWithinAt analyticAt_fst
analyticWithinAt_id 📖	mathematical	—	AnalyticWithinAt	—	AnalyticAt.analyticWithinAt analyticAt_id
analyticWithinAt_snd 📖	mathematical	—	AnalyticWithinAt Prod.normedAddCommGroup Prod.normedSpace NontriviallyNormedField.toNormedField NormedAddCommGroup.toSeminormedAddCommGroup	—	AnalyticAt.analyticWithinAt analyticAt_snd

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