BHSP_2

:: BHSP_2 semantic presentation

begin

definition

attr seq is convergent means :: BHSP_2:def 1
ex g being ( ( ) ( ) Point of ( ( ) ( ) set ) ) st
for r being ( ( ) ( ext-real V27() real ) Real) st r : ( ( ) ( ext-real V27() real ) Real) > 0 : ( ( ) ( V4() ext-real non positive non negative epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V27() real V125() V126() V127() V128() V129() V130() V131() V132() V133() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) ) holds
ex m being ( ( ) ( ext-real epsilon-transitive epsilon-connected ordinal natural V27() real V125() V126() V127() V128() V129() V130() V131() V132() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) ) st
for n being ( ( ) ( ext-real epsilon-transitive epsilon-connected ordinal natural V27() real V125() V126() V127() V128() V129() V130() V131() V132() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) ) st n : ( ( ) ( ext-real epsilon-transitive epsilon-connected ordinal natural V27() real V125() V126() V127() V128() V129() V130() V131() V132() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) ) >= m : ( ( ) ( ext-real epsilon-transitive epsilon-connected ordinal natural V27() real V125() V126() V127() V128() V129() V130() V131() V132() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) ) holds
dist ((seq : ( ( ) ( ) Element of X : ( ( ) ( ) NORMSTR ) ) . n : ( ( ) ( ext-real epsilon-transitive epsilon-connected ordinal natural V27() real V125() V126() V127() V128() V129() V130() V131() V132() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of the carrier of X : ( ( ) ( ) NORMSTR ) : ( ( ) ( ) set ) ) ,g : ( ( ) ( ) Point of ( ( ) ( V4() ) set ) ) ) : ( ( ) ( ext-real V27() real ) Element of REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) < r : ( ( ) ( ext-real V27() real ) Real) ;

end;

theorem :: BHSP_2:1

theorem :: BHSP_2:2

theorem :: BHSP_2:3

theorem :: BHSP_2:4

theorem :: BHSP_2:5

theorem :: BHSP_2:6

theorem :: BHSP_2:7

theorem :: BHSP_2:8

theorem :: BHSP_2:9

definition

func lim seq -> ( ( ) ( ) Point of ( ( ) ( ) set ) ) means :: BHSP_2:def 2
for r being ( ( ) ( ext-real V27() real ) Real) st r : ( ( ) ( ext-real V27() real ) Real) > 0 : ( ( ) ( V4() ext-real non positive non negative epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V27() real V125() V126() V127() V128() V129() V130() V131() V132() V133() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) ) holds
ex m being ( ( ) ( ext-real epsilon-transitive epsilon-connected ordinal natural V27() real V125() V126() V127() V128() V129() V130() V131() V132() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) ) st
for n being ( ( ) ( ext-real epsilon-transitive epsilon-connected ordinal natural V27() real V125() V126() V127() V128() V129() V130() V131() V132() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) ) st n : ( ( ) ( ext-real epsilon-transitive epsilon-connected ordinal natural V27() real V125() V126() V127() V128() V129() V130() V131() V132() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) ) >= m : ( ( ) ( ext-real epsilon-transitive epsilon-connected ordinal natural V27() real V125() V126() V127() V128() V129() V130() V131() V132() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) ) holds
dist ((seq : ( ( ) ( ) Element of X : ( ( ) ( ) NORMSTR ) ) . n : ( ( ) ( ext-real epsilon-transitive epsilon-connected ordinal natural V27() real V125() V126() V127() V128() V129() V130() V131() V132() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of the carrier of X : ( ( ) ( ) NORMSTR ) : ( ( ) ( ) set ) ) ,it : ( ( V12() V30(K33(X : ( ( ) ( ) NORMSTR ) ,X : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ,X : ( ( ) ( ) NORMSTR ) ) ) ( Relation-like K33(X : ( ( ) ( ) NORMSTR ) ,X : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) -defined X : ( ( ) ( ) NORMSTR ) -valued V12() V30(K33(X : ( ( ) ( ) NORMSTR ) ,X : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ,X : ( ( ) ( ) NORMSTR ) ) ) Element of K32(K33(K33(X : ( ( ) ( ) NORMSTR ) ,X : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ,X : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real V27() real ) Element of REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) < r : ( ( ) ( ext-real V27() real ) Real) ;

end;

theorem :: BHSP_2:10

theorem :: BHSP_2:11

theorem :: BHSP_2:12

theorem :: BHSP_2:13

for X being ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace)
for seq1, seq2 being ( ( V12() V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) -valued V12() V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) st seq1 : ( ( V12() V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) -valued V12() V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) is convergent & seq2 : ( ( V12() V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) -valued V12() V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) is convergent holds
lim (seq1 : ( ( V12() V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) -valued V12() V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) + seq2 : ( ( V12() V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) -valued V12() V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) ) : ( ( V12() V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) -valued V12() V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) Element of K32(K33(NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) Point of ( ( ) ( V4() ) set ) ) = (lim seq1 : ( ( V12() V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) -valued V12() V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) ) : ( ( ) ( ) Point of ( ( ) ( V4() ) set ) ) + (lim seq2 : ( ( V12() V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) -valued V12() V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) ) : ( ( ) ( ) Point of ( ( ) ( V4() ) set ) ) : ( ( ) ( ) Element of the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ;

theorem :: BHSP_2:14

for X being ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace)
for seq1, seq2 being ( ( V12() V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) -valued V12() V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) st seq1 : ( ( V12() V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) -valued V12() V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) is convergent & seq2 : ( ( V12() V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) -valued V12() V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) is convergent holds
lim (seq1 : ( ( V12() V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) -valued V12() V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) - seq2 : ( ( V12() V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) -valued V12() V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) ) : ( ( V12() V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) -valued V12() V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) Element of K32(K33(NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) Point of ( ( ) ( V4() ) set ) ) = (lim seq1 : ( ( V12() V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) -valued V12() V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) ) : ( ( ) ( ) Point of ( ( ) ( V4() ) set ) ) - (lim seq2 : ( ( V12() V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) -valued V12() V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) ) : ( ( ) ( ) Point of ( ( ) ( V4() ) set ) ) : ( ( ) ( ) Element of the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ;

theorem :: BHSP_2:15

theorem :: BHSP_2:16

for X being ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace)
for seq being ( ( V12() V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) -valued V12() V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) st seq : ( ( V12() V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) -valued V12() V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) is convergent holds
lim (- seq : ( ( V12() V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) -valued V12() V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) ) : ( ( V12() ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) -valued V12() V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) Element of K32(K33(NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) Point of ( ( ) ( V4() ) set ) ) = - (lim seq : ( ( V12() V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) -valued V12() V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) ) : ( ( ) ( ) Point of ( ( ) ( V4() ) set ) ) : ( ( ) ( ) Element of the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ;

theorem :: BHSP_2:17

theorem :: BHSP_2:18

theorem :: BHSP_2:19

definition

func ||.seq.|| -> ( ( V12() V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) ) ( Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V127() V128() V129() V133() ) set ) -valued V12() V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) ) Real_Sequence) means :: BHSP_2:def 3
for n being ( ( ) ( ext-real epsilon-transitive epsilon-connected ordinal natural V27() real V125() V126() V127() V128() V129() V130() V131() V132() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) ) holds it : ( ( V12() V30(K33(X : ( ( ) ( ) NORMSTR ) ,X : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ,X : ( ( ) ( ) NORMSTR ) ) ) ( Relation-like K33(X : ( ( ) ( ) NORMSTR ) ,X : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) -defined X : ( ( ) ( ) NORMSTR ) -valued V12() V30(K33(X : ( ( ) ( ) NORMSTR ) ,X : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ,X : ( ( ) ( ) NORMSTR ) ) ) Element of K32(K33(K33(X : ( ( ) ( ) NORMSTR ) ,X : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ,X : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . n : ( ( ) ( ext-real epsilon-transitive epsilon-connected ordinal natural V27() real V125() V126() V127() V128() V129() V130() V131() V132() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real V27() real ) Element of REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) = ||.(seq : ( ( ) ( ) Element of X : ( ( ) ( ) NORMSTR ) ) . n : ( ( ) ( ext-real epsilon-transitive epsilon-connected ordinal natural V27() real V125() V126() V127() V128() V129() V130() V131() V132() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of the carrier of X : ( ( ) ( ) NORMSTR ) : ( ( ) ( ) set ) ) .|| : ( ( ) ( ext-real V27() real ) Element of REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) ;

end;

theorem :: BHSP_2:20

theorem :: BHSP_2:21

theorem :: BHSP_2:22

definition

func dist (seq,x) -> ( ( V12() V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) ) ( Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V127() V128() V129() V133() ) set ) -valued V12() V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) ) Real_Sequence) means :: BHSP_2:def 4
for n being ( ( ) ( ext-real epsilon-transitive epsilon-connected ordinal natural V27() real V125() V126() V127() V128() V129() V130() V131() V132() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) ) holds it : ( ( V12() V30(K33(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ,X : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ,X : ( ( ) ( ) NORMSTR ) ) ) ( Relation-like K33(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ,X : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) -defined X : ( ( ) ( ) NORMSTR ) -valued V12() V30(K33(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ,X : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ,X : ( ( ) ( ) NORMSTR ) ) ) Element of K32(K33(K33(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ,X : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ,X : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . n : ( ( ) ( ext-real epsilon-transitive epsilon-connected ordinal natural V27() real V125() V126() V127() V128() V129() V130() V131() V132() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real V27() real ) Element of REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) = dist ((seq : ( ( ) ( ) Element of X : ( ( ) ( ) NORMSTR ) ) . n : ( ( ) ( ext-real epsilon-transitive epsilon-connected ordinal natural V27() real V125() V126() V127() V128() V129() V130() V131() V132() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of the carrier of X : ( ( ) ( ) NORMSTR ) : ( ( ) ( ) set ) ) ,x : ( ( V12() V30(K33(X : ( ( ) ( ) NORMSTR ) ,X : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ,X : ( ( ) ( ) NORMSTR ) ) ) ( Relation-like K33(X : ( ( ) ( ) NORMSTR ) ,X : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) -defined X : ( ( ) ( ) NORMSTR ) -valued V12() V30(K33(X : ( ( ) ( ) NORMSTR ) ,X : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ,X : ( ( ) ( ) NORMSTR ) ) ) Element of K32(K33(K33(X : ( ( ) ( ) NORMSTR ) ,X : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ,X : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real V27() real ) Element of REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) ;

end;

theorem :: BHSP_2:23

theorem :: BHSP_2:24

theorem :: BHSP_2:25

for X being ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace)
for g1, g2 being ( ( ) ( ) Point of ( ( ) ( V4() ) set ) )
for seq1, seq2 being ( ( V12() V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) -valued V12() V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) st seq1 : ( ( V12() V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) -valued V12() V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) is convergent & lim seq1 : ( ( V12() V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) -valued V12() V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) : ( ( ) ( ) Point of ( ( ) ( V4() ) set ) ) = g1 : ( ( ) ( ) Point of ( ( ) ( V4() ) set ) ) & seq2 : ( ( V12() V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) -valued V12() V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) is convergent & lim seq2 : ( ( V12() V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) -valued V12() V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) : ( ( ) ( ) Point of ( ( ) ( V4() ) set ) ) = g2 : ( ( ) ( ) Point of ( ( ) ( V4() ) set ) ) holds
( ||.(seq1 : ( ( V12() V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) -valued V12() V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) + seq2 : ( ( V12() V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) -valued V12() V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) ) : ( ( V12() V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) -valued V12() V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) Element of K32(K33(NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) .|| : ( ( V12() V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) ) ( Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V127() V128() V129() V133() ) set ) -valued V12() V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) ) Real_Sequence) is convergent & lim ||.(seq1 : ( ( V12() V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) -valued V12() V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) + seq2 : ( ( V12() V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) -valued V12() V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) ) : ( ( V12() V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) -valued V12() V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) Element of K32(K33(NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) .|| : ( ( V12() V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) ) ( Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V127() V128() V129() V133() ) set ) -valued V12() V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) ) Real_Sequence) : ( ( ) ( ext-real V27() real ) Element of REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) = ||.(g1 : ( ( ) ( ) Point of ( ( ) ( V4() ) set ) ) + g2 : ( ( ) ( ) Point of ( ( ) ( V4() ) set ) ) ) : ( ( ) ( ) Element of the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) .|| : ( ( ) ( ext-real V27() real ) Element of REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) ) ;

theorem :: BHSP_2:26

for X being ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace)
for g1, g2 being ( ( ) ( ) Point of ( ( ) ( V4() ) set ) )
for seq1, seq2 being ( ( V12() V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) -valued V12() V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) st seq1 : ( ( V12() V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) -valued V12() V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) is convergent & lim seq1 : ( ( V12() V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) -valued V12() V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) : ( ( ) ( ) Point of ( ( ) ( V4() ) set ) ) = g1 : ( ( ) ( ) Point of ( ( ) ( V4() ) set ) ) & seq2 : ( ( V12() V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) -valued V12() V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) is convergent & lim seq2 : ( ( V12() V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) -valued V12() V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) : ( ( ) ( ) Point of ( ( ) ( V4() ) set ) ) = g2 : ( ( ) ( ) Point of ( ( ) ( V4() ) set ) ) holds
( ||.((seq1 : ( ( V12() V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) -valued V12() V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) + seq2 : ( ( V12() V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) -valued V12() V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) ) : ( ( V12() V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) -valued V12() V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) Element of K32(K33(NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) - (g1 : ( ( ) ( ) Point of ( ( ) ( V4() ) set ) ) + g2 : ( ( ) ( ) Point of ( ( ) ( V4() ) set ) ) ) : ( ( ) ( ) Element of the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) : ( ( V12() V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) -valued V12() V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) Element of K32(K33(NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) .|| : ( ( V12() V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) ) ( Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V127() V128() V129() V133() ) set ) -valued V12() V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) ) Real_Sequence) is convergent & lim ||.((seq1 : ( ( V12() V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) -valued V12() V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) + seq2 : ( ( V12() V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) -valued V12() V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) ) : ( ( V12() V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) -valued V12() V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) Element of K32(K33(NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) - (g1 : ( ( ) ( ) Point of ( ( ) ( V4() ) set ) ) + g2 : ( ( ) ( ) Point of ( ( ) ( V4() ) set ) ) ) : ( ( ) ( ) Element of the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) : ( ( V12() V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) -valued V12() V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) Element of K32(K33(NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) .|| : ( ( V12() V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) ) ( Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V127() V128() V129() V133() ) set ) -valued V12() V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) ) Real_Sequence) : ( ( ) ( ext-real V27() real ) Element of REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) = 0 : ( ( ) ( V4() ext-real non positive non negative epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V27() real V125() V126() V127() V128() V129() V130() V131() V132() V133() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) ) ) ;

theorem :: BHSP_2:27

for X being ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace)
for g1, g2 being ( ( ) ( ) Point of ( ( ) ( V4() ) set ) )
for seq1, seq2 being ( ( V12() V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) -valued V12() V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) st seq1 : ( ( V12() V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) -valued V12() V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) is convergent & lim seq1 : ( ( V12() V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) -valued V12() V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) : ( ( ) ( ) Point of ( ( ) ( V4() ) set ) ) = g1 : ( ( ) ( ) Point of ( ( ) ( V4() ) set ) ) & seq2 : ( ( V12() V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) -valued V12() V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) is convergent & lim seq2 : ( ( V12() V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) -valued V12() V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) : ( ( ) ( ) Point of ( ( ) ( V4() ) set ) ) = g2 : ( ( ) ( ) Point of ( ( ) ( V4() ) set ) ) holds
( ||.(seq1 : ( ( V12() V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) -valued V12() V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) - seq2 : ( ( V12() V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) -valued V12() V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) ) : ( ( V12() V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) -valued V12() V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) Element of K32(K33(NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) .|| : ( ( V12() V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) ) ( Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V127() V128() V129() V133() ) set ) -valued V12() V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) ) Real_Sequence) is convergent & lim ||.(seq1 : ( ( V12() V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) -valued V12() V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) - seq2 : ( ( V12() V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) -valued V12() V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) ) : ( ( V12() V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) -valued V12() V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) Element of K32(K33(NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) .|| : ( ( V12() V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) ) ( Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V127() V128() V129() V133() ) set ) -valued V12() V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) ) Real_Sequence) : ( ( ) ( ext-real V27() real ) Element of REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) = ||.(g1 : ( ( ) ( ) Point of ( ( ) ( V4() ) set ) ) - g2 : ( ( ) ( ) Point of ( ( ) ( V4() ) set ) ) ) : ( ( ) ( ) Element of the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) .|| : ( ( ) ( ext-real V27() real ) Element of REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) ) ;

theorem :: BHSP_2:28

for X being ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace)
for g1, g2 being ( ( ) ( ) Point of ( ( ) ( V4() ) set ) )
for seq1, seq2 being ( ( V12() V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) -valued V12() V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) st seq1 : ( ( V12() V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) -valued V12() V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) is convergent & lim seq1 : ( ( V12() V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) -valued V12() V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) : ( ( ) ( ) Point of ( ( ) ( V4() ) set ) ) = g1 : ( ( ) ( ) Point of ( ( ) ( V4() ) set ) ) & seq2 : ( ( V12() V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) -valued V12() V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) is convergent & lim seq2 : ( ( V12() V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) -valued V12() V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) : ( ( ) ( ) Point of ( ( ) ( V4() ) set ) ) = g2 : ( ( ) ( ) Point of ( ( ) ( V4() ) set ) ) holds
( ||.((seq1 : ( ( V12() V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) -valued V12() V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) - seq2 : ( ( V12() V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) -valued V12() V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) ) : ( ( V12() V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) -valued V12() V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) Element of K32(K33(NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) - (g1 : ( ( ) ( ) Point of ( ( ) ( V4() ) set ) ) - g2 : ( ( ) ( ) Point of ( ( ) ( V4() ) set ) ) ) : ( ( ) ( ) Element of the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) : ( ( V12() V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) -valued V12() V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) Element of K32(K33(NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) .|| : ( ( V12() V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) ) ( Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V127() V128() V129() V133() ) set ) -valued V12() V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) ) Real_Sequence) is convergent & lim ||.((seq1 : ( ( V12() V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) -valued V12() V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) - seq2 : ( ( V12() V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) -valued V12() V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) ) : ( ( V12() V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) -valued V12() V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) Element of K32(K33(NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) - (g1 : ( ( ) ( ) Point of ( ( ) ( V4() ) set ) ) - g2 : ( ( ) ( ) Point of ( ( ) ( V4() ) set ) ) ) : ( ( ) ( ) Element of the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) : ( ( V12() V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) -valued V12() V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) Element of K32(K33(NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) .|| : ( ( V12() V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) ) ( Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V127() V128() V129() V133() ) set ) -valued V12() V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) ) Real_Sequence) : ( ( ) ( ext-real V27() real ) Element of REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) = 0 : ( ( ) ( V4() ext-real non positive non negative epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V27() real V125() V126() V127() V128() V129() V130() V131() V132() V133() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) ) ) ;

theorem :: BHSP_2:29

theorem :: BHSP_2:30

for X being ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace)
for g being ( ( ) ( ) Point of ( ( ) ( V4() ) set ) )
for a being ( ( ) ( ext-real V27() real ) Real)
for seq being ( ( V12() V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) -valued V12() V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) st seq : ( ( V12() V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) -valued V12() V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) is convergent & lim seq : ( ( V12() V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) -valued V12() V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) : ( ( ) ( ) Point of ( ( ) ( V4() ) set ) ) = g : ( ( ) ( ) Point of ( ( ) ( V4() ) set ) ) holds
( ||.((a : ( ( ) ( ext-real V27() real ) Real) * seq : ( ( V12() V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) -valued V12() V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) ) : ( ( V12() V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) -valued V12() V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) Element of K32(K33(NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) - (a : ( ( ) ( ext-real V27() real ) Real) * g : ( ( ) ( ) Point of ( ( ) ( V4() ) set ) ) ) : ( ( ) ( ) Element of the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) : ( ( V12() V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) -valued V12() V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) Element of K32(K33(NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) .|| : ( ( V12() V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) ) ( Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V127() V128() V129() V133() ) set ) -valued V12() V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) ) Real_Sequence) is convergent & lim ||.((a : ( ( ) ( ext-real V27() real ) Real) * seq : ( ( V12() V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) -valued V12() V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) ) : ( ( V12() V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) -valued V12() V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) Element of K32(K33(NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) - (a : ( ( ) ( ext-real V27() real ) Real) * g : ( ( ) ( ) Point of ( ( ) ( V4() ) set ) ) ) : ( ( ) ( ) Element of the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) : ( ( V12() V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) -valued V12() V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) Element of K32(K33(NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) .|| : ( ( V12() V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) ) ( Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V127() V128() V129() V133() ) set ) -valued V12() V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) ) Real_Sequence) : ( ( ) ( ext-real V27() real ) Element of REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) = 0 : ( ( ) ( V4() ext-real non positive non negative epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V27() real V125() V126() V127() V128() V129() V130() V131() V132() V133() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) ) ) ;

theorem :: BHSP_2:31

theorem :: BHSP_2:32

for X being ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace)
for g being ( ( ) ( ) Point of ( ( ) ( V4() ) set ) )
for seq being ( ( V12() V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) -valued V12() V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) st seq : ( ( V12() V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) -valued V12() V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) is convergent & lim seq : ( ( V12() V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) -valued V12() V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) : ( ( ) ( ) Point of ( ( ) ( V4() ) set ) ) = g : ( ( ) ( ) Point of ( ( ) ( V4() ) set ) ) holds
( ||.((- seq : ( ( V12() V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) -valued V12() V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) ) : ( ( V12() ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) -valued V12() V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) Element of K32(K33(NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) - (- g : ( ( ) ( ) Point of ( ( ) ( V4() ) set ) ) ) : ( ( ) ( ) Element of the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) : ( ( V12() V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) -valued V12() V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) Element of K32(K33(NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) .|| : ( ( V12() V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) ) ( Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V127() V128() V129() V133() ) set ) -valued V12() V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) ) Real_Sequence) is convergent & lim ||.((- seq : ( ( V12() V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) -valued V12() V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) ) : ( ( V12() ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) -valued V12() V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) Element of K32(K33(NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) - (- g : ( ( ) ( ) Point of ( ( ) ( V4() ) set ) ) ) : ( ( ) ( ) Element of the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) : ( ( V12() V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) -valued V12() V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) Element of K32(K33(NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) .|| : ( ( V12() V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) ) ( Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V127() V128() V129() V133() ) set ) -valued V12() V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) ) Real_Sequence) : ( ( ) ( ext-real V27() real ) Element of REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) = 0 : ( ( ) ( V4() ext-real non positive non negative epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V27() real V125() V126() V127() V128() V129() V130() V131() V132() V133() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) ) ) ;

theorem :: BHSP_2:33

for X being ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace)
for g, x being ( ( ) ( ) Point of ( ( ) ( V4() ) set ) )
for seq being ( ( V12() V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) -valued V12() V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) st seq : ( ( V12() V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) -valued V12() V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) is convergent & lim seq : ( ( V12() V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) -valued V12() V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) : ( ( ) ( ) Point of ( ( ) ( V4() ) set ) ) = g : ( ( ) ( ) Point of ( ( ) ( V4() ) set ) ) holds
( ||.((seq : ( ( V12() V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) -valued V12() V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) + x : ( ( ) ( ) Point of ( ( ) ( V4() ) set ) ) ) : ( ( V12() V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) -valued V12() V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) Element of K32(K33(NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) - (g : ( ( ) ( ) Point of ( ( ) ( V4() ) set ) ) + x : ( ( ) ( ) Point of ( ( ) ( V4() ) set ) ) ) : ( ( ) ( ) Element of the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) : ( ( V12() V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) -valued V12() V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) Element of K32(K33(NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) .|| : ( ( V12() V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) ) ( Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V127() V128() V129() V133() ) set ) -valued V12() V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) ) Real_Sequence) is convergent & lim ||.((seq : ( ( V12() V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) -valued V12() V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) + x : ( ( ) ( ) Point of ( ( ) ( V4() ) set ) ) ) : ( ( V12() V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) -valued V12() V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) Element of K32(K33(NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) - (g : ( ( ) ( ) Point of ( ( ) ( V4() ) set ) ) + x : ( ( ) ( ) Point of ( ( ) ( V4() ) set ) ) ) : ( ( ) ( ) Element of the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) : ( ( V12() V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) -valued V12() V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) Element of K32(K33(NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) .|| : ( ( V12() V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) ) ( Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V127() V128() V129() V133() ) set ) -valued V12() V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) ) Real_Sequence) : ( ( ) ( ext-real V27() real ) Element of REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) = 0 : ( ( ) ( V4() ext-real non positive non negative epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V27() real V125() V126() V127() V128() V129() V130() V131() V132() V133() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) ) ) ;

theorem :: BHSP_2:34

theorem :: BHSP_2:35

for X being ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace)
for g, x being ( ( ) ( ) Point of ( ( ) ( V4() ) set ) )
for seq being ( ( V12() V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) -valued V12() V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) st seq : ( ( V12() V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) -valued V12() V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) is convergent & lim seq : ( ( V12() V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) -valued V12() V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) : ( ( ) ( ) Point of ( ( ) ( V4() ) set ) ) = g : ( ( ) ( ) Point of ( ( ) ( V4() ) set ) ) holds
( ||.((seq : ( ( V12() V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) -valued V12() V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) - x : ( ( ) ( ) Point of ( ( ) ( V4() ) set ) ) ) : ( ( V12() V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) -valued V12() V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) Element of K32(K33(NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) - (g : ( ( ) ( ) Point of ( ( ) ( V4() ) set ) ) - x : ( ( ) ( ) Point of ( ( ) ( V4() ) set ) ) ) : ( ( ) ( ) Element of the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) : ( ( V12() V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) -valued V12() V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) Element of K32(K33(NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) .|| : ( ( V12() V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) ) ( Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V127() V128() V129() V133() ) set ) -valued V12() V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) ) Real_Sequence) is convergent & lim ||.((seq : ( ( V12() V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) -valued V12() V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) - x : ( ( ) ( ) Point of ( ( ) ( V4() ) set ) ) ) : ( ( V12() V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) -valued V12() V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) Element of K32(K33(NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) - (g : ( ( ) ( ) Point of ( ( ) ( V4() ) set ) ) - x : ( ( ) ( ) Point of ( ( ) ( V4() ) set ) ) ) : ( ( ) ( ) Element of the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) : ( ( V12() V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) -valued V12() V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) Element of K32(K33(NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) .|| : ( ( V12() V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) ) ( Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V127() V128() V129() V133() ) set ) -valued V12() V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) ) Real_Sequence) : ( ( ) ( ext-real V27() real ) Element of REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) = 0 : ( ( ) ( V4() ext-real non positive non negative epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V27() real V125() V126() V127() V128() V129() V130() V131() V132() V133() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) ) ) ;

theorem :: BHSP_2:36

for X being ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace)
for g1, g2 being ( ( ) ( ) Point of ( ( ) ( V4() ) set ) )
for seq1, seq2 being ( ( V12() V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) -valued V12() V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) st seq1 : ( ( V12() V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) -valued V12() V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) is convergent & lim seq1 : ( ( V12() V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) -valued V12() V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) : ( ( ) ( ) Point of ( ( ) ( V4() ) set ) ) = g1 : ( ( ) ( ) Point of ( ( ) ( V4() ) set ) ) & seq2 : ( ( V12() V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) -valued V12() V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) is convergent & lim seq2 : ( ( V12() V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) -valued V12() V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) : ( ( ) ( ) Point of ( ( ) ( V4() ) set ) ) = g2 : ( ( ) ( ) Point of ( ( ) ( V4() ) set ) ) holds
( dist ((seq1 : ( ( V12() V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) -valued V12() V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) + seq2 : ( ( V12() V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) -valued V12() V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) ) : ( ( V12() V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) -valued V12() V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) Element of K32(K33(NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,(g1 : ( ( ) ( ) Point of ( ( ) ( V4() ) set ) ) + g2 : ( ( ) ( ) Point of ( ( ) ( V4() ) set ) ) ) : ( ( ) ( ) Element of the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) : ( ( V12() V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) ) ( Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V127() V128() V129() V133() ) set ) -valued V12() V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) ) Real_Sequence) is convergent & lim (dist ((seq1 : ( ( V12() V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) -valued V12() V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) + seq2 : ( ( V12() V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) -valued V12() V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) ) : ( ( V12() V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) -valued V12() V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) Element of K32(K33(NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,(g1 : ( ( ) ( ) Point of ( ( ) ( V4() ) set ) ) + g2 : ( ( ) ( ) Point of ( ( ) ( V4() ) set ) ) ) : ( ( ) ( ) Element of the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) )) : ( ( V12() V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) ) ( Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V127() V128() V129() V133() ) set ) -valued V12() V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) ) Real_Sequence) : ( ( ) ( ext-real V27() real ) Element of REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) = 0 : ( ( ) ( V4() ext-real non positive non negative epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V27() real V125() V126() V127() V128() V129() V130() V131() V132() V133() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) ) ) ;

theorem :: BHSP_2:37

for X being ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace)
for g1, g2 being ( ( ) ( ) Point of ( ( ) ( V4() ) set ) )
for seq1, seq2 being ( ( V12() V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) -valued V12() V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) st seq1 : ( ( V12() V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) -valued V12() V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) is convergent & lim seq1 : ( ( V12() V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) -valued V12() V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) : ( ( ) ( ) Point of ( ( ) ( V4() ) set ) ) = g1 : ( ( ) ( ) Point of ( ( ) ( V4() ) set ) ) & seq2 : ( ( V12() V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) -valued V12() V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) is convergent & lim seq2 : ( ( V12() V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) -valued V12() V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) : ( ( ) ( ) Point of ( ( ) ( V4() ) set ) ) = g2 : ( ( ) ( ) Point of ( ( ) ( V4() ) set ) ) holds
( dist ((seq1 : ( ( V12() V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) -valued V12() V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) - seq2 : ( ( V12() V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) -valued V12() V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) ) : ( ( V12() V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) -valued V12() V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) Element of K32(K33(NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,(g1 : ( ( ) ( ) Point of ( ( ) ( V4() ) set ) ) - g2 : ( ( ) ( ) Point of ( ( ) ( V4() ) set ) ) ) : ( ( ) ( ) Element of the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) : ( ( V12() V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) ) ( Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V127() V128() V129() V133() ) set ) -valued V12() V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) ) Real_Sequence) is convergent & lim (dist ((seq1 : ( ( V12() V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) -valued V12() V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) - seq2 : ( ( V12() V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) -valued V12() V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) ) : ( ( V12() V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) -defined the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) -valued V12() V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) ) Element of K32(K33(NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,(g1 : ( ( ) ( ) Point of ( ( ) ( V4() ) set ) ) - g2 : ( ( ) ( ) Point of ( ( ) ( V4() ) set ) ) ) : ( ( ) ( ) Element of the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) )) : ( ( V12() V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) ) ( Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( V127() V128() V129() V133() ) set ) -valued V12() V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) ) Real_Sequence) : ( ( ) ( ext-real V27() real ) Element of REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) = 0 : ( ( ) ( V4() ext-real non positive non negative epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V27() real V125() V126() V127() V128() V129() V130() V131() V132() V133() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V127() V128() V129() V130() V131() V132() V133() ) Element of K32(REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) : ( ( ) ( ) set ) ) ) ) ;

theorem :: BHSP_2:38

theorem :: BHSP_2:39

definition

func Ball (x,r) -> ( ( ) ( ) Subset of ) equals :: BHSP_2:def 5
{ y : ( ( ) ( ) Point of ( ( ) ( V4() ) set ) ) where y is ( ( ) ( ) Point of ( ( ) ( ) set ) ) : ||.(x : ( ( ) ( ) Element of X : ( ( ) ( ) NORMSTR ) ) - y : ( ( ) ( ) Point of ( ( ) ( V4() ) set ) ) ) : ( ( ) ( ) Element of the carrier of X : ( ( ) ( ) NORMSTR ) : ( ( ) ( ) set ) ) .|| : ( ( ) ( ext-real V27() real ) Element of REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) < r : ( ( V12() V30(K33(X : ( ( ) ( ) NORMSTR ) ,X : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ,X : ( ( ) ( ) NORMSTR ) ) ) ( Relation-like K33(X : ( ( ) ( ) NORMSTR ) ,X : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) -defined X : ( ( ) ( ) NORMSTR ) -valued V12() V30(K33(X : ( ( ) ( ) NORMSTR ) ,X : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ,X : ( ( ) ( ) NORMSTR ) ) ) Element of K32(K33(K33(X : ( ( ) ( ) NORMSTR ) ,X : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ,X : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) } ;

func cl_Ball (x,r) -> ( ( ) ( ) Subset of ) equals :: BHSP_2:def 6
{ y : ( ( ) ( ) Point of ( ( ) ( V4() ) set ) ) where y is ( ( ) ( ) Point of ( ( ) ( ) set ) ) : ||.(x : ( ( ) ( ) Element of X : ( ( ) ( ) NORMSTR ) ) - y : ( ( ) ( ) Point of ( ( ) ( V4() ) set ) ) ) : ( ( ) ( ) Element of the carrier of X : ( ( ) ( ) NORMSTR ) : ( ( ) ( ) set ) ) .|| : ( ( ) ( ext-real V27() real ) Element of REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) <= r : ( ( V12() V30(K33(X : ( ( ) ( ) NORMSTR ) ,X : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ,X : ( ( ) ( ) NORMSTR ) ) ) ( Relation-like K33(X : ( ( ) ( ) NORMSTR ) ,X : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) -defined X : ( ( ) ( ) NORMSTR ) -valued V12() V30(K33(X : ( ( ) ( ) NORMSTR ) ,X : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ,X : ( ( ) ( ) NORMSTR ) ) ) Element of K32(K33(K33(X : ( ( ) ( ) NORMSTR ) ,X : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ,X : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) } ;

func Sphere (x,r) -> ( ( ) ( ) Subset of ) equals :: BHSP_2:def 7
{ y : ( ( ) ( ) Point of ( ( ) ( V4() ) set ) ) where y is ( ( ) ( ) Point of ( ( ) ( ) set ) ) : ||.(x : ( ( ) ( ) Element of X : ( ( ) ( ) NORMSTR ) ) - y : ( ( ) ( ) Point of ( ( ) ( V4() ) set ) ) ) : ( ( ) ( ) Element of the carrier of X : ( ( ) ( ) NORMSTR ) : ( ( ) ( ) set ) ) .|| : ( ( ) ( ext-real V27() real ) Element of REAL : ( ( ) ( V127() V128() V129() V133() ) set ) ) = r : ( ( V12() V30(K33(X : ( ( ) ( ) NORMSTR ) ,X : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ,X : ( ( ) ( ) NORMSTR ) ) ) ( Relation-like K33(X : ( ( ) ( ) NORMSTR ) ,X : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) -defined X : ( ( ) ( ) NORMSTR ) -valued V12() V30(K33(X : ( ( ) ( ) NORMSTR ) ,X : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ,X : ( ( ) ( ) NORMSTR ) ) ) Element of K32(K33(K33(X : ( ( ) ( ) NORMSTR ) ,X : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ,X : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) } ;

end;

theorem :: BHSP_2:40

theorem :: BHSP_2:41

theorem :: BHSP_2:42

theorem :: BHSP_2:43

theorem :: BHSP_2:44

theorem :: BHSP_2:45

theorem :: BHSP_2:46

theorem :: BHSP_2:47

theorem :: BHSP_2:48

theorem :: BHSP_2:49

theorem :: BHSP_2:50

theorem :: BHSP_2:51

theorem :: BHSP_2:52

theorem :: BHSP_2:53

theorem :: BHSP_2:54

for X being ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace)
for x being ( ( ) ( ) Point of ( ( ) ( V4() ) set ) )
for r being ( ( ) ( ext-real V27() real ) Real) holds Ball (x : ( ( ) ( ) Point of ( ( ) ( V4() ) set ) ) ,r : ( ( ) ( ext-real V27() real ) Real) ) : ( ( ) ( ) Subset of ) c= cl_Ball (x : ( ( ) ( ) Point of ( ( ) ( V4() ) set ) ) ,r : ( ( ) ( ext-real V27() real ) Real) ) : ( ( ) ( ) Subset of ) ;

theorem :: BHSP_2:55

for X being ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace)
for x being ( ( ) ( ) Point of ( ( ) ( V4() ) set ) )
for r being ( ( ) ( ext-real V27() real ) Real) holds Sphere (x : ( ( ) ( ) Point of ( ( ) ( V4() ) set ) ) ,r : ( ( ) ( ext-real V27() real ) Real) ) : ( ( ) ( ) Subset of ) c= cl_Ball (x : ( ( ) ( ) Point of ( ( ) ( V4() ) set ) ) ,r : ( ( ) ( ext-real V27() real ) Real) ) : ( ( ) ( ) Subset of ) ;

theorem :: BHSP_2:56

for X being ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace)
for x being ( ( ) ( ) Point of ( ( ) ( V4() ) set ) )
for r being ( ( ) ( ext-real V27() real ) Real) holds (Ball (x : ( ( ) ( ) Point of ( ( ) ( V4() ) set ) ) ,r : ( ( ) ( ext-real V27() real ) Real) )) : ( ( ) ( ) Subset of ) \/ (Sphere (x : ( ( ) ( ) Point of ( ( ) ( V4() ) set ) ) ,r : ( ( ) ( ext-real V27() real ) Real) )) : ( ( ) ( ) Subset of ) : ( ( ) ( ) Element of K32( the carrier of b₁ : ( ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) ( non empty V83() V108() V109() V110() V111() V112() V113() V114() RealUnitarySpace-like ) RealUnitarySpace) : ( ( ) ( V4() ) set ) ) : ( ( ) ( ) set ) ) = cl_Ball (x : ( ( ) ( ) Point of ( ( ) ( V4() ) set ) ) ,r : ( ( ) ( ext-real V27() real ) Real) ) : ( ( ) ( ) Subset of ) ;