:: LOPBAN_3 semantic presentation

begin

theorem :: LOPBAN_3:1
for X being ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) NORMSTR )
for seq being ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) NORMSTR ) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) NORMSTR ) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) NORMSTR ) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) st ( for n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) holds seq : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) NORMSTR ) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) NORMSTR ) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) NORMSTR ) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) NORMSTR ) : ( ( ) ( V4() ) set ) ) = 0. X : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) NORMSTR ) : ( ( ) ( V79(b1 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) NORMSTR ) ) right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) NORMSTR ) : ( ( ) ( V4() ) set ) ) ) holds
for m being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) holds (Partial_Sums seq : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) NORMSTR ) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) NORMSTR ) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) NORMSTR ) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) ) : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) NORMSTR ) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) NORMSTR ) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) NORMSTR ) : ( ( ) ( V4() ) set ) ) ) Element of K32(K33(NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) NORMSTR ) : ( ( ) ( V4() ) set ) ) : ( ( ) ( V4() ) set ) ) : ( ( ) ( V4() ) set ) ) . m : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) NORMSTR ) : ( ( ) ( V4() ) set ) ) = 0. X : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) NORMSTR ) : ( ( ) ( V79(b1 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) NORMSTR ) ) right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty right_complementable add-associative right_zeroed ) NORMSTR ) : ( ( ) ( V4() ) set ) ) ;

definition
let X be ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) ;
let seq be ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) ;
attr seq is summable means :: LOPBAN_3:def 1
 Partial_Sums seq : ( ( ) ( ) VectSpStr over X : ( ( ) ( ) Normed_AlgebraStr ) ) : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of X : ( ( ) ( ) Normed_AlgebraStr ) : ( ( ) ( ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of X : ( ( ) ( ) Normed_AlgebraStr ) : ( ( ) ( ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of X : ( ( ) ( ) Normed_AlgebraStr ) : ( ( ) ( ) set ) ) ) Element of K32(K33(NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of X : ( ( ) ( ) Normed_AlgebraStr ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V4() ) set ) ) is convergent ;
end;

registration
let X be ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) ;
cluster V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V4() ) set ) ) summable for ( ( ) ( ) Element of K32(K33(NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V4() ) set ) ) : ( ( ) ( V4() ) set ) ) : ( ( ) ( V4() ) set ) ) ;
end;

definition
let X be ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) ;
let seq be ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) ;
func Sum seq -> ( ( ) ( right_complementable ) Element of ( ( ) ( V4() ) set ) ) equals :: LOPBAN_3:def 2
 lim (Partial_Sums seq : ( ( ) ( ) VectSpStr over X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) NORMSTR ) ) ) : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V4() ) set ) ) ) Element of K32(K33(NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V4() ) set ) ) : ( ( ) ( V4() ) set ) ) : ( ( ) ( V4() ) set ) ) : ( ( ) ( right_complementable ) Element of the carrier of X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V4() ) set ) ) ;
end;

definition
let X be ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) ;
let seq be ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) ;
attr seq is norm_summable means :: LOPBAN_3:def 3
||.seq : ( ( ) ( ) VectSpStr over X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) NORMSTR ) ) .|| : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) complex-valued ext-real-valued real-valued ) Element of K32(K33(NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ,REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( V4() ) set ) ) is summable ;
end;

theorem :: LOPBAN_3:2
for X being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace)
for seq being ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) )
for m being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) holds 0 : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V24() real ext-real non positive non negative V45() V46() V52() V53() V54() V55() V56() V57() V58() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) <= ||.seq : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) .|| : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) complex-valued ext-real-valued real-valued ) Element of K32(K33(NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ,REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( V4() ) set ) ) . m : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) ;

theorem :: LOPBAN_3:3
for X being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace)
for x, y, z being ( ( ) ( right_complementable ) Element of ( ( ) ( V4() ) set ) ) holds ||.(x : ( ( ) ( right_complementable ) Element of ( ( ) ( V4() ) set ) ) - y : ( ( ) ( right_complementable ) Element of ( ( ) ( V4() ) set ) ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) .|| : ( ( ) ( V24() real ext-real non negative ) Element of REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) = ||.((x : ( ( ) ( right_complementable ) Element of ( ( ) ( V4() ) set ) ) - z : ( ( ) ( right_complementable ) Element of ( ( ) ( V4() ) set ) ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) + (z : ( ( ) ( right_complementable ) Element of ( ( ) ( V4() ) set ) ) - y : ( ( ) ( right_complementable ) Element of ( ( ) ( V4() ) set ) ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) .|| : ( ( ) ( V24() real ext-real non negative ) Element of REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) ;

theorem :: LOPBAN_3:4
for X being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace)
for seq being ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) st seq : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) is convergent holds
for s being ( ( ) ( V24() real ext-real ) Real) st 0 : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V24() real ext-real non positive non negative V45() V46() V52() V53() V54() V55() V56() V57() V58() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) < s : ( ( ) ( V24() real ext-real ) Real) holds
ex n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) st
for m being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) st n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) <= m : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) holds
||.((seq : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) . m : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) - (seq : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) .|| : ( ( ) ( V24() real ext-real non negative ) Element of REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) < s : ( ( ) ( V24() real ext-real ) Real) ;

theorem :: LOPBAN_3:5
for X being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace)
for seq being ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) holds
( seq : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) is Cauchy_sequence_by_Norm iff for p being ( ( ) ( V24() real ext-real ) Real) st p : ( ( ) ( V24() real ext-real ) Real) > 0 : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V24() real ext-real non positive non negative V45() V46() V52() V53() V54() V55() V56() V57() V58() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) holds
ex n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) st
for m being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) st n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) <= m : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) holds
||.((seq : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) . m : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) - (seq : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) .|| : ( ( ) ( V24() real ext-real non negative ) Element of REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) < p : ( ( ) ( V24() real ext-real ) Real) ) ;

theorem :: LOPBAN_3:6
for X being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace)
for seq being ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) st ( for n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) holds seq : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) = 0. X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V79(b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) ) right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) holds
for m being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) holds (Partial_Sums ||.seq : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) .|| : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) complex-valued ext-real-valued real-valued ) Element of K32(K33(NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ,REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( V4() ) set ) ) ) : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) complex-valued ext-real-valued real-valued ) Element of K32(K33(NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ,REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( V4() ) set ) ) . m : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) = 0 : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V24() real ext-real non positive non negative V45() V46() V52() V53() V54() V55() V56() V57() V58() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) ;

theorem :: LOPBAN_3:7
for X being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace)
for seq, seq1 being ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) st seq1 : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) is ( ( ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) subsequence of seq : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) ) & seq : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) is convergent holds
seq1 : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) is convergent ;

theorem :: LOPBAN_3:8
for X being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace)
for seq, seq1 being ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) st seq1 : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) is ( ( ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) subsequence of seq : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) ) & seq : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) is convergent holds
lim seq1 : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) = lim seq : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ;

theorem :: LOPBAN_3:9
for X being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace)
for seq, seq1 being ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) )
for k being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) st seq : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) is convergent holds
( seq : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) ^\ k : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) : ( ( ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) subsequence of b2 : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) ) is convergent & lim (seq : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) ^\ k : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) ) : ( ( ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) subsequence of b2 : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) = lim seq : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) ;

theorem :: LOPBAN_3:10
for X being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace)
for seq, seq1 being ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) st seq : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) is convergent & ex k being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) st seq : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) = seq1 : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) ^\ k : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) : ( ( ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) subsequence of b3 : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) ) holds
seq1 : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) is convergent ;

theorem :: LOPBAN_3:11
for X being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace)
for seq, seq1 being ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) st seq : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) is convergent & ex k being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) st seq : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) = seq1 : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) ^\ k : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) : ( ( ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) subsequence of b3 : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) ) holds
lim seq1 : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) = lim seq : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ;

theorem :: LOPBAN_3:12
for X being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace)
for seq being ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) st seq : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) is constant holds
seq : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) is convergent ;

theorem :: LOPBAN_3:13
for X being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace)
for seq being ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) st ( for n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) holds seq : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) = 0. X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V79(b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) ) right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) holds
seq : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) is norm_summable ;

registration
let X be ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) ;
cluster V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V4() ) set ) ) norm_summable for ( ( ) ( ) Element of K32(K33(NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V4() ) set ) ) : ( ( ) ( V4() ) set ) ) : ( ( ) ( V4() ) set ) ) ;
end;

theorem :: LOPBAN_3:14
for X being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace)
for s being ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) st s : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) is summable holds
( s : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) is convergent & lim s : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) = 0. X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V79(b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) ) right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) ;

theorem :: LOPBAN_3:15
for X being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace)
for s1, s2 being ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) holds (Partial_Sums s1 : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) ) : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) Element of K32(K33(NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) : ( ( ) ( V4() ) set ) ) : ( ( ) ( V4() ) set ) ) + (Partial_Sums s2 : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) ) : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) Element of K32(K33(NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) : ( ( ) ( V4() ) set ) ) : ( ( ) ( V4() ) set ) ) : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) Element of K32(K33(NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) : ( ( ) ( V4() ) set ) ) : ( ( ) ( V4() ) set ) ) = Partial_Sums (s1 : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) + s2 : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) ) : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) Element of K32(K33(NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) : ( ( ) ( V4() ) set ) ) : ( ( ) ( V4() ) set ) ) : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) Element of K32(K33(NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) : ( ( ) ( V4() ) set ) ) : ( ( ) ( V4() ) set ) ) ;

theorem :: LOPBAN_3:16
for X being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace)
for s1, s2 being ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) holds (Partial_Sums s1 : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) ) : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) Element of K32(K33(NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) : ( ( ) ( V4() ) set ) ) : ( ( ) ( V4() ) set ) ) - (Partial_Sums s2 : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) ) : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) Element of K32(K33(NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) : ( ( ) ( V4() ) set ) ) : ( ( ) ( V4() ) set ) ) : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) Element of K32(K33(NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) : ( ( ) ( V4() ) set ) ) : ( ( ) ( V4() ) set ) ) = Partial_Sums (s1 : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) - s2 : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) ) : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) Element of K32(K33(NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) : ( ( ) ( V4() ) set ) ) : ( ( ) ( V4() ) set ) ) : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) Element of K32(K33(NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) : ( ( ) ( V4() ) set ) ) : ( ( ) ( V4() ) set ) ) ;

registration
let X be ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) ;
let seq be ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) norm_summable ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) norm_summable ) sequence of ( ( ) ( V4() ) set ) ) ;
cluster ||.seq : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V4() ) set ) ) norm_summable ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V4() ) set ) ) norm_summable ) Element of K32(K33(NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V4() ) set ) ) : ( ( ) ( V4() ) set ) ) : ( ( ) ( V4() ) set ) ) .|| : ( ( Relation-like Function-like ) ( Relation-like REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) set ) -> Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) summable for ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ;
end;

registration
let X be ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) ;
cluster Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V4() ) set ) ) summable -> Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V4() ) set ) ) convergent for ( ( ) ( ) Element of K32(K33(NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V4() ) set ) ) : ( ( ) ( V4() ) set ) ) : ( ( ) ( V4() ) set ) ) ;
end;

theorem :: LOPBAN_3:17
for X being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace)
for seq1, seq2 being ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) st seq1 : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) is summable & seq2 : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) is summable holds
( seq1 : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) + seq2 : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) Element of K32(K33(NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) : ( ( ) ( V4() ) set ) ) : ( ( ) ( V4() ) set ) ) is summable & Sum (seq1 : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) + seq2 : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) ) : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) Element of K32(K33(NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) : ( ( ) ( V4() ) set ) ) : ( ( ) ( V4() ) set ) ) : ( ( ) ( right_complementable ) Element of ( ( ) ( V4() ) set ) ) = (Sum seq1 : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) ) : ( ( ) ( right_complementable ) Element of ( ( ) ( V4() ) set ) ) + (Sum seq2 : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) ) : ( ( ) ( right_complementable ) Element of ( ( ) ( V4() ) set ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) ;

theorem :: LOPBAN_3:18
for X being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace)
for seq1, seq2 being ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) st seq1 : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) is summable & seq2 : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) is summable holds
( seq1 : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) - seq2 : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) Element of K32(K33(NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) : ( ( ) ( V4() ) set ) ) : ( ( ) ( V4() ) set ) ) is summable & Sum (seq1 : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) - seq2 : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) ) : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) Element of K32(K33(NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) : ( ( ) ( V4() ) set ) ) : ( ( ) ( V4() ) set ) ) : ( ( ) ( right_complementable ) Element of ( ( ) ( V4() ) set ) ) = (Sum seq1 : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) ) : ( ( ) ( right_complementable ) Element of ( ( ) ( V4() ) set ) ) - (Sum seq2 : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) ) : ( ( ) ( right_complementable ) Element of ( ( ) ( V4() ) set ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) ;

registration
let X be ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) ;
let seq1, seq2 be ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) summable ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) convergent summable ) sequence of ( ( ) ( V4() ) set ) ) ;
cluster seq1 : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V4() ) set ) ) summable ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V4() ) set ) ) convergent summable ) Element of K32(K33(NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V4() ) set ) ) : ( ( ) ( V4() ) set ) ) : ( ( ) ( V4() ) set ) ) + seq2 : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V4() ) set ) ) summable ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V4() ) set ) ) convergent summable ) Element of K32(K33(NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V4() ) set ) ) : ( ( ) ( V4() ) set ) ) : ( ( ) ( V4() ) set ) ) : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V4() ) set ) ) ) Element of K32(K33(NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V4() ) set ) ) : ( ( ) ( V4() ) set ) ) : ( ( ) ( V4() ) set ) ) -> Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V4() ) set ) ) summable ;
cluster seq1 : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V4() ) set ) ) summable ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V4() ) set ) ) convergent summable ) Element of K32(K33(NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V4() ) set ) ) : ( ( ) ( V4() ) set ) ) : ( ( ) ( V4() ) set ) ) - seq2 : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V4() ) set ) ) summable ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V4() ) set ) ) convergent summable ) Element of K32(K33(NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V4() ) set ) ) : ( ( ) ( V4() ) set ) ) : ( ( ) ( V4() ) set ) ) : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V4() ) set ) ) ) Element of K32(K33(NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V4() ) set ) ) : ( ( ) ( V4() ) set ) ) : ( ( ) ( V4() ) set ) ) -> Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V4() ) set ) ) summable ;
end;

theorem :: LOPBAN_3:19
for X being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace)
for seq being ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) )
for z being ( ( ) ( V24() real ext-real ) Real) holds Partial_Sums (z : ( ( ) ( V24() real ext-real ) Real) * seq : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) ) : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) Element of K32(K33(NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) : ( ( ) ( V4() ) set ) ) : ( ( ) ( V4() ) set ) ) : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) Element of K32(K33(NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) : ( ( ) ( V4() ) set ) ) : ( ( ) ( V4() ) set ) ) = z : ( ( ) ( V24() real ext-real ) Real) * (Partial_Sums seq : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) ) : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) Element of K32(K33(NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) : ( ( ) ( V4() ) set ) ) : ( ( ) ( V4() ) set ) ) : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) Element of K32(K33(NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) : ( ( ) ( V4() ) set ) ) : ( ( ) ( V4() ) set ) ) ;

theorem :: LOPBAN_3:20
for X being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace)
for seq being ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) summable ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) convergent summable ) sequence of ( ( ) ( V4() ) set ) )
for z being ( ( ) ( V24() real ext-real ) Real) holds
( z : ( ( ) ( V24() real ext-real ) Real) * seq : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) summable ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) convergent summable ) sequence of ( ( ) ( V4() ) set ) ) : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) Element of K32(K33(NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) : ( ( ) ( V4() ) set ) ) : ( ( ) ( V4() ) set ) ) is summable & Sum (z : ( ( ) ( V24() real ext-real ) Real) * seq : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) summable ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) convergent summable ) sequence of ( ( ) ( V4() ) set ) ) ) : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) Element of K32(K33(NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) : ( ( ) ( V4() ) set ) ) : ( ( ) ( V4() ) set ) ) : ( ( ) ( right_complementable ) Element of ( ( ) ( V4() ) set ) ) = z : ( ( ) ( V24() real ext-real ) Real) * (Sum seq : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) summable ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) convergent summable ) sequence of ( ( ) ( V4() ) set ) ) ) : ( ( ) ( right_complementable ) Element of ( ( ) ( V4() ) set ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) ;

registration
let X be ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) ;
let z be ( ( ) ( V24() real ext-real ) Real) ;
let seq be ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) summable ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) convergent summable ) sequence of ( ( ) ( V4() ) set ) ) ;
cluster z : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) * seq : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V4() ) set ) ) summable ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V4() ) set ) ) convergent summable ) Element of K32(K33(NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V4() ) set ) ) : ( ( ) ( V4() ) set ) ) : ( ( ) ( V4() ) set ) ) : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V4() ) set ) ) ) Element of K32(K33(NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V4() ) set ) ) : ( ( ) ( V4() ) set ) ) : ( ( ) ( V4() ) set ) ) -> Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V4() ) set ) ) summable ;
end;

theorem :: LOPBAN_3:21
for X being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace)
for s, s1 being ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) st ( for n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) holds s1 : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) = s : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) . 0 : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V24() real ext-real non positive non negative V45() V46() V52() V53() V54() V55() V56() V57() V58() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) holds
Partial_Sums (s : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) ^\ 1 : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) ) : ( ( ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) subsequence of b2 : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) ) : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) Element of K32(K33(NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) : ( ( ) ( V4() ) set ) ) : ( ( ) ( V4() ) set ) ) = ((Partial_Sums s : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) ) : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) Element of K32(K33(NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) : ( ( ) ( V4() ) set ) ) : ( ( ) ( V4() ) set ) ) ^\ 1 : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) ) : ( ( ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) subsequence of Partial_Sums b2 : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) Element of K32(K33(NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) : ( ( ) ( V4() ) set ) ) : ( ( ) ( V4() ) set ) ) ) - s1 : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) Element of K32(K33(NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) : ( ( ) ( V4() ) set ) ) : ( ( ) ( V4() ) set ) ) ;

theorem :: LOPBAN_3:22
for X being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace)
for s being ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) st s : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) is summable holds
for n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) holds s : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) ^\ n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) : ( ( ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) subsequence of b2 : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) ) is summable ;

registration
let X be ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) ;
let seq be ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) summable ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) convergent summable ) sequence of ( ( ) ( V4() ) set ) ) ;
let n be ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) ;
cluster seq : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V4() ) set ) ) summable ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V4() ) set ) ) convergent summable ) Element of K32(K33(NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V4() ) set ) ) : ( ( ) ( V4() ) set ) ) : ( ( ) ( V4() ) set ) ) ^\ n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) : ( ( Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) ) ( Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) ) set ) -> Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V4() ) set ) ) summable for ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) ;
end;

theorem :: LOPBAN_3:23
for X being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace)
for seq being ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) holds
( Partial_Sums ||.seq : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) .|| : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) complex-valued ext-real-valued real-valued ) Element of K32(K33(NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ,REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( V4() ) set ) ) : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) complex-valued ext-real-valued real-valued ) Element of K32(K33(NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ,REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( V4() ) set ) ) is bounded_above iff seq : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) is norm_summable ) ;

registration
let X be ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) ;
let seq be ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) norm_summable ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) norm_summable ) sequence of ( ( ) ( V4() ) set ) ) ;
cluster Partial_Sums ||.seq : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V4() ) set ) ) norm_summable ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V4() ) set ) ) norm_summable ) Element of K32(K33(NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V4() ) set ) ) : ( ( ) ( V4() ) set ) ) : ( ( ) ( V4() ) set ) ) .|| : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) complex-valued ext-real-valued real-valued summable ) Element of K32(K33(NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ,REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( V4() ) set ) ) : ( ( Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) complex-valued ) ( Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) complex-valued ) set ) -> Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) bounded_above for ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) ;
end;

theorem :: LOPBAN_3:24
for X being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like complete ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like complete ) RealBanachSpace)
for seq being ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like complete ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like complete ) RealBanachSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like complete ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like complete ) RealBanachSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like complete ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like complete ) RealBanachSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) holds
( seq : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like complete ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like complete ) RealBanachSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like complete ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like complete ) RealBanachSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like complete ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like complete ) RealBanachSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) is summable iff for p being ( ( ) ( V24() real ext-real ) Real) st 0 : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V24() real ext-real non positive non negative V45() V46() V52() V53() V54() V55() V56() V57() V58() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) < p : ( ( ) ( V24() real ext-real ) Real) holds
ex n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) st
for m being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) st n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) <= m : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) holds
||.(((Partial_Sums seq : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like complete ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like complete ) RealBanachSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like complete ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like complete ) RealBanachSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like complete ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like complete ) RealBanachSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) ) : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like complete ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like complete ) RealBanachSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like complete ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like complete ) RealBanachSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like complete ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like complete ) RealBanachSpace) : ( ( ) ( V4() ) set ) ) ) Element of K32(K33(NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like complete ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like complete ) RealBanachSpace) : ( ( ) ( V4() ) set ) ) : ( ( ) ( V4() ) set ) ) : ( ( ) ( V4() ) set ) ) . m : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like complete ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like complete ) RealBanachSpace) : ( ( ) ( V4() ) set ) ) - ((Partial_Sums seq : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like complete ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like complete ) RealBanachSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like complete ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like complete ) RealBanachSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like complete ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like complete ) RealBanachSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) ) : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like complete ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like complete ) RealBanachSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like complete ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like complete ) RealBanachSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like complete ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like complete ) RealBanachSpace) : ( ( ) ( V4() ) set ) ) ) Element of K32(K33(NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like complete ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like complete ) RealBanachSpace) : ( ( ) ( V4() ) set ) ) : ( ( ) ( V4() ) set ) ) : ( ( ) ( V4() ) set ) ) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like complete ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like complete ) RealBanachSpace) : ( ( ) ( V4() ) set ) ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like complete ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like complete ) RealBanachSpace) : ( ( ) ( V4() ) set ) ) .|| : ( ( ) ( V24() real ext-real non negative ) Element of REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) < p : ( ( ) ( V24() real ext-real ) Real) ) ;

theorem :: LOPBAN_3:25
for X being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace)
for s being ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) )
for n, m being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) st n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) <= m : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) holds
||.(((Partial_Sums s : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) ) : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) Element of K32(K33(NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) : ( ( ) ( V4() ) set ) ) : ( ( ) ( V4() ) set ) ) . m : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) - ((Partial_Sums s : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) ) : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) Element of K32(K33(NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) : ( ( ) ( V4() ) set ) ) : ( ( ) ( V4() ) set ) ) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) .|| : ( ( ) ( V24() real ext-real non negative ) Element of REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) <= abs (((Partial_Sums ||.s : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) .|| : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) complex-valued ext-real-valued real-valued ) Element of K32(K33(NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ,REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( V4() ) set ) ) ) : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) complex-valued ext-real-valued real-valued ) Element of K32(K33(NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ,REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( V4() ) set ) ) . m : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) ) : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) - ((Partial_Sums ||.s : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) .|| : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) complex-valued ext-real-valued real-valued ) Element of K32(K33(NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ,REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( V4() ) set ) ) ) : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) complex-valued ext-real-valued real-valued ) Element of K32(K33(NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ,REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( V4() ) set ) ) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) ) : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) ) : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) ;

theorem :: LOPBAN_3:26
for X being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like complete ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like complete ) RealBanachSpace)
for seq being ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like complete ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like complete ) RealBanachSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like complete ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like complete ) RealBanachSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like complete ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like complete ) RealBanachSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) st seq : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like complete ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like complete ) RealBanachSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like complete ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like complete ) RealBanachSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like complete ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like complete ) RealBanachSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) is norm_summable holds
seq : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like complete ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like complete ) RealBanachSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like complete ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like complete ) RealBanachSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like complete ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like complete ) RealBanachSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) is summable ;

theorem :: LOPBAN_3:27
for X being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace)
for rseq1 being ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence)
for seq2 being ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) st rseq1 : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) is summable & ex m being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) st
for n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) st m : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) <= n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) holds
||.(seq2 : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) .|| : ( ( ) ( V24() real ext-real non negative ) Element of REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) <= rseq1 : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) holds
seq2 : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) is norm_summable ;

theorem :: LOPBAN_3:28
for X being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace)
for seq1, seq2 being ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) st ( for n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) holds
( 0 : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V24() real ext-real non positive non negative V45() V46() V52() V53() V54() V55() V56() V57() V58() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) <= ||.seq1 : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) .|| : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) complex-valued ext-real-valued real-valued ) Element of K32(K33(NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ,REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( V4() ) set ) ) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) & ||.seq1 : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) .|| : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) complex-valued ext-real-valued real-valued ) Element of K32(K33(NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ,REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( V4() ) set ) ) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) <= ||.seq2 : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) .|| : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) complex-valued ext-real-valued real-valued ) Element of K32(K33(NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ,REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( V4() ) set ) ) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) ) ) & seq2 : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) is norm_summable holds
( seq1 : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) is norm_summable & Sum ||.seq1 : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) .|| : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) complex-valued ext-real-valued real-valued ) Element of K32(K33(NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ,REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( V4() ) set ) ) : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) <= Sum ||.seq2 : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) .|| : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) complex-valued ext-real-valued real-valued ) Element of K32(K33(NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ,REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( V4() ) set ) ) : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) ) ;

theorem :: LOPBAN_3:29
for X being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace)
for seq being ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) st ( for n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) holds ||.seq : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) .|| : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) complex-valued ext-real-valued real-valued ) Element of K32(K33(NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ,REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( V4() ) set ) ) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) > 0 : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V24() real ext-real non positive non negative V45() V46() V52() V53() V54() V55() V56() V57() V58() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) ) & ex m being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) st
for n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) st n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) >= m : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) holds
(||.seq : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) .|| : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) complex-valued ext-real-valued real-valued ) Element of K32(K33(NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ,REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( V4() ) set ) ) . (n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) + 1 : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) ) : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) / (||.seq : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) .|| : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) complex-valued ext-real-valued real-valued ) Element of K32(K33(NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ,REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( V4() ) set ) ) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) ) : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) >= 1 : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) holds
not seq : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) is norm_summable ;

theorem :: LOPBAN_3:30
for X being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace)
for seq being ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) )
for rseq1 being ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) st ( for n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) holds rseq1 : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) = n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) -root (||.seq : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) .|| : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) complex-valued ext-real-valued real-valued ) Element of K32(K33(NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ,REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( V4() ) set ) ) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) ) : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) ) & rseq1 : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) is convergent & lim rseq1 : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) < 1 : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) holds
seq : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) is norm_summable ;

theorem :: LOPBAN_3:31
for X being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace)
for seq being ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) )
for rseq1 being ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) st ( for n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) holds rseq1 : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) = n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) -root (||.seq : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) .|| : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) complex-valued ext-real-valued real-valued ) Element of K32(K33(NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ,REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( V4() ) set ) ) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) ) : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) ) & ex m being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) st
for n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) st m : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) <= n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) holds
rseq1 : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) >= 1 : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) holds
not ||.seq : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) .|| : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) complex-valued ext-real-valued real-valued ) Element of K32(K33(NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ,REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( V4() ) set ) ) is summable ;

theorem :: LOPBAN_3:32
for X being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace)
for seq being ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) )
for rseq1 being ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) st ( for n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) holds rseq1 : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) = n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) -root (||.seq : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) .|| : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) complex-valued ext-real-valued real-valued ) Element of K32(K33(NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ,REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( V4() ) set ) ) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) ) : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) ) & rseq1 : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) is convergent & lim rseq1 : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) > 1 : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) holds
not seq : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) is norm_summable ;

theorem :: LOPBAN_3:33
for X being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace)
for seq being ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) )
for rseq1 being ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) st ||.seq : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) .|| : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) complex-valued ext-real-valued real-valued ) Element of K32(K33(NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ,REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( V4() ) set ) ) is non-increasing & ( for n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) holds rseq1 : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) = (2 : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) to_power n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) ) : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) * (||.seq : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) .|| : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) complex-valued ext-real-valued real-valued ) Element of K32(K33(NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ,REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( V4() ) set ) ) . (2 : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) to_power n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) ) : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) ) : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) ) holds
( seq : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) is norm_summable iff rseq1 : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) is summable ) ;

theorem :: LOPBAN_3:34
for X being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace)
for seq being ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) )
for p being ( ( ) ( V24() real ext-real ) Real) st p : ( ( ) ( V24() real ext-real ) Real) > 1 : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) & ( for n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) st n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) >= 1 : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) holds
||.seq : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) .|| : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) complex-valued ext-real-valued real-valued ) Element of K32(K33(NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ,REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( V4() ) set ) ) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) = 1 : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) / (n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) to_power p : ( ( ) ( V24() real ext-real ) Real) ) : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) ) holds
seq : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) is norm_summable ;

theorem :: LOPBAN_3:35
for X being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace)
for seq being ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) )
for p being ( ( ) ( V24() real ext-real ) Real) st p : ( ( ) ( V24() real ext-real ) Real) <= 1 : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) & ( for n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) st n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) >= 1 : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) holds
||.seq : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) .|| : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) complex-valued ext-real-valued real-valued ) Element of K32(K33(NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ,REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( V4() ) set ) ) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) = 1 : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) / (n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) to_power p : ( ( ) ( V24() real ext-real ) Real) ) : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) ) holds
not seq : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) is norm_summable ;

theorem :: LOPBAN_3:36
for X being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace)
for seq being ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) )
for rseq1 being ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) st ( for n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) holds
( seq : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) <> 0. X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V79(b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) ) right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) & rseq1 : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) = (||.seq : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) .|| : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) complex-valued ext-real-valued real-valued ) Element of K32(K33(NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ,REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( V4() ) set ) ) . (n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) + 1 : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) ) : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) / (||.seq : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) .|| : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) complex-valued ext-real-valued real-valued ) Element of K32(K33(NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ,REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( V4() ) set ) ) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) ) : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) ) ) & rseq1 : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) is convergent & lim rseq1 : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) complex-valued ext-real-valued real-valued ) Real_Sequence) : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) < 1 : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) holds
seq : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) is norm_summable ;

theorem :: LOPBAN_3:37
for X being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace)
for seq being ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) st ( for n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) holds seq : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) <> 0. X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V79(b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) ) right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) & ex m being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) st
for n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) st n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) >= m : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) holds
(||.seq : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) .|| : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) complex-valued ext-real-valued real-valued ) Element of K32(K33(NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ,REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( V4() ) set ) ) . (n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) + 1 : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) ) : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) / (||.seq : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) .|| : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) complex-valued ext-real-valued real-valued ) Element of K32(K33(NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ,REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( V4() ) set ) ) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) ) : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) >= 1 : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) holds
not seq : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) RealNormSpace) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) is norm_summable ;

registration
let X be ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like complete ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like complete ) RealBanachSpace) ;
cluster Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V4() ) set ) ) norm_summable -> Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V4() ) set ) ) summable for ( ( ) ( ) Element of K32(K33(NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V4() ) set ) ) : ( ( ) ( V4() ) set ) ) : ( ( ) ( V4() ) set ) ) ;
end;

begin

theorem :: LOPBAN_3:38
for X being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive right_unital Banach_Algebra-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive left-distributive right_unital left_unital complete Banach_Algebra-like_1 Banach_Algebra-like_2 Banach_Algebra-like_3 Banach_Algebra-like ) Banach_Algebra)
for x, y, z being ( ( ) ( right_complementable ) Element of ( ( ) ( V4() ) set ) )
for a, b being ( ( ) ( V24() real ext-real ) Real) holds
( x : ( ( ) ( right_complementable ) Element of ( ( ) ( V4() ) set ) ) + y : ( ( ) ( right_complementable ) Element of ( ( ) ( V4() ) set ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive right_unital Banach_Algebra-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive left-distributive right_unital left_unital complete Banach_Algebra-like_1 Banach_Algebra-like_2 Banach_Algebra-like_3 Banach_Algebra-like ) Banach_Algebra) : ( ( ) ( V4() ) set ) ) = y : ( ( ) ( right_complementable ) Element of ( ( ) ( V4() ) set ) ) + x : ( ( ) ( right_complementable ) Element of ( ( ) ( V4() ) set ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive right_unital Banach_Algebra-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive left-distributive right_unital left_unital complete Banach_Algebra-like_1 Banach_Algebra-like_2 Banach_Algebra-like_3 Banach_Algebra-like ) Banach_Algebra) : ( ( ) ( V4() ) set ) ) & (x : ( ( ) ( right_complementable ) Element of ( ( ) ( V4() ) set ) ) + y : ( ( ) ( right_complementable ) Element of ( ( ) ( V4() ) set ) ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive right_unital Banach_Algebra-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive left-distributive right_unital left_unital complete Banach_Algebra-like_1 Banach_Algebra-like_2 Banach_Algebra-like_3 Banach_Algebra-like ) Banach_Algebra) : ( ( ) ( V4() ) set ) ) + z : ( ( ) ( right_complementable ) Element of ( ( ) ( V4() ) set ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive right_unital Banach_Algebra-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive left-distributive right_unital left_unital complete Banach_Algebra-like_1 Banach_Algebra-like_2 Banach_Algebra-like_3 Banach_Algebra-like ) Banach_Algebra) : ( ( ) ( V4() ) set ) ) = x : ( ( ) ( right_complementable ) Element of ( ( ) ( V4() ) set ) ) + (y : ( ( ) ( right_complementable ) Element of ( ( ) ( V4() ) set ) ) + z : ( ( ) ( right_complementable ) Element of ( ( ) ( V4() ) set ) ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive right_unital Banach_Algebra-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive left-distributive right_unital left_unital complete Banach_Algebra-like_1 Banach_Algebra-like_2 Banach_Algebra-like_3 Banach_Algebra-like ) Banach_Algebra) : ( ( ) ( V4() ) set ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive right_unital Banach_Algebra-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive left-distributive right_unital left_unital complete Banach_Algebra-like_1 Banach_Algebra-like_2 Banach_Algebra-like_3 Banach_Algebra-like ) Banach_Algebra) : ( ( ) ( V4() ) set ) ) & x : ( ( ) ( right_complementable ) Element of ( ( ) ( V4() ) set ) ) + (0. X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive right_unital Banach_Algebra-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive left-distributive right_unital left_unital complete Banach_Algebra-like_1 Banach_Algebra-like_2 Banach_Algebra-like_3 Banach_Algebra-like ) Banach_Algebra) ) : ( ( ) ( V79(b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive right_unital Banach_Algebra-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive left-distributive right_unital left_unital complete Banach_Algebra-like_1 Banach_Algebra-like_2 Banach_Algebra-like_3 Banach_Algebra-like ) Banach_Algebra) ) right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive right_unital Banach_Algebra-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive left-distributive right_unital left_unital complete Banach_Algebra-like_1 Banach_Algebra-like_2 Banach_Algebra-like_3 Banach_Algebra-like ) Banach_Algebra) : ( ( ) ( V4() ) set ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive right_unital Banach_Algebra-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive left-distributive right_unital left_unital complete Banach_Algebra-like_1 Banach_Algebra-like_2 Banach_Algebra-like_3 Banach_Algebra-like ) Banach_Algebra) : ( ( ) ( V4() ) set ) ) = x : ( ( ) ( right_complementable ) Element of ( ( ) ( V4() ) set ) ) & ex t being ( ( ) ( right_complementable ) Element of ( ( ) ( V4() ) set ) ) st x : ( ( ) ( right_complementable ) Element of ( ( ) ( V4() ) set ) ) + t : ( ( ) ( right_complementable ) Element of ( ( ) ( V4() ) set ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive right_unital Banach_Algebra-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive left-distributive right_unital left_unital complete Banach_Algebra-like_1 Banach_Algebra-like_2 Banach_Algebra-like_3 Banach_Algebra-like ) Banach_Algebra) : ( ( ) ( V4() ) set ) ) = 0. X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive right_unital Banach_Algebra-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive left-distributive right_unital left_unital complete Banach_Algebra-like_1 Banach_Algebra-like_2 Banach_Algebra-like_3 Banach_Algebra-like ) Banach_Algebra) : ( ( ) ( V79(b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive right_unital Banach_Algebra-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive left-distributive right_unital left_unital complete Banach_Algebra-like_1 Banach_Algebra-like_2 Banach_Algebra-like_3 Banach_Algebra-like ) Banach_Algebra) ) right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive right_unital Banach_Algebra-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive left-distributive right_unital left_unital complete Banach_Algebra-like_1 Banach_Algebra-like_2 Banach_Algebra-like_3 Banach_Algebra-like ) Banach_Algebra) : ( ( ) ( V4() ) set ) ) & (x : ( ( ) ( right_complementable ) Element of ( ( ) ( V4() ) set ) ) * y : ( ( ) ( right_complementable ) Element of ( ( ) ( V4() ) set ) ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive right_unital Banach_Algebra-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive left-distributive right_unital left_unital complete Banach_Algebra-like_1 Banach_Algebra-like_2 Banach_Algebra-like_3 Banach_Algebra-like ) Banach_Algebra) : ( ( ) ( V4() ) set ) ) * z : ( ( ) ( right_complementable ) Element of ( ( ) ( V4() ) set ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive right_unital Banach_Algebra-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive left-distributive right_unital left_unital complete Banach_Algebra-like_1 Banach_Algebra-like_2 Banach_Algebra-like_3 Banach_Algebra-like ) Banach_Algebra) : ( ( ) ( V4() ) set ) ) = x : ( ( ) ( right_complementable ) Element of ( ( ) ( V4() ) set ) ) * (y : ( ( ) ( right_complementable ) Element of ( ( ) ( V4() ) set ) ) * z : ( ( ) ( right_complementable ) Element of ( ( ) ( V4() ) set ) ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive right_unital Banach_Algebra-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive left-distributive right_unital left_unital complete Banach_Algebra-like_1 Banach_Algebra-like_2 Banach_Algebra-like_3 Banach_Algebra-like ) Banach_Algebra) : ( ( ) ( V4() ) set ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive right_unital Banach_Algebra-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive left-distributive right_unital left_unital complete Banach_Algebra-like_1 Banach_Algebra-like_2 Banach_Algebra-like_3 Banach_Algebra-like ) Banach_Algebra) : ( ( ) ( V4() ) set ) ) & 1 : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) * x : ( ( ) ( right_complementable ) Element of ( ( ) ( V4() ) set ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive right_unital Banach_Algebra-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive left-distributive right_unital left_unital complete Banach_Algebra-like_1 Banach_Algebra-like_2 Banach_Algebra-like_3 Banach_Algebra-like ) Banach_Algebra) : ( ( ) ( V4() ) set ) ) = x : ( ( ) ( right_complementable ) Element of ( ( ) ( V4() ) set ) ) & 0 : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V24() real ext-real non positive non negative V45() V46() V52() V53() V54() V55() V56() V57() V58() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) * x : ( ( ) ( right_complementable ) Element of ( ( ) ( V4() ) set ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive right_unital Banach_Algebra-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive left-distributive right_unital left_unital complete Banach_Algebra-like_1 Banach_Algebra-like_2 Banach_Algebra-like_3 Banach_Algebra-like ) Banach_Algebra) : ( ( ) ( V4() ) set ) ) = 0. X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive right_unital Banach_Algebra-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive left-distributive right_unital left_unital complete Banach_Algebra-like_1 Banach_Algebra-like_2 Banach_Algebra-like_3 Banach_Algebra-like ) Banach_Algebra) : ( ( ) ( V79(b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive right_unital Banach_Algebra-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive left-distributive right_unital left_unital complete Banach_Algebra-like_1 Banach_Algebra-like_2 Banach_Algebra-like_3 Banach_Algebra-like ) Banach_Algebra) ) right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive right_unital Banach_Algebra-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive left-distributive right_unital left_unital complete Banach_Algebra-like_1 Banach_Algebra-like_2 Banach_Algebra-like_3 Banach_Algebra-like ) Banach_Algebra) : ( ( ) ( V4() ) set ) ) & a : ( ( ) ( V24() real ext-real ) Real) * (0. X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive right_unital Banach_Algebra-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive left-distributive right_unital left_unital complete Banach_Algebra-like_1 Banach_Algebra-like_2 Banach_Algebra-like_3 Banach_Algebra-like ) Banach_Algebra) ) : ( ( ) ( V79(b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive right_unital Banach_Algebra-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive left-distributive right_unital left_unital complete Banach_Algebra-like_1 Banach_Algebra-like_2 Banach_Algebra-like_3 Banach_Algebra-like ) Banach_Algebra) ) right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive right_unital Banach_Algebra-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive left-distributive right_unital left_unital complete Banach_Algebra-like_1 Banach_Algebra-like_2 Banach_Algebra-like_3 Banach_Algebra-like ) Banach_Algebra) : ( ( ) ( V4() ) set ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive right_unital Banach_Algebra-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive left-distributive right_unital left_unital complete Banach_Algebra-like_1 Banach_Algebra-like_2 Banach_Algebra-like_3 Banach_Algebra-like ) Banach_Algebra) : ( ( ) ( V4() ) set ) ) = 0. X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive right_unital Banach_Algebra-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive left-distributive right_unital left_unital complete Banach_Algebra-like_1 Banach_Algebra-like_2 Banach_Algebra-like_3 Banach_Algebra-like ) Banach_Algebra) : ( ( ) ( V79(b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive right_unital Banach_Algebra-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive left-distributive right_unital left_unital complete Banach_Algebra-like_1 Banach_Algebra-like_2 Banach_Algebra-like_3 Banach_Algebra-like ) Banach_Algebra) ) right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive right_unital Banach_Algebra-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive left-distributive right_unital left_unital complete Banach_Algebra-like_1 Banach_Algebra-like_2 Banach_Algebra-like_3 Banach_Algebra-like ) Banach_Algebra) : ( ( ) ( V4() ) set ) ) & (- 1 : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) ) : ( ( ) ( V24() real ext-real non positive ) Element of REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) * x : ( ( ) ( right_complementable ) Element of ( ( ) ( V4() ) set ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive right_unital Banach_Algebra-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive left-distributive right_unital left_unital complete Banach_Algebra-like_1 Banach_Algebra-like_2 Banach_Algebra-like_3 Banach_Algebra-like ) Banach_Algebra) : ( ( ) ( V4() ) set ) ) = - x : ( ( ) ( right_complementable ) Element of ( ( ) ( V4() ) set ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive right_unital Banach_Algebra-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive left-distributive right_unital left_unital complete Banach_Algebra-like_1 Banach_Algebra-like_2 Banach_Algebra-like_3 Banach_Algebra-like ) Banach_Algebra) : ( ( ) ( V4() ) set ) ) & x : ( ( ) ( right_complementable ) Element of ( ( ) ( V4() ) set ) ) * (1. X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive right_unital Banach_Algebra-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive left-distributive right_unital left_unital complete Banach_Algebra-like_1 Banach_Algebra-like_2 Banach_Algebra-like_3 Banach_Algebra-like ) Banach_Algebra) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive right_unital Banach_Algebra-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive left-distributive right_unital left_unital complete Banach_Algebra-like_1 Banach_Algebra-like_2 Banach_Algebra-like_3 Banach_Algebra-like ) Banach_Algebra) : ( ( ) ( V4() ) set ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive right_unital Banach_Algebra-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive left-distributive right_unital left_unital complete Banach_Algebra-like_1 Banach_Algebra-like_2 Banach_Algebra-like_3 Banach_Algebra-like ) Banach_Algebra) : ( ( ) ( V4() ) set ) ) = x : ( ( ) ( right_complementable ) Element of ( ( ) ( V4() ) set ) ) & (1. X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive right_unital Banach_Algebra-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive left-distributive right_unital left_unital complete Banach_Algebra-like_1 Banach_Algebra-like_2 Banach_Algebra-like_3 Banach_Algebra-like ) Banach_Algebra) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive right_unital Banach_Algebra-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive left-distributive right_unital left_unital complete Banach_Algebra-like_1 Banach_Algebra-like_2 Banach_Algebra-like_3 Banach_Algebra-like ) Banach_Algebra) : ( ( ) ( V4() ) set ) ) * x : ( ( ) ( right_complementable ) Element of ( ( ) ( V4() ) set ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive right_unital Banach_Algebra-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive left-distributive right_unital left_unital complete Banach_Algebra-like_1 Banach_Algebra-like_2 Banach_Algebra-like_3 Banach_Algebra-like ) Banach_Algebra) : ( ( ) ( V4() ) set ) ) = x : ( ( ) ( right_complementable ) Element of ( ( ) ( V4() ) set ) ) & x : ( ( ) ( right_complementable ) Element of ( ( ) ( V4() ) set ) ) * (y : ( ( ) ( right_complementable ) Element of ( ( ) ( V4() ) set ) ) + z : ( ( ) ( right_complementable ) Element of ( ( ) ( V4() ) set ) ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive right_unital Banach_Algebra-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive left-distributive right_unital left_unital complete Banach_Algebra-like_1 Banach_Algebra-like_2 Banach_Algebra-like_3 Banach_Algebra-like ) Banach_Algebra) : ( ( ) ( V4() ) set ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive right_unital Banach_Algebra-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive left-distributive right_unital left_unital complete Banach_Algebra-like_1 Banach_Algebra-like_2 Banach_Algebra-like_3 Banach_Algebra-like ) Banach_Algebra) : ( ( ) ( V4() ) set ) ) = (x : ( ( ) ( right_complementable ) Element of ( ( ) ( V4() ) set ) ) * y : ( ( ) ( right_complementable ) Element of ( ( ) ( V4() ) set ) ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive right_unital Banach_Algebra-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive left-distributive right_unital left_unital complete Banach_Algebra-like_1 Banach_Algebra-like_2 Banach_Algebra-like_3 Banach_Algebra-like ) Banach_Algebra) : ( ( ) ( V4() ) set ) ) + (x : ( ( ) ( right_complementable ) Element of ( ( ) ( V4() ) set ) ) * z : ( ( ) ( right_complementable ) Element of ( ( ) ( V4() ) set ) ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive right_unital Banach_Algebra-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive left-distributive right_unital left_unital complete Banach_Algebra-like_1 Banach_Algebra-like_2 Banach_Algebra-like_3 Banach_Algebra-like ) Banach_Algebra) : ( ( ) ( V4() ) set ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive right_unital Banach_Algebra-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive left-distributive right_unital left_unital complete Banach_Algebra-like_1 Banach_Algebra-like_2 Banach_Algebra-like_3 Banach_Algebra-like ) Banach_Algebra) : ( ( ) ( V4() ) set ) ) & (y : ( ( ) ( right_complementable ) Element of ( ( ) ( V4() ) set ) ) + z : ( ( ) ( right_complementable ) Element of ( ( ) ( V4() ) set ) ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive right_unital Banach_Algebra-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive left-distributive right_unital left_unital complete Banach_Algebra-like_1 Banach_Algebra-like_2 Banach_Algebra-like_3 Banach_Algebra-like ) Banach_Algebra) : ( ( ) ( V4() ) set ) ) * x : ( ( ) ( right_complementable ) Element of ( ( ) ( V4() ) set ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive right_unital Banach_Algebra-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive left-distributive right_unital left_unital complete Banach_Algebra-like_1 Banach_Algebra-like_2 Banach_Algebra-like_3 Banach_Algebra-like ) Banach_Algebra) : ( ( ) ( V4() ) set ) ) = (y : ( ( ) ( right_complementable ) Element of ( ( ) ( V4() ) set ) ) * x : ( ( ) ( right_complementable ) Element of ( ( ) ( V4() ) set ) ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive right_unital Banach_Algebra-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive left-distributive right_unital left_unital complete Banach_Algebra-like_1 Banach_Algebra-like_2 Banach_Algebra-like_3 Banach_Algebra-like ) Banach_Algebra) : ( ( ) ( V4() ) set ) ) + (z : ( ( ) ( right_complementable ) Element of ( ( ) ( V4() ) set ) ) * x : ( ( ) ( right_complementable ) Element of ( ( ) ( V4() ) set ) ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive right_unital Banach_Algebra-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive left-distributive right_unital left_unital complete Banach_Algebra-like_1 Banach_Algebra-like_2 Banach_Algebra-like_3 Banach_Algebra-like ) Banach_Algebra) : ( ( ) ( V4() ) set ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive right_unital Banach_Algebra-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive left-distributive right_unital left_unital complete Banach_Algebra-like_1 Banach_Algebra-like_2 Banach_Algebra-like_3 Banach_Algebra-like ) Banach_Algebra) : ( ( ) ( V4() ) set ) ) & a : ( ( ) ( V24() real ext-real ) Real) * (x : ( ( ) ( right_complementable ) Element of ( ( ) ( V4() ) set ) ) * y : ( ( ) ( right_complementable ) Element of ( ( ) ( V4() ) set ) ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive right_unital Banach_Algebra-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive left-distributive right_unital left_unital complete Banach_Algebra-like_1 Banach_Algebra-like_2 Banach_Algebra-like_3 Banach_Algebra-like ) Banach_Algebra) : ( ( ) ( V4() ) set ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive right_unital Banach_Algebra-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive left-distributive right_unital left_unital complete Banach_Algebra-like_1 Banach_Algebra-like_2 Banach_Algebra-like_3 Banach_Algebra-like ) Banach_Algebra) : ( ( ) ( V4() ) set ) ) = (a : ( ( ) ( V24() real ext-real ) Real) * x : ( ( ) ( right_complementable ) Element of ( ( ) ( V4() ) set ) ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive right_unital Banach_Algebra-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive left-distributive right_unital left_unital complete Banach_Algebra-like_1 Banach_Algebra-like_2 Banach_Algebra-like_3 Banach_Algebra-like ) Banach_Algebra) : ( ( ) ( V4() ) set ) ) * y : ( ( ) ( right_complementable ) Element of ( ( ) ( V4() ) set ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive right_unital Banach_Algebra-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive left-distributive right_unital left_unital complete Banach_Algebra-like_1 Banach_Algebra-like_2 Banach_Algebra-like_3 Banach_Algebra-like ) Banach_Algebra) : ( ( ) ( V4() ) set ) ) & a : ( ( ) ( V24() real ext-real ) Real) * (x : ( ( ) ( right_complementable ) Element of ( ( ) ( V4() ) set ) ) + y : ( ( ) ( right_complementable ) Element of ( ( ) ( V4() ) set ) ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive right_unital Banach_Algebra-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive left-distributive right_unital left_unital complete Banach_Algebra-like_1 Banach_Algebra-like_2 Banach_Algebra-like_3 Banach_Algebra-like ) Banach_Algebra) : ( ( ) ( V4() ) set ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive right_unital Banach_Algebra-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive left-distributive right_unital left_unital complete Banach_Algebra-like_1 Banach_Algebra-like_2 Banach_Algebra-like_3 Banach_Algebra-like ) Banach_Algebra) : ( ( ) ( V4() ) set ) ) = (a : ( ( ) ( V24() real ext-real ) Real) * x : ( ( ) ( right_complementable ) Element of ( ( ) ( V4() ) set ) ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive right_unital Banach_Algebra-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive left-distributive right_unital left_unital complete Banach_Algebra-like_1 Banach_Algebra-like_2 Banach_Algebra-like_3 Banach_Algebra-like ) Banach_Algebra) : ( ( ) ( V4() ) set ) ) + (a : ( ( ) ( V24() real ext-real ) Real) * y : ( ( ) ( right_complementable ) Element of ( ( ) ( V4() ) set ) ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive right_unital Banach_Algebra-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive left-distributive right_unital left_unital complete Banach_Algebra-like_1 Banach_Algebra-like_2 Banach_Algebra-like_3 Banach_Algebra-like ) Banach_Algebra) : ( ( ) ( V4() ) set ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive right_unital Banach_Algebra-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive left-distributive right_unital left_unital complete Banach_Algebra-like_1 Banach_Algebra-like_2 Banach_Algebra-like_3 Banach_Algebra-like ) Banach_Algebra) : ( ( ) ( V4() ) set ) ) & (a : ( ( ) ( V24() real ext-real ) Real) + b : ( ( ) ( V24() real ext-real ) Real) ) : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) * x : ( ( ) ( right_complementable ) Element of ( ( ) ( V4() ) set ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive right_unital Banach_Algebra-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive left-distributive right_unital left_unital complete Banach_Algebra-like_1 Banach_Algebra-like_2 Banach_Algebra-like_3 Banach_Algebra-like ) Banach_Algebra) : ( ( ) ( V4() ) set ) ) = (a : ( ( ) ( V24() real ext-real ) Real) * x : ( ( ) ( right_complementable ) Element of ( ( ) ( V4() ) set ) ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive right_unital Banach_Algebra-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive left-distributive right_unital left_unital complete Banach_Algebra-like_1 Banach_Algebra-like_2 Banach_Algebra-like_3 Banach_Algebra-like ) Banach_Algebra) : ( ( ) ( V4() ) set ) ) + (b : ( ( ) ( V24() real ext-real ) Real) * x : ( ( ) ( right_complementable ) Element of ( ( ) ( V4() ) set ) ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive right_unital Banach_Algebra-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive left-distributive right_unital left_unital complete Banach_Algebra-like_1 Banach_Algebra-like_2 Banach_Algebra-like_3 Banach_Algebra-like ) Banach_Algebra) : ( ( ) ( V4() ) set ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive right_unital Banach_Algebra-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive left-distributive right_unital left_unital complete Banach_Algebra-like_1 Banach_Algebra-like_2 Banach_Algebra-like_3 Banach_Algebra-like ) Banach_Algebra) : ( ( ) ( V4() ) set ) ) & (a : ( ( ) ( V24() real ext-real ) Real) * b : ( ( ) ( V24() real ext-real ) Real) ) : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) * x : ( ( ) ( right_complementable ) Element of ( ( ) ( V4() ) set ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive right_unital Banach_Algebra-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive left-distributive right_unital left_unital complete Banach_Algebra-like_1 Banach_Algebra-like_2 Banach_Algebra-like_3 Banach_Algebra-like ) Banach_Algebra) : ( ( ) ( V4() ) set ) ) = a : ( ( ) ( V24() real ext-real ) Real) * (b : ( ( ) ( V24() real ext-real ) Real) * x : ( ( ) ( right_complementable ) Element of ( ( ) ( V4() ) set ) ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive right_unital Banach_Algebra-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive left-distributive right_unital left_unital complete Banach_Algebra-like_1 Banach_Algebra-like_2 Banach_Algebra-like_3 Banach_Algebra-like ) Banach_Algebra) : ( ( ) ( V4() ) set ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive right_unital Banach_Algebra-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive left-distributive right_unital left_unital complete Banach_Algebra-like_1 Banach_Algebra-like_2 Banach_Algebra-like_3 Banach_Algebra-like ) Banach_Algebra) : ( ( ) ( V4() ) set ) ) & (a : ( ( ) ( V24() real ext-real ) Real) * b : ( ( ) ( V24() real ext-real ) Real) ) : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) * (x : ( ( ) ( right_complementable ) Element of ( ( ) ( V4() ) set ) ) * y : ( ( ) ( right_complementable ) Element of ( ( ) ( V4() ) set ) ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive right_unital Banach_Algebra-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive left-distributive right_unital left_unital complete Banach_Algebra-like_1 Banach_Algebra-like_2 Banach_Algebra-like_3 Banach_Algebra-like ) Banach_Algebra) : ( ( ) ( V4() ) set ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive right_unital Banach_Algebra-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive left-distributive right_unital left_unital complete Banach_Algebra-like_1 Banach_Algebra-like_2 Banach_Algebra-like_3 Banach_Algebra-like ) Banach_Algebra) : ( ( ) ( V4() ) set ) ) = (a : ( ( ) ( V24() real ext-real ) Real) * x : ( ( ) ( right_complementable ) Element of ( ( ) ( V4() ) set ) ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive right_unital Banach_Algebra-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive left-distributive right_unital left_unital complete Banach_Algebra-like_1 Banach_Algebra-like_2 Banach_Algebra-like_3 Banach_Algebra-like ) Banach_Algebra) : ( ( ) ( V4() ) set ) ) * (b : ( ( ) ( V24() real ext-real ) Real) * y : ( ( ) ( right_complementable ) Element of ( ( ) ( V4() ) set ) ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive right_unital Banach_Algebra-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive left-distributive right_unital left_unital complete Banach_Algebra-like_1 Banach_Algebra-like_2 Banach_Algebra-like_3 Banach_Algebra-like ) Banach_Algebra) : ( ( ) ( V4() ) set ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive right_unital Banach_Algebra-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive left-distributive right_unital left_unital complete Banach_Algebra-like_1 Banach_Algebra-like_2 Banach_Algebra-like_3 Banach_Algebra-like ) Banach_Algebra) : ( ( ) ( V4() ) set ) ) & a : ( ( ) ( V24() real ext-real ) Real) * (x : ( ( ) ( right_complementable ) Element of ( ( ) ( V4() ) set ) ) * y : ( ( ) ( right_complementable ) Element of ( ( ) ( V4() ) set ) ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive right_unital Banach_Algebra-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive left-distributive right_unital left_unital complete Banach_Algebra-like_1 Banach_Algebra-like_2 Banach_Algebra-like_3 Banach_Algebra-like ) Banach_Algebra) : ( ( ) ( V4() ) set ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive right_unital Banach_Algebra-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive left-distributive right_unital left_unital complete Banach_Algebra-like_1 Banach_Algebra-like_2 Banach_Algebra-like_3 Banach_Algebra-like ) Banach_Algebra) : ( ( ) ( V4() ) set ) ) = x : ( ( ) ( right_complementable ) Element of ( ( ) ( V4() ) set ) ) * (a : ( ( ) ( V24() real ext-real ) Real) * y : ( ( ) ( right_complementable ) Element of ( ( ) ( V4() ) set ) ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive right_unital Banach_Algebra-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive left-distributive right_unital left_unital complete Banach_Algebra-like_1 Banach_Algebra-like_2 Banach_Algebra-like_3 Banach_Algebra-like ) Banach_Algebra) : ( ( ) ( V4() ) set ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive right_unital Banach_Algebra-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive left-distributive right_unital left_unital complete Banach_Algebra-like_1 Banach_Algebra-like_2 Banach_Algebra-like_3 Banach_Algebra-like ) Banach_Algebra) : ( ( ) ( V4() ) set ) ) & (0. X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive right_unital Banach_Algebra-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive left-distributive right_unital left_unital complete Banach_Algebra-like_1 Banach_Algebra-like_2 Banach_Algebra-like_3 Banach_Algebra-like ) Banach_Algebra) ) : ( ( ) ( V79(b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive right_unital Banach_Algebra-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive left-distributive right_unital left_unital complete Banach_Algebra-like_1 Banach_Algebra-like_2 Banach_Algebra-like_3 Banach_Algebra-like ) Banach_Algebra) ) right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive right_unital Banach_Algebra-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive left-distributive right_unital left_unital complete Banach_Algebra-like_1 Banach_Algebra-like_2 Banach_Algebra-like_3 Banach_Algebra-like ) Banach_Algebra) : ( ( ) ( V4() ) set ) ) * x : ( ( ) ( right_complementable ) Element of ( ( ) ( V4() ) set ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive right_unital Banach_Algebra-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive left-distributive right_unital left_unital complete Banach_Algebra-like_1 Banach_Algebra-like_2 Banach_Algebra-like_3 Banach_Algebra-like ) Banach_Algebra) : ( ( ) ( V4() ) set ) ) = 0. X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive right_unital Banach_Algebra-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive left-distributive right_unital left_unital complete Banach_Algebra-like_1 Banach_Algebra-like_2 Banach_Algebra-like_3 Banach_Algebra-like ) Banach_Algebra) : ( ( ) ( V79(b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive right_unital Banach_Algebra-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive left-distributive right_unital left_unital complete Banach_Algebra-like_1 Banach_Algebra-like_2 Banach_Algebra-like_3 Banach_Algebra-like ) Banach_Algebra) ) right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive right_unital Banach_Algebra-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive left-distributive right_unital left_unital complete Banach_Algebra-like_1 Banach_Algebra-like_2 Banach_Algebra-like_3 Banach_Algebra-like ) Banach_Algebra) : ( ( ) ( V4() ) set ) ) & x : ( ( ) ( right_complementable ) Element of ( ( ) ( V4() ) set ) ) * (0. X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive right_unital Banach_Algebra-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive left-distributive right_unital left_unital complete Banach_Algebra-like_1 Banach_Algebra-like_2 Banach_Algebra-like_3 Banach_Algebra-like ) Banach_Algebra) ) : ( ( ) ( V79(b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive right_unital Banach_Algebra-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive left-distributive right_unital left_unital complete Banach_Algebra-like_1 Banach_Algebra-like_2 Banach_Algebra-like_3 Banach_Algebra-like ) Banach_Algebra) ) right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive right_unital Banach_Algebra-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive left-distributive right_unital left_unital complete Banach_Algebra-like_1 Banach_Algebra-like_2 Banach_Algebra-like_3 Banach_Algebra-like ) Banach_Algebra) : ( ( ) ( V4() ) set ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive right_unital Banach_Algebra-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive left-distributive right_unital left_unital complete Banach_Algebra-like_1 Banach_Algebra-like_2 Banach_Algebra-like_3 Banach_Algebra-like ) Banach_Algebra) : ( ( ) ( V4() ) set ) ) = 0. X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive right_unital Banach_Algebra-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive left-distributive right_unital left_unital complete Banach_Algebra-like_1 Banach_Algebra-like_2 Banach_Algebra-like_3 Banach_Algebra-like ) Banach_Algebra) : ( ( ) ( V79(b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive right_unital Banach_Algebra-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive left-distributive right_unital left_unital complete Banach_Algebra-like_1 Banach_Algebra-like_2 Banach_Algebra-like_3 Banach_Algebra-like ) Banach_Algebra) ) right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive right_unital Banach_Algebra-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive left-distributive right_unital left_unital complete Banach_Algebra-like_1 Banach_Algebra-like_2 Banach_Algebra-like_3 Banach_Algebra-like ) Banach_Algebra) : ( ( ) ( V4() ) set ) ) & x : ( ( ) ( right_complementable ) Element of ( ( ) ( V4() ) set ) ) * (y : ( ( ) ( right_complementable ) Element of ( ( ) ( V4() ) set ) ) - z : ( ( ) ( right_complementable ) Element of ( ( ) ( V4() ) set ) ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive right_unital Banach_Algebra-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive left-distributive right_unital left_unital complete Banach_Algebra-like_1 Banach_Algebra-like_2 Banach_Algebra-like_3 Banach_Algebra-like ) Banach_Algebra) : ( ( ) ( V4() ) set ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive right_unital Banach_Algebra-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive left-distributive right_unital left_unital complete Banach_Algebra-like_1 Banach_Algebra-like_2 Banach_Algebra-like_3 Banach_Algebra-like ) Banach_Algebra) : ( ( ) ( V4() ) set ) ) = (x : ( ( ) ( right_complementable ) Element of ( ( ) ( V4() ) set ) ) * y : ( ( ) ( right_complementable ) Element of ( ( ) ( V4() ) set ) ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive right_unital Banach_Algebra-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive left-distributive right_unital left_unital complete Banach_Algebra-like_1 Banach_Algebra-like_2 Banach_Algebra-like_3 Banach_Algebra-like ) Banach_Algebra) : ( ( ) ( V4() ) set ) ) - (x : ( ( ) ( right_complementable ) Element of ( ( ) ( V4() ) set ) ) * z : ( ( ) ( right_complementable ) Element of ( ( ) ( V4() ) set ) ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive right_unital Banach_Algebra-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive left-distributive right_unital left_unital complete Banach_Algebra-like_1 Banach_Algebra-like_2 Banach_Algebra-like_3 Banach_Algebra-like ) Banach_Algebra) : ( ( ) ( V4() ) set ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive right_unital Banach_Algebra-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive left-distributive right_unital left_unital complete Banach_Algebra-like_1 Banach_Algebra-like_2 Banach_Algebra-like_3 Banach_Algebra-like ) Banach_Algebra) : ( ( ) ( V4() ) set ) ) & (y : ( ( ) ( right_complementable ) Element of ( ( ) ( V4() ) set ) ) - z : ( ( ) ( right_complementable ) Element of ( ( ) ( V4() ) set ) ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive right_unital Banach_Algebra-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive left-distributive right_unital left_unital complete Banach_Algebra-like_1 Banach_Algebra-like_2 Banach_Algebra-like_3 Banach_Algebra-like ) Banach_Algebra) : ( ( ) ( V4() ) set ) ) * x : ( ( ) ( right_complementable ) Element of ( ( ) ( V4() ) set ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive right_unital Banach_Algebra-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive left-distributive right_unital left_unital complete Banach_Algebra-like_1 Banach_Algebra-like_2 Banach_Algebra-like_3 Banach_Algebra-like ) Banach_Algebra) : ( ( ) ( V4() ) set ) ) = (y : ( ( ) ( right_complementable ) Element of ( ( ) ( V4() ) set ) ) * x : ( ( ) ( right_complementable ) Element of ( ( ) ( V4() ) set ) ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive right_unital Banach_Algebra-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive left-distributive right_unital left_unital complete Banach_Algebra-like_1 Banach_Algebra-like_2 Banach_Algebra-like_3 Banach_Algebra-like ) Banach_Algebra) : ( ( ) ( V4() ) set ) ) - (z : ( ( ) ( right_complementable ) Element of ( ( ) ( V4() ) set ) ) * x : ( ( ) ( right_complementable ) Element of ( ( ) ( V4() ) set ) ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive right_unital Banach_Algebra-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive left-distributive right_unital left_unital complete Banach_Algebra-like_1 Banach_Algebra-like_2 Banach_Algebra-like_3 Banach_Algebra-like ) Banach_Algebra) : ( ( ) ( V4() ) set ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive right_unital Banach_Algebra-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive left-distributive right_unital left_unital complete Banach_Algebra-like_1 Banach_Algebra-like_2 Banach_Algebra-like_3 Banach_Algebra-like ) Banach_Algebra) : ( ( ) ( V4() ) set ) ) & (x : ( ( ) ( right_complementable ) Element of ( ( ) ( V4() ) set ) ) + y : ( ( ) ( right_complementable ) Element of ( ( ) ( V4() ) set ) ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive right_unital Banach_Algebra-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive left-distributive right_unital left_unital complete Banach_Algebra-like_1 Banach_Algebra-like_2 Banach_Algebra-like_3 Banach_Algebra-like ) Banach_Algebra) : ( ( ) ( V4() ) set ) ) - z : ( ( ) ( right_complementable ) Element of ( ( ) ( V4() ) set ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive right_unital Banach_Algebra-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive left-distributive right_unital left_unital complete Banach_Algebra-like_1 Banach_Algebra-like_2 Banach_Algebra-like_3 Banach_Algebra-like ) Banach_Algebra) : ( ( ) ( V4() ) set ) ) = x : ( ( ) ( right_complementable ) Element of ( ( ) ( V4() ) set ) ) + (y : ( ( ) ( right_complementable ) Element of ( ( ) ( V4() ) set ) ) - z : ( ( ) ( right_complementable ) Element of ( ( ) ( V4() ) set ) ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive right_unital Banach_Algebra-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive left-distributive right_unital left_unital complete Banach_Algebra-like_1 Banach_Algebra-like_2 Banach_Algebra-like_3 Banach_Algebra-like ) Banach_Algebra) : ( ( ) ( V4() ) set ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive right_unital Banach_Algebra-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive left-distributive right_unital left_unital complete Banach_Algebra-like_1 Banach_Algebra-like_2 Banach_Algebra-like_3 Banach_Algebra-like ) Banach_Algebra) : ( ( ) ( V4() ) set ) ) & (x : ( ( ) ( right_complementable ) Element of ( ( ) ( V4() ) set ) ) - y : ( ( ) ( right_complementable ) Element of ( ( ) ( V4() ) set ) ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive right_unital Banach_Algebra-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive left-distributive right_unital left_unital complete Banach_Algebra-like_1 Banach_Algebra-like_2 Banach_Algebra-like_3 Banach_Algebra-like ) Banach_Algebra) : ( ( ) ( V4() ) set ) ) + z : ( ( ) ( right_complementable ) Element of ( ( ) ( V4() ) set ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive right_unital Banach_Algebra-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive left-distributive right_unital left_unital complete Banach_Algebra-like_1 Banach_Algebra-like_2 Banach_Algebra-like_3 Banach_Algebra-like ) Banach_Algebra) : ( ( ) ( V4() ) set ) ) = x : ( ( ) ( right_complementable ) Element of ( ( ) ( V4() ) set ) ) - (y : ( ( ) ( right_complementable ) Element of ( ( ) ( V4() ) set ) ) - z : ( ( ) ( right_complementable ) Element of ( ( ) ( V4() ) set ) ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive right_unital Banach_Algebra-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive left-distributive right_unital left_unital complete Banach_Algebra-like_1 Banach_Algebra-like_2 Banach_Algebra-like_3 Banach_Algebra-like ) Banach_Algebra) : ( ( ) ( V4() ) set ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive right_unital Banach_Algebra-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive left-distributive right_unital left_unital complete Banach_Algebra-like_1 Banach_Algebra-like_2 Banach_Algebra-like_3 Banach_Algebra-like ) Banach_Algebra) : ( ( ) ( V4() ) set ) ) & (x : ( ( ) ( right_complementable ) Element of ( ( ) ( V4() ) set ) ) - y : ( ( ) ( right_complementable ) Element of ( ( ) ( V4() ) set ) ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive right_unital Banach_Algebra-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive left-distributive right_unital left_unital complete Banach_Algebra-like_1 Banach_Algebra-like_2 Banach_Algebra-like_3 Banach_Algebra-like ) Banach_Algebra) : ( ( ) ( V4() ) set ) ) - z : ( ( ) ( right_complementable ) Element of ( ( ) ( V4() ) set ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive right_unital Banach_Algebra-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive left-distributive right_unital left_unital complete Banach_Algebra-like_1 Banach_Algebra-like_2 Banach_Algebra-like_3 Banach_Algebra-like ) Banach_Algebra) : ( ( ) ( V4() ) set ) ) = x : ( ( ) ( right_complementable ) Element of ( ( ) ( V4() ) set ) ) - (y : ( ( ) ( right_complementable ) Element of ( ( ) ( V4() ) set ) ) + z : ( ( ) ( right_complementable ) Element of ( ( ) ( V4() ) set ) ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive right_unital Banach_Algebra-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive left-distributive right_unital left_unital complete Banach_Algebra-like_1 Banach_Algebra-like_2 Banach_Algebra-like_3 Banach_Algebra-like ) Banach_Algebra) : ( ( ) ( V4() ) set ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive right_unital Banach_Algebra-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive left-distributive right_unital left_unital complete Banach_Algebra-like_1 Banach_Algebra-like_2 Banach_Algebra-like_3 Banach_Algebra-like ) Banach_Algebra) : ( ( ) ( V4() ) set ) ) & x : ( ( ) ( right_complementable ) Element of ( ( ) ( V4() ) set ) ) + y : ( ( ) ( right_complementable ) Element of ( ( ) ( V4() ) set ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive right_unital Banach_Algebra-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive left-distributive right_unital left_unital complete Banach_Algebra-like_1 Banach_Algebra-like_2 Banach_Algebra-like_3 Banach_Algebra-like ) Banach_Algebra) : ( ( ) ( V4() ) set ) ) = (x : ( ( ) ( right_complementable ) Element of ( ( ) ( V4() ) set ) ) - z : ( ( ) ( right_complementable ) Element of ( ( ) ( V4() ) set ) ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive right_unital Banach_Algebra-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive left-distributive right_unital left_unital complete Banach_Algebra-like_1 Banach_Algebra-like_2 Banach_Algebra-like_3 Banach_Algebra-like ) Banach_Algebra) : ( ( ) ( V4() ) set ) ) + (z : ( ( ) ( right_complementable ) Element of ( ( ) ( V4() ) set ) ) + y : ( ( ) ( right_complementable ) Element of ( ( ) ( V4() ) set ) ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive right_unital Banach_Algebra-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive left-distributive right_unital left_unital complete Banach_Algebra-like_1 Banach_Algebra-like_2 Banach_Algebra-like_3 Banach_Algebra-like ) Banach_Algebra) : ( ( ) ( V4() ) set ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive right_unital Banach_Algebra-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive left-distributive right_unital left_unital complete Banach_Algebra-like_1 Banach_Algebra-like_2 Banach_Algebra-like_3 Banach_Algebra-like ) Banach_Algebra) : ( ( ) ( V4() ) set ) ) & x : ( ( ) ( right_complementable ) Element of ( ( ) ( V4() ) set ) ) - y : ( ( ) ( right_complementable ) Element of ( ( ) ( V4() ) set ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive right_unital Banach_Algebra-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive left-distributive right_unital left_unital complete Banach_Algebra-like_1 Banach_Algebra-like_2 Banach_Algebra-like_3 Banach_Algebra-like ) Banach_Algebra) : ( ( ) ( V4() ) set ) ) = (x : ( ( ) ( right_complementable ) Element of ( ( ) ( V4() ) set ) ) - z : ( ( ) ( right_complementable ) Element of ( ( ) ( V4() ) set ) ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive right_unital Banach_Algebra-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive left-distributive right_unital left_unital complete Banach_Algebra-like_1 Banach_Algebra-like_2 Banach_Algebra-like_3 Banach_Algebra-like ) Banach_Algebra) : ( ( ) ( V4() ) set ) ) + (z : ( ( ) ( right_complementable ) Element of ( ( ) ( V4() ) set ) ) - y : ( ( ) ( right_complementable ) Element of ( ( ) ( V4() ) set ) ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive right_unital Banach_Algebra-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive left-distributive right_unital left_unital complete Banach_Algebra-like_1 Banach_Algebra-like_2 Banach_Algebra-like_3 Banach_Algebra-like ) Banach_Algebra) : ( ( ) ( V4() ) set ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive right_unital Banach_Algebra-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive left-distributive right_unital left_unital complete Banach_Algebra-like_1 Banach_Algebra-like_2 Banach_Algebra-like_3 Banach_Algebra-like ) Banach_Algebra) : ( ( ) ( V4() ) set ) ) & x : ( ( ) ( right_complementable ) Element of ( ( ) ( V4() ) set ) ) = (x : ( ( ) ( right_complementable ) Element of ( ( ) ( V4() ) set ) ) - y : ( ( ) ( right_complementable ) Element of ( ( ) ( V4() ) set ) ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive right_unital Banach_Algebra-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive left-distributive right_unital left_unital complete Banach_Algebra-like_1 Banach_Algebra-like_2 Banach_Algebra-like_3 Banach_Algebra-like ) Banach_Algebra) : ( ( ) ( V4() ) set ) ) + y : ( ( ) ( right_complementable ) Element of ( ( ) ( V4() ) set ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive right_unital Banach_Algebra-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive left-distributive right_unital left_unital complete Banach_Algebra-like_1 Banach_Algebra-like_2 Banach_Algebra-like_3 Banach_Algebra-like ) Banach_Algebra) : ( ( ) ( V4() ) set ) ) & x : ( ( ) ( right_complementable ) Element of ( ( ) ( V4() ) set ) ) = y : ( ( ) ( right_complementable ) Element of ( ( ) ( V4() ) set ) ) - (y : ( ( ) ( right_complementable ) Element of ( ( ) ( V4() ) set ) ) - x : ( ( ) ( right_complementable ) Element of ( ( ) ( V4() ) set ) ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive right_unital Banach_Algebra-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive left-distributive right_unital left_unital complete Banach_Algebra-like_1 Banach_Algebra-like_2 Banach_Algebra-like_3 Banach_Algebra-like ) Banach_Algebra) : ( ( ) ( V4() ) set ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive right_unital Banach_Algebra-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive left-distributive right_unital left_unital complete Banach_Algebra-like_1 Banach_Algebra-like_2 Banach_Algebra-like_3 Banach_Algebra-like ) Banach_Algebra) : ( ( ) ( V4() ) set ) ) & ( ||.x : ( ( ) ( right_complementable ) Element of ( ( ) ( V4() ) set ) ) .|| : ( ( ) ( V24() real ext-real non negative ) Element of REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) = 0 : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V24() real ext-real non positive non negative V45() V46() V52() V53() V54() V55() V56() V57() V58() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) implies x : ( ( ) ( right_complementable ) Element of ( ( ) ( V4() ) set ) ) = 0. X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive right_unital Banach_Algebra-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive left-distributive right_unital left_unital complete Banach_Algebra-like_1 Banach_Algebra-like_2 Banach_Algebra-like_3 Banach_Algebra-like ) Banach_Algebra) : ( ( ) ( V79(b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive right_unital Banach_Algebra-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive left-distributive right_unital left_unital complete Banach_Algebra-like_1 Banach_Algebra-like_2 Banach_Algebra-like_3 Banach_Algebra-like ) Banach_Algebra) ) right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive right_unital Banach_Algebra-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive left-distributive right_unital left_unital complete Banach_Algebra-like_1 Banach_Algebra-like_2 Banach_Algebra-like_3 Banach_Algebra-like ) Banach_Algebra) : ( ( ) ( V4() ) set ) ) ) & ( x : ( ( ) ( right_complementable ) Element of ( ( ) ( V4() ) set ) ) = 0. X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive right_unital Banach_Algebra-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive left-distributive right_unital left_unital complete Banach_Algebra-like_1 Banach_Algebra-like_2 Banach_Algebra-like_3 Banach_Algebra-like ) Banach_Algebra) : ( ( ) ( V79(b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive right_unital Banach_Algebra-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive left-distributive right_unital left_unital complete Banach_Algebra-like_1 Banach_Algebra-like_2 Banach_Algebra-like_3 Banach_Algebra-like ) Banach_Algebra) ) right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive right_unital Banach_Algebra-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive left-distributive right_unital left_unital complete Banach_Algebra-like_1 Banach_Algebra-like_2 Banach_Algebra-like_3 Banach_Algebra-like ) Banach_Algebra) : ( ( ) ( V4() ) set ) ) implies ||.x : ( ( ) ( right_complementable ) Element of ( ( ) ( V4() ) set ) ) .|| : ( ( ) ( V24() real ext-real non negative ) Element of REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) = 0 : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V24() real ext-real non positive non negative V45() V46() V52() V53() V54() V55() V56() V57() V58() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) ) & ||.(a : ( ( ) ( V24() real ext-real ) Real) * x : ( ( ) ( right_complementable ) Element of ( ( ) ( V4() ) set ) ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive right_unital Banach_Algebra-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive left-distributive right_unital left_unital complete Banach_Algebra-like_1 Banach_Algebra-like_2 Banach_Algebra-like_3 Banach_Algebra-like ) Banach_Algebra) : ( ( ) ( V4() ) set ) ) .|| : ( ( ) ( V24() real ext-real non negative ) Element of REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) = (abs a : ( ( ) ( V24() real ext-real ) Real) ) : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) * ||.x : ( ( ) ( right_complementable ) Element of ( ( ) ( V4() ) set ) ) .|| : ( ( ) ( V24() real ext-real non negative ) Element of REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V24() real ext-real ) Element of REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) & ||.(x : ( ( ) ( right_complementable ) Element of ( ( ) ( V4() ) set ) ) + y : ( ( ) ( right_complementable ) Element of ( ( ) ( V4() ) set ) ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive right_unital Banach_Algebra-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive left-distributive right_unital left_unital complete Banach_Algebra-like_1 Banach_Algebra-like_2 Banach_Algebra-like_3 Banach_Algebra-like ) Banach_Algebra) : ( ( ) ( V4() ) set ) ) .|| : ( ( ) ( V24() real ext-real non negative ) Element of REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) <= ||.x : ( ( ) ( right_complementable ) Element of ( ( ) ( V4() ) set ) ) .|| : ( ( ) ( V24() real ext-real non negative ) Element of REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) + ||.y : ( ( ) ( right_complementable ) Element of ( ( ) ( V4() ) set ) ) .|| : ( ( ) ( V24() real ext-real non negative ) Element of REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V24() real ext-real non negative ) Element of REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) & ||.(x : ( ( ) ( right_complementable ) Element of ( ( ) ( V4() ) set ) ) * y : ( ( ) ( right_complementable ) Element of ( ( ) ( V4() ) set ) ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive right_unital Banach_Algebra-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive left-distributive right_unital left_unital complete Banach_Algebra-like_1 Banach_Algebra-like_2 Banach_Algebra-like_3 Banach_Algebra-like ) Banach_Algebra) : ( ( ) ( V4() ) set ) ) .|| : ( ( ) ( V24() real ext-real non negative ) Element of REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) <= ||.x : ( ( ) ( right_complementable ) Element of ( ( ) ( V4() ) set ) ) .|| : ( ( ) ( V24() real ext-real non negative ) Element of REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) * ||.y : ( ( ) ( right_complementable ) Element of ( ( ) ( V4() ) set ) ) .|| : ( ( ) ( V24() real ext-real non negative ) Element of REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V24() real ext-real non negative ) Element of REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) & ||.(1. X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive right_unital Banach_Algebra-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive left-distributive right_unital left_unital complete Banach_Algebra-like_1 Banach_Algebra-like_2 Banach_Algebra-like_3 Banach_Algebra-like ) Banach_Algebra) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive right_unital Banach_Algebra-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive left-distributive right_unital left_unital complete Banach_Algebra-like_1 Banach_Algebra-like_2 Banach_Algebra-like_3 Banach_Algebra-like ) Banach_Algebra) : ( ( ) ( V4() ) set ) ) .|| : ( ( ) ( V24() real ext-real non negative ) Element of REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) = 1 : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) & X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive right_unital Banach_Algebra-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive left-distributive right_unital left_unital complete Banach_Algebra-like_1 Banach_Algebra-like_2 Banach_Algebra-like_3 Banach_Algebra-like ) Banach_Algebra) is complete ) ;

registration
cluster non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive right_unital Banach_Algebra-like -> non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive right_unital well-unital Banach_Algebra-like for ( ( ) ( ) Normed_AlgebraStr ) ;
end;

definition
let X be ( ( non empty associative well-unital ) ( non empty associative well-unital ) multLoopStr ) ;
let v be ( ( ) ( ) Element of ( ( ) ( V4() ) set ) ) ;
redefine attr v is invertible means :: LOPBAN_3:def 4
 ex w being ( ( ) ( ) Element of ( ( ) ( V4() ) set ) ) st
( v : ( ( V4() ) ( V4() ) set ) * w : ( ( ) ( ) Element of ( ( ) ( V4() ) set ) ) : ( ( ) ( right_complementable ) Element of the carrier of X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V4() ) set ) ) = 1. X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) NORMSTR ) : ( ( ) ( right_complementable ) Element of the carrier of X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V4() ) set ) ) & w : ( ( ) ( ) Element of ( ( ) ( V4() ) set ) ) * v : ( ( V4() ) ( V4() ) set ) : ( ( ) ( right_complementable ) Element of the carrier of X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V4() ) set ) ) = 1. X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) NORMSTR ) : ( ( ) ( right_complementable ) Element of the carrier of X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V4() ) set ) ) );
end;

definition
let X be ( ( non empty ) ( non empty ) Normed_AlgebraStr ) ;
let S be ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of X : ( ( non empty ) ( non empty ) Normed_AlgebraStr ) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of X : ( ( non empty ) ( non empty ) Normed_AlgebraStr ) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of X : ( ( non empty ) ( non empty ) Normed_AlgebraStr ) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) ;
let a be ( ( ) ( ) Element of ( ( ) ( V4() ) set ) ) ;
func a * S -> ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) means :: LOPBAN_3:def 5
 for n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) holds it : ( ( Function-like V30(K33(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ,X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) NORMSTR ) ) : ( ( ) ( ) set ) ,X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) NORMSTR ) ) ) ( Relation-like K33(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ,X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) NORMSTR ) ) : ( ( ) ( ) set ) -defined X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) NORMSTR ) -valued Function-like V30(K33(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ,X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) NORMSTR ) ) : ( ( ) ( ) set ) ,X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) NORMSTR ) ) ) Element of K32(K33(K33(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ,X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) NORMSTR ) ) : ( ( ) ( ) set ) ,X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V4() ) set ) ) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) : ( ( ) ( right_complementable ) Element of the carrier of X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V4() ) set ) ) = a : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) * (S : ( ( V4() ) ( V4() ) set ) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) ) : ( ( ) ( right_complementable ) Element of the carrier of X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V4() ) set ) ) : ( ( ) ( right_complementable ) Element of the carrier of X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V4() ) set ) ) ;
end;

definition
let X be ( ( non empty ) ( non empty ) Normed_AlgebraStr ) ;
let S be ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of X : ( ( non empty ) ( non empty ) Normed_AlgebraStr ) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of X : ( ( non empty ) ( non empty ) Normed_AlgebraStr ) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of X : ( ( non empty ) ( non empty ) Normed_AlgebraStr ) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) ;
let a be ( ( ) ( ) Element of ( ( ) ( V4() ) set ) ) ;
func S * a -> ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) means :: LOPBAN_3:def 6
 for n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) holds it : ( ( Function-like V30(K33(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ,X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) NORMSTR ) ) : ( ( ) ( ) set ) ,X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) NORMSTR ) ) ) ( Relation-like K33(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ,X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) NORMSTR ) ) : ( ( ) ( ) set ) -defined X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) NORMSTR ) -valued Function-like V30(K33(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ,X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) NORMSTR ) ) : ( ( ) ( ) set ) ,X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) NORMSTR ) ) ) Element of K32(K33(K33(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ,X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) NORMSTR ) ) : ( ( ) ( ) set ) ,X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V4() ) set ) ) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) : ( ( ) ( right_complementable ) Element of the carrier of X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V4() ) set ) ) = (S : ( ( V4() ) ( V4() ) set ) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) ) : ( ( ) ( right_complementable ) Element of the carrier of X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V4() ) set ) ) * a : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) : ( ( ) ( right_complementable ) Element of the carrier of X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V4() ) set ) ) ;
end;

definition
let X be ( ( non empty ) ( non empty ) Normed_AlgebraStr ) ;
let seq1, seq2 be ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of X : ( ( non empty ) ( non empty ) Normed_AlgebraStr ) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of X : ( ( non empty ) ( non empty ) Normed_AlgebraStr ) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of X : ( ( non empty ) ( non empty ) Normed_AlgebraStr ) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) ;
func seq1 * seq2 -> ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) means :: LOPBAN_3:def 7
 for n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) holds it : ( ( Function-like V30(K33(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ,X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) NORMSTR ) ) : ( ( ) ( ) set ) ,X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) NORMSTR ) ) ) ( Relation-like K33(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ,X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) NORMSTR ) ) : ( ( ) ( ) set ) -defined X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) NORMSTR ) -valued Function-like V30(K33(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ,X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) NORMSTR ) ) : ( ( ) ( ) set ) ,X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) NORMSTR ) ) ) Element of K32(K33(K33(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ,X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) NORMSTR ) ) : ( ( ) ( ) set ) ,X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V4() ) set ) ) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) : ( ( ) ( right_complementable ) Element of the carrier of X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V4() ) set ) ) = (seq1 : ( ( V4() ) ( V4() ) set ) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) ) : ( ( ) ( right_complementable ) Element of the carrier of X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V4() ) set ) ) * (seq2 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) ) : ( ( ) ( right_complementable ) Element of the carrier of X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V4() ) set ) ) : ( ( ) ( right_complementable ) Element of the carrier of X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V4() ) set ) ) ;
end;

notation
let X be ( ( non empty associative well-unital ) ( non empty associative well-unital ) multLoopStr ) ;
let x be ( ( ) ( ) Element of ( ( ) ( V4() ) set ) ) ;
synonym x " for / x;
end;

definition
let X be ( ( non empty associative well-unital ) ( non empty associative well-unital ) multLoopStr ) ;
let x be ( ( ) ( ) Element of ( ( ) ( V4() ) set ) ) ;
assume x : ( ( ) ( ) Element of ( ( ) ( V4() ) set ) ) is invertible ;
redefine func / x means :: LOPBAN_3:def 8
( x : ( ( V4() ) ( V4() ) set ) * it : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) : ( ( ) ( right_complementable ) Element of the carrier of X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V4() ) set ) ) = 1. X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) NORMSTR ) : ( ( ) ( right_complementable ) Element of the carrier of X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V4() ) set ) ) & it : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) * x : ( ( V4() ) ( V4() ) set ) : ( ( ) ( right_complementable ) Element of the carrier of X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V4() ) set ) ) = 1. X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) NORMSTR ) : ( ( ) ( right_complementable ) Element of the carrier of X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V4() ) set ) ) );
end;

definition
let X be ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive right_unital Banach_Algebra-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive left-distributive right_unital well-unital left_unital complete Banach_Algebra-like_1 Banach_Algebra-like_2 Banach_Algebra-like_3 Banach_Algebra-like ) Banach_Algebra) ;
let z be ( ( ) ( right_complementable ) Element of ( ( ) ( V4() ) set ) ) ;
func z GeoSeq -> ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) means :: LOPBAN_3:def 9
( it : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) . 0 : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V24() real ext-real non positive non negative V45() V46() V52() V53() V54() V55() V56() V57() V58() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) : ( ( ) ( right_complementable ) Element of the carrier of X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V4() ) set ) ) = 1. X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) NORMSTR ) : ( ( ) ( right_complementable ) Element of the carrier of X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V4() ) set ) ) & ( for n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) holds it : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) . (n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) + 1 : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) : ( ( ) ( right_complementable ) Element of the carrier of X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V4() ) set ) ) = (it : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) ) : ( ( ) ( right_complementable ) Element of the carrier of X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V4() ) set ) ) * z : ( ( V4() ) ( V4() ) set ) : ( ( ) ( right_complementable ) Element of the carrier of X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V4() ) set ) ) ) );
end;

definition
let X be ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive right_unital Banach_Algebra-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive left-distributive right_unital well-unital left_unital complete Banach_Algebra-like_1 Banach_Algebra-like_2 Banach_Algebra-like_3 Banach_Algebra-like ) Banach_Algebra) ;
let z be ( ( ) ( right_complementable ) Element of ( ( ) ( V4() ) set ) ) ;
let n be ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) ;
func z #N n -> ( ( ) ( right_complementable ) Element of ( ( ) ( V4() ) set ) ) equals :: LOPBAN_3:def 10
(z : ( ( V4() ) ( V4() ) set ) GeoSeq) : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V24() real ext-real V45() V46() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) : ( ( ) ( right_complementable ) Element of the carrier of X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like ) NORMSTR ) : ( ( ) ( V4() ) set ) ) ;
end;

theorem :: LOPBAN_3:39
for X being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive right_unital Banach_Algebra-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive left-distributive right_unital well-unital left_unital complete Banach_Algebra-like_1 Banach_Algebra-like_2 Banach_Algebra-like_3 Banach_Algebra-like ) Banach_Algebra)
for z being ( ( ) ( right_complementable ) Element of ( ( ) ( V4() ) set ) ) holds z : ( ( ) ( right_complementable ) Element of ( ( ) ( V4() ) set ) ) #N 0 : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V24() real ext-real non positive non negative V45() V46() V52() V53() V54() V55() V56() V57() V58() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) : ( ( ) ( right_complementable ) Element of ( ( ) ( V4() ) set ) ) = 1. X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive right_unital Banach_Algebra-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive left-distributive right_unital well-unital left_unital complete Banach_Algebra-like_1 Banach_Algebra-like_2 Banach_Algebra-like_3 Banach_Algebra-like ) Banach_Algebra) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive right_unital Banach_Algebra-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive left-distributive right_unital well-unital left_unital complete Banach_Algebra-like_1 Banach_Algebra-like_2 Banach_Algebra-like_3 Banach_Algebra-like ) Banach_Algebra) : ( ( ) ( V4() ) set ) ) ;

theorem :: LOPBAN_3:40
for X being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive right_unital Banach_Algebra-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive left-distributive right_unital well-unital left_unital complete Banach_Algebra-like_1 Banach_Algebra-like_2 Banach_Algebra-like_3 Banach_Algebra-like ) Banach_Algebra)
for z being ( ( ) ( right_complementable ) Element of ( ( ) ( V4() ) set ) ) st ||.z : ( ( ) ( right_complementable ) Element of ( ( ) ( V4() ) set ) ) .|| : ( ( ) ( V24() real ext-real non negative ) Element of REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) < 1 : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) holds
( z : ( ( ) ( right_complementable ) Element of ( ( ) ( V4() ) set ) ) GeoSeq : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive right_unital Banach_Algebra-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive left-distributive right_unital well-unital left_unital complete Banach_Algebra-like_1 Banach_Algebra-like_2 Banach_Algebra-like_3 Banach_Algebra-like ) Banach_Algebra) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive right_unital Banach_Algebra-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive left-distributive right_unital well-unital left_unital complete Banach_Algebra-like_1 Banach_Algebra-like_2 Banach_Algebra-like_3 Banach_Algebra-like ) Banach_Algebra) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive right_unital Banach_Algebra-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive left-distributive right_unital well-unital left_unital complete Banach_Algebra-like_1 Banach_Algebra-like_2 Banach_Algebra-like_3 Banach_Algebra-like ) Banach_Algebra) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) is summable & z : ( ( ) ( right_complementable ) Element of ( ( ) ( V4() ) set ) ) GeoSeq : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive right_unital Banach_Algebra-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive left-distributive right_unital well-unital left_unital complete Banach_Algebra-like_1 Banach_Algebra-like_2 Banach_Algebra-like_3 Banach_Algebra-like ) Banach_Algebra) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive right_unital Banach_Algebra-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive left-distributive right_unital well-unital left_unital complete Banach_Algebra-like_1 Banach_Algebra-like_2 Banach_Algebra-like_3 Banach_Algebra-like ) Banach_Algebra) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive right_unital Banach_Algebra-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive left-distributive right_unital well-unital left_unital complete Banach_Algebra-like_1 Banach_Algebra-like_2 Banach_Algebra-like_3 Banach_Algebra-like ) Banach_Algebra) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) is norm_summable ) ;

theorem :: LOPBAN_3:41
for X being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive right_unital Banach_Algebra-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive left-distributive right_unital well-unital left_unital complete Banach_Algebra-like_1 Banach_Algebra-like_2 Banach_Algebra-like_3 Banach_Algebra-like ) Banach_Algebra)
for x being ( ( ) ( right_complementable ) Point of ( ( ) ( V4() ) set ) ) st ||.((1. X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive right_unital Banach_Algebra-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive left-distributive right_unital well-unital left_unital complete Banach_Algebra-like_1 Banach_Algebra-like_2 Banach_Algebra-like_3 Banach_Algebra-like ) Banach_Algebra) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive right_unital Banach_Algebra-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive left-distributive right_unital well-unital left_unital complete Banach_Algebra-like_1 Banach_Algebra-like_2 Banach_Algebra-like_3 Banach_Algebra-like ) Banach_Algebra) : ( ( ) ( V4() ) set ) ) - x : ( ( ) ( right_complementable ) Point of ( ( ) ( V4() ) set ) ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive right_unital Banach_Algebra-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive left-distributive right_unital well-unital left_unital complete Banach_Algebra-like_1 Banach_Algebra-like_2 Banach_Algebra-like_3 Banach_Algebra-like ) Banach_Algebra) : ( ( ) ( V4() ) set ) ) .|| : ( ( ) ( V24() real ext-real non negative ) Element of REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) < 1 : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) holds
( ((1. X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive right_unital Banach_Algebra-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive left-distributive right_unital well-unital left_unital complete Banach_Algebra-like_1 Banach_Algebra-like_2 Banach_Algebra-like_3 Banach_Algebra-like ) Banach_Algebra) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive right_unital Banach_Algebra-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive left-distributive right_unital well-unital left_unital complete Banach_Algebra-like_1 Banach_Algebra-like_2 Banach_Algebra-like_3 Banach_Algebra-like ) Banach_Algebra) : ( ( ) ( V4() ) set ) ) - x : ( ( ) ( right_complementable ) Point of ( ( ) ( V4() ) set ) ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive right_unital Banach_Algebra-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive left-distributive right_unital well-unital left_unital complete Banach_Algebra-like_1 Banach_Algebra-like_2 Banach_Algebra-like_3 Banach_Algebra-like ) Banach_Algebra) : ( ( ) ( V4() ) set ) ) GeoSeq : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive right_unital Banach_Algebra-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive left-distributive right_unital well-unital left_unital complete Banach_Algebra-like_1 Banach_Algebra-like_2 Banach_Algebra-like_3 Banach_Algebra-like ) Banach_Algebra) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive right_unital Banach_Algebra-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive left-distributive right_unital well-unital left_unital complete Banach_Algebra-like_1 Banach_Algebra-like_2 Banach_Algebra-like_3 Banach_Algebra-like ) Banach_Algebra) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive right_unital Banach_Algebra-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive left-distributive right_unital well-unital left_unital complete Banach_Algebra-like_1 Banach_Algebra-like_2 Banach_Algebra-like_3 Banach_Algebra-like ) Banach_Algebra) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) is summable & ((1. X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive right_unital Banach_Algebra-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive left-distributive right_unital well-unital left_unital complete Banach_Algebra-like_1 Banach_Algebra-like_2 Banach_Algebra-like_3 Banach_Algebra-like ) Banach_Algebra) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive right_unital Banach_Algebra-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive left-distributive right_unital well-unital left_unital complete Banach_Algebra-like_1 Banach_Algebra-like_2 Banach_Algebra-like_3 Banach_Algebra-like ) Banach_Algebra) : ( ( ) ( V4() ) set ) ) - x : ( ( ) ( right_complementable ) Point of ( ( ) ( V4() ) set ) ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive right_unital Banach_Algebra-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive left-distributive right_unital well-unital left_unital complete Banach_Algebra-like_1 Banach_Algebra-like_2 Banach_Algebra-like_3 Banach_Algebra-like ) Banach_Algebra) : ( ( ) ( V4() ) set ) ) GeoSeq : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive right_unital Banach_Algebra-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive left-distributive right_unital well-unital left_unital complete Banach_Algebra-like_1 Banach_Algebra-like_2 Banach_Algebra-like_3 Banach_Algebra-like ) Banach_Algebra) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive right_unital Banach_Algebra-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive left-distributive right_unital well-unital left_unital complete Banach_Algebra-like_1 Banach_Algebra-like_2 Banach_Algebra-like_3 Banach_Algebra-like ) Banach_Algebra) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive right_unital Banach_Algebra-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive left-distributive right_unital well-unital left_unital complete Banach_Algebra-like_1 Banach_Algebra-like_2 Banach_Algebra-like_3 Banach_Algebra-like ) Banach_Algebra) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) is norm_summable ) ;

theorem :: LOPBAN_3:42
for X being ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive right_unital Banach_Algebra-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive left-distributive right_unital well-unital left_unital complete Banach_Algebra-like_1 Banach_Algebra-like_2 Banach_Algebra-like_3 Banach_Algebra-like ) Banach_Algebra)
for x being ( ( ) ( right_complementable ) Point of ( ( ) ( V4() ) set ) ) st ||.((1. X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive right_unital Banach_Algebra-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive left-distributive right_unital well-unital left_unital complete Banach_Algebra-like_1 Banach_Algebra-like_2 Banach_Algebra-like_3 Banach_Algebra-like ) Banach_Algebra) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive right_unital Banach_Algebra-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive left-distributive right_unital well-unital left_unital complete Banach_Algebra-like_1 Banach_Algebra-like_2 Banach_Algebra-like_3 Banach_Algebra-like ) Banach_Algebra) : ( ( ) ( V4() ) set ) ) - x : ( ( ) ( right_complementable ) Point of ( ( ) ( V4() ) set ) ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive right_unital Banach_Algebra-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive left-distributive right_unital well-unital left_unital complete Banach_Algebra-like_1 Banach_Algebra-like_2 Banach_Algebra-like_3 Banach_Algebra-like ) Banach_Algebra) : ( ( ) ( V4() ) set ) ) .|| : ( ( ) ( V24() real ext-real non negative ) Element of REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) < 1 : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal natural V24() real ext-real positive non negative V45() V46() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) holds
( x : ( ( ) ( right_complementable ) Point of ( ( ) ( V4() ) set ) ) is invertible & x : ( ( ) ( right_complementable ) Point of ( ( ) ( V4() ) set ) ) " : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive right_unital Banach_Algebra-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive left-distributive right_unital well-unital left_unital complete Banach_Algebra-like_1 Banach_Algebra-like_2 Banach_Algebra-like_3 Banach_Algebra-like ) Banach_Algebra) : ( ( ) ( V4() ) set ) ) = Sum (((1. X : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive right_unital Banach_Algebra-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive left-distributive right_unital well-unital left_unital complete Banach_Algebra-like_1 Banach_Algebra-like_2 Banach_Algebra-like_3 Banach_Algebra-like ) Banach_Algebra) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive right_unital Banach_Algebra-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive left-distributive right_unital well-unital left_unital complete Banach_Algebra-like_1 Banach_Algebra-like_2 Banach_Algebra-like_3 Banach_Algebra-like ) Banach_Algebra) : ( ( ) ( V4() ) set ) ) - x : ( ( ) ( right_complementable ) Point of ( ( ) ( V4() ) set ) ) ) : ( ( ) ( right_complementable ) Element of the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive right_unital Banach_Algebra-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive left-distributive right_unital well-unital left_unital complete Banach_Algebra-like_1 Banach_Algebra-like_2 Banach_Algebra-like_3 Banach_Algebra-like ) Banach_Algebra) : ( ( ) ( V4() ) set ) ) GeoSeq) : ( ( Function-like V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive right_unital Banach_Algebra-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive left-distributive right_unital well-unital left_unital complete Banach_Algebra-like_1 Banach_Algebra-like_2 Banach_Algebra-like_3 Banach_Algebra-like ) Banach_Algebra) : ( ( ) ( V4() ) set ) ) ) ( V4() Relation-like NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) -defined the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive right_unital Banach_Algebra-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive left-distributive right_unital well-unital left_unital complete Banach_Algebra-like_1 Banach_Algebra-like_2 Banach_Algebra-like_3 Banach_Algebra-like ) Banach_Algebra) : ( ( ) ( V4() ) set ) -valued Function-like V29( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) ) V30( NAT : ( ( ) ( V4() epsilon-transitive epsilon-connected ordinal V52() V53() V54() V55() V56() V57() V58() ) Element of K32(REAL : ( ( ) ( V4() V47() V52() V53() V54() V58() ) set ) ) : ( ( ) ( V4() ) set ) ) , the carrier of b1 : ( ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive right_unital Banach_Algebra-like ) ( non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V133() V134() RealNormSpace-like V150() associative right-distributive left-distributive right_unital well-unital left_unital complete Banach_Algebra-like_1 Banach_Algebra-like_2 Banach_Algebra-like_3 Banach_Algebra-like ) Banach_Algebra) : ( ( ) ( V4() ) set ) ) ) sequence of ( ( ) ( V4() ) set ) ) : ( ( ) ( right_complementable ) Element of ( ( ) ( V4() ) set ) ) ) ;