:: COMPLSP2 semantic presentation

begin

definition
let z be ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ;
func z *' -> ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) means :: COMPLSP2:def 1
( len it : ( ( ) ( ) set ) : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) = len z : ( ( Relation-like Function-like real-valued ) ( Relation-like Function-like complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) & ( for i being ( ( natural ) ( natural complex ext-real ) Nat) st 1 : ( ( ) ( non zero natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) <= i : ( ( natural ) ( natural complex ext-real ) Nat) & i : ( ( natural ) ( natural complex ext-real ) Nat) <= len z : ( ( Relation-like Function-like real-valued ) ( Relation-like Function-like complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) holds
it : ( ( ) ( ) set ) . i : ( ( natural ) ( natural complex ext-real ) Nat) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) = (z : ( ( Relation-like Function-like real-valued ) ( Relation-like Function-like complex-valued ext-real-valued real-valued ) set ) . i : ( ( natural ) ( natural complex ext-real ) Nat) ) : ( ( ) ( complex V34() ext-real ) Element of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) *' : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ) );
end;

theorem :: COMPLSP2:1
for i being ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) )
for x, y being ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) st i : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) in dom (x : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) + y : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) : ( ( ) ( V64() V65() V66() V67() V68() V69() ) Element of bool NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) holds
(x : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) + y : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) . i : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) = (x : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) . i : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) + (y : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) . i : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ;

theorem :: COMPLSP2:2
for i being ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) )
for x, y being ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) st i : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) in dom (x : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) - y : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) : ( ( ) ( V64() V65() V66() V67() V68() V69() ) Element of bool NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) holds
(x : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) - y : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) . i : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) = (x : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) . i : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) - (y : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) . i : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ;

definition
let i be ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) ;
let R1, R2 be ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() V46(i : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-valued ) Element of i : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) -tuples_on COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) : ( ( ) ( functional non zero FinSequence-membered ) FinSequenceSet of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ) ;
:: original: -
redefine func R1 - R2 -> ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() V46(i : ( ( Relation-like Function-like real-valued ) ( Relation-like Function-like complex-valued ext-real-valued real-valued ) set ) ) FinSequence-like FinSubsequence-like complex-valued ) Element of i : ( ( Relation-like Function-like real-valued ) ( Relation-like Function-like complex-valued ext-real-valued real-valued ) set ) -tuples_on COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ) ;
end;

definition
let i be ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) ;
let R1, R2 be ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() V46(i : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-valued ) Element of i : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) -tuples_on COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) : ( ( ) ( functional non zero FinSequence-membered ) FinSequenceSet of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ) ;
:: original: +
redefine func R1 + R2 -> ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() V46(i : ( ( Relation-like Function-like real-valued ) ( Relation-like Function-like complex-valued ext-real-valued real-valued ) set ) ) FinSequence-like FinSubsequence-like complex-valued ) Element of i : ( ( Relation-like Function-like real-valued ) ( Relation-like Function-like complex-valued ext-real-valued real-valued ) set ) -tuples_on COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ) ;
end;

definition
let i be ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) ;
let R be ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() V46(i : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-valued ) Element of i : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) -tuples_on COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) : ( ( ) ( functional non zero FinSequence-membered ) FinSequenceSet of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ) ;
let r be ( ( complex ) ( complex ) number ) ;
:: original: *
redefine func r * R -> ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() V46(i : ( ( Relation-like Function-like real-valued ) ( Relation-like Function-like complex-valued ext-real-valued real-valued ) set ) ) FinSequence-like FinSubsequence-like complex-valued ) Element of i : ( ( Relation-like Function-like real-valued ) ( Relation-like Function-like complex-valued ext-real-valued real-valued ) set ) -tuples_on COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ) ;
end;

theorem :: COMPLSP2:3
for a being ( ( complex ) ( complex ) number )
for x being ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) holds len (a : ( ( complex ) ( complex ) number ) * x : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) = len x : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) ;

theorem :: COMPLSP2:4
for x being ( ( Relation-like Function-like FinSequence-like complex-yielding ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined Function-like V39() FinSequence-like FinSubsequence-like complex-yielding ) FinSequence) holds dom x : ( ( Relation-like Function-like FinSequence-like complex-yielding ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined Function-like V39() FinSequence-like FinSubsequence-like complex-yielding ) FinSequence) : ( ( ) ( V64() V65() V66() V67() V68() V69() ) Element of bool NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) = dom (- x : ( ( Relation-like Function-like FinSequence-like complex-yielding ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined Function-like V39() FinSequence-like FinSubsequence-like complex-yielding ) FinSequence) ) : ( ( Relation-like Function-like complex-valued ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) set ) : ( ( ) ( V64() V65() V66() V67() V68() V69() ) Element of bool NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: COMPLSP2:5
for x being ( ( Relation-like Function-like FinSequence-like complex-valued ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence) holds len (- x : ( ( Relation-like Function-like FinSequence-like complex-valued ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence) ) : ( ( Relation-like Function-like complex-valued ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) set ) : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) = len x : ( ( Relation-like Function-like FinSequence-like complex-valued ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence) : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) ;

theorem :: COMPLSP2:6
for x1, x2 being ( ( Relation-like Function-like FinSequence-like complex-valued ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence) st len x1 : ( ( Relation-like Function-like FinSequence-like complex-valued ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence) : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) = len x2 : ( ( Relation-like Function-like FinSequence-like complex-valued ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence) : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) holds
len (x1 : ( ( Relation-like Function-like FinSequence-like complex-valued ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence) + x2 : ( ( Relation-like Function-like FinSequence-like complex-valued ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence) ) : ( ( Relation-like Function-like ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) set ) : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) = len x1 : ( ( Relation-like Function-like FinSequence-like complex-valued ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence) : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) ;

theorem :: COMPLSP2:7
for x1, x2 being ( ( Relation-like Function-like FinSequence-like complex-valued ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence) st len x1 : ( ( Relation-like Function-like FinSequence-like complex-valued ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence) : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) = len x2 : ( ( Relation-like Function-like FinSequence-like complex-valued ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence) : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) holds
len (x1 : ( ( Relation-like Function-like FinSequence-like complex-valued ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence) - x2 : ( ( Relation-like Function-like FinSequence-like complex-valued ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence) ) : ( ( Relation-like Function-like ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) set ) : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) = len x1 : ( ( Relation-like Function-like FinSequence-like complex-valued ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence) : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) ;

theorem :: COMPLSP2:8
for f being ( ( Relation-like Function-like FinSequence-like complex-valued ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence) holds f : ( ( Relation-like Function-like FinSequence-like complex-valued ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence) is ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) Element of COMPLEX (len f : ( ( Relation-like Function-like FinSequence-like complex-valued ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence) ) : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ) ;

theorem :: COMPLSP2:9
for i being ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) )
for R1, R2 being ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() V46(b1 : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-valued ) Element of i : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) -tuples_on COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) : ( ( ) ( functional non zero FinSequence-membered ) FinSequenceSet of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ) holds R1 : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() V46(b1 : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-valued ) Element of b1 : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) -tuples_on COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) : ( ( ) ( functional non zero FinSequence-membered ) FinSequenceSet of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ) - R2 : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() V46(b1 : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-valued ) Element of b1 : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) -tuples_on COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) : ( ( ) ( functional non zero FinSequence-membered ) FinSequenceSet of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() V46(b1 : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-valued ) Element of b1 : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) -tuples_on COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) : ( ( ) ( functional non zero FinSequence-membered ) FinSequenceSet of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ) = R1 : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() V46(b1 : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-valued ) Element of b1 : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) -tuples_on COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) : ( ( ) ( functional non zero FinSequence-membered ) FinSequenceSet of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ) + (- R2 : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() V46(b1 : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-valued ) Element of b1 : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) -tuples_on COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) : ( ( ) ( functional non zero FinSequence-membered ) FinSequenceSet of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ;

theorem :: COMPLSP2:10
for x, y being ( ( Relation-like Function-like FinSequence-like complex-valued ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence) st len x : ( ( Relation-like Function-like FinSequence-like complex-valued ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence) : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) = len y : ( ( Relation-like Function-like FinSequence-like complex-valued ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence) : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) holds
x : ( ( Relation-like Function-like FinSequence-like complex-valued ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence) - y : ( ( Relation-like Function-like FinSequence-like complex-valued ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence) : ( ( Relation-like Function-like ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) set ) = x : ( ( Relation-like Function-like FinSequence-like complex-valued ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence) + (- y : ( ( Relation-like Function-like FinSequence-like complex-valued ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence) ) : ( ( Relation-like Function-like complex-valued ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) set ) : ( ( Relation-like Function-like ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) set ) ;

theorem :: COMPLSP2:11
for i being ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) )
for R being ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() V46(b1 : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-valued ) Element of i : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) -tuples_on COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) : ( ( ) ( functional non zero FinSequence-membered ) FinSequenceSet of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ) holds (- 1 : ( ( ) ( non zero natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non zero complex V34() V35() V36() ext-real ) Element of INT : ( ( ) ( non zero V39() V64() V65() V66() V67() V68() V70() ) set ) ) * R : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() V46(b1 : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-valued ) Element of b1 : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) -tuples_on COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) : ( ( ) ( functional non zero FinSequence-membered ) FinSequenceSet of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() V46(b1 : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-valued ) Element of b1 : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) -tuples_on COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) : ( ( ) ( functional non zero FinSequence-membered ) FinSequenceSet of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ) = - R : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() V46(b1 : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-valued ) Element of b1 : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) -tuples_on COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) : ( ( ) ( functional non zero FinSequence-membered ) FinSequenceSet of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ;

theorem :: COMPLSP2:12
for x being ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) holds (- 1 : ( ( ) ( non zero natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non zero complex V34() V35() V36() ext-real ) Element of INT : ( ( ) ( non zero V39() V64() V65() V66() V67() V68() V70() ) set ) ) * x : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) = - x : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ;

theorem :: COMPLSP2:13
for i being ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) )
for x being ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) holds (- x : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) . i : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) = - (x : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) . i : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ;

definition
let i be ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) ;
let R be ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() V46(i : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-valued ) Element of i : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) -tuples_on COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) : ( ( ) ( functional non zero FinSequence-membered ) FinSequenceSet of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ) ;
:: original: -
redefine func - R -> ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() V46(i : ( ( Relation-like Function-like real-valued ) ( Relation-like Function-like complex-valued ext-real-valued real-valued ) set ) ) FinSequence-like FinSubsequence-like complex-valued ) Element of i : ( ( Relation-like Function-like real-valued ) ( Relation-like Function-like complex-valued ext-real-valued real-valued ) set ) -tuples_on COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ) ;
end;

theorem :: COMPLSP2:14
for i, j being ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) )
for c being ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) )
for R being ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() V46(b1 : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-valued ) Element of i : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) -tuples_on COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) : ( ( ) ( functional non zero FinSequence-membered ) FinSequenceSet of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ) st c : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) = R : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() V46(b1 : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-valued ) Element of b1 : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) -tuples_on COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) : ( ( ) ( functional non zero FinSequence-membered ) FinSequenceSet of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ) . j : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) holds
(- R : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() V46(b1 : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-valued ) Element of b1 : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) -tuples_on COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) : ( ( ) ( functional non zero FinSequence-membered ) FinSequenceSet of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() V46(b1 : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-valued ) Element of b1 : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) -tuples_on COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) : ( ( ) ( functional non zero FinSequence-membered ) FinSequenceSet of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ) . j : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) = - c : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ;

theorem :: COMPLSP2:15
for x being ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) )
for a being ( ( complex ) ( complex ) number ) holds dom (a : ( ( complex ) ( complex ) number ) * x : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) : ( ( ) ( V64() V65() V66() V67() V68() V69() ) Element of bool NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) = dom x : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) : ( ( ) ( V64() V65() V66() V67() V68() V69() ) Element of bool NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: COMPLSP2:16
for x being ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) )
for i being ( ( natural ) ( natural complex ext-real ) Nat)
for a being ( ( complex ) ( complex ) number ) holds (a : ( ( complex ) ( complex ) number ) * x : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) . i : ( ( natural ) ( natural complex ext-real ) Nat) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) = a : ( ( complex ) ( complex ) number ) * (x : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) . i : ( ( natural ) ( natural complex ext-real ) Nat) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ;

theorem :: COMPLSP2:17
for x being ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) )
for a being ( ( complex ) ( complex ) number ) holds (a : ( ( complex ) ( complex ) number ) * x : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) *' : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) = (a : ( ( complex ) ( complex ) number ) *') : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) * (x : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) *') : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ;

theorem :: COMPLSP2:18
for i, j being ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) )
for R1, R2 being ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() V46(b1 : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-valued ) Element of i : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) -tuples_on COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) : ( ( ) ( functional non zero FinSequence-membered ) FinSequenceSet of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ) holds (R1 : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() V46(b1 : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-valued ) Element of b1 : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) -tuples_on COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) : ( ( ) ( functional non zero FinSequence-membered ) FinSequenceSet of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ) + R2 : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() V46(b1 : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-valued ) Element of b1 : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) -tuples_on COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) : ( ( ) ( functional non zero FinSequence-membered ) FinSequenceSet of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() V46(b1 : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-valued ) Element of b1 : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) -tuples_on COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) : ( ( ) ( functional non zero FinSequence-membered ) FinSequenceSet of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ) . j : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) = (R1 : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() V46(b1 : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-valued ) Element of b1 : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) -tuples_on COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) : ( ( ) ( functional non zero FinSequence-membered ) FinSequenceSet of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ) . j : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) + (R2 : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() V46(b1 : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-valued ) Element of b1 : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) -tuples_on COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) : ( ( ) ( functional non zero FinSequence-membered ) FinSequenceSet of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ) . j : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ;

theorem :: COMPLSP2:19
for x1, x2 being ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) st len x1 : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) = len x2 : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) holds
(x1 : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) + x2 : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) *' : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) = (x1 : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) *') : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) + (x2 : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) *') : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ;

theorem :: COMPLSP2:20
for i, j being ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) )
for R1, R2 being ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() V46(b1 : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-valued ) Element of i : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) -tuples_on COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) : ( ( ) ( functional non zero FinSequence-membered ) FinSequenceSet of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ) holds (R1 : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() V46(b1 : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-valued ) Element of b1 : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) -tuples_on COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) : ( ( ) ( functional non zero FinSequence-membered ) FinSequenceSet of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ) - R2 : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() V46(b1 : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-valued ) Element of b1 : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) -tuples_on COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) : ( ( ) ( functional non zero FinSequence-membered ) FinSequenceSet of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() V46(b1 : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-valued ) Element of b1 : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) -tuples_on COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) : ( ( ) ( functional non zero FinSequence-membered ) FinSequenceSet of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ) . j : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) = (R1 : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() V46(b1 : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-valued ) Element of b1 : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) -tuples_on COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) : ( ( ) ( functional non zero FinSequence-membered ) FinSequenceSet of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ) . j : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) - (R2 : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() V46(b1 : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) ) FinSequence-like FinSubsequence-like complex-valued ) Element of b1 : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) -tuples_on COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) : ( ( ) ( functional non zero FinSequence-membered ) FinSequenceSet of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ) . j : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ;

theorem :: COMPLSP2:21
for x1, x2 being ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) st len x1 : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) = len x2 : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) holds
(x1 : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) - x2 : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) *' : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) = (x1 : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) *') : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) - (x2 : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) *') : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ;

theorem :: COMPLSP2:22
for z being ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) holds (z : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) *') : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) *' : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) = z : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ;

theorem :: COMPLSP2:23
for z being ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) holds (- z : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) *' : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) = - (z : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) *') : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ;

theorem :: COMPLSP2:24
for z being ( ( complex ) ( complex ) number ) holds z : ( ( complex ) ( complex ) number ) + (z : ( ( complex ) ( complex ) number ) *') : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) = 2 : ( ( ) ( non zero natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) * (Re z : ( ( complex ) ( complex ) number ) ) : ( ( ) ( complex V34() ext-real ) Element of REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ) : ( ( ) ( complex V34() ext-real ) Element of REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ) ;

theorem :: COMPLSP2:25
for i being ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) )
for x, y being ( ( Relation-like Function-like FinSequence-like complex-valued ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence) st len x : ( ( Relation-like Function-like FinSequence-like complex-valued ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence) : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) = len y : ( ( Relation-like Function-like FinSequence-like complex-valued ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence) : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) holds
(x : ( ( Relation-like Function-like FinSequence-like complex-valued ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence) - y : ( ( Relation-like Function-like FinSequence-like complex-valued ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence) ) : ( ( Relation-like Function-like ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) set ) . i : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) = (x : ( ( Relation-like Function-like FinSequence-like complex-valued ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence) . i : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) - (y : ( ( Relation-like Function-like FinSequence-like complex-valued ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence) . i : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ;

theorem :: COMPLSP2:26
for i being ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) )
for x, y being ( ( Relation-like Function-like FinSequence-like complex-valued ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence) st len x : ( ( Relation-like Function-like FinSequence-like complex-valued ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence) : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) = len y : ( ( Relation-like Function-like FinSequence-like complex-valued ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence) : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) holds
(x : ( ( Relation-like Function-like FinSequence-like complex-valued ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence) + y : ( ( Relation-like Function-like FinSequence-like complex-valued ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence) ) : ( ( Relation-like Function-like ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) set ) . i : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) = (x : ( ( Relation-like Function-like FinSequence-like complex-valued ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence) . i : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) + (y : ( ( Relation-like Function-like FinSequence-like complex-valued ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence) . i : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ;

definition
let z be ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ;
func Re z -> ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) FinSequence of REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ) equals :: COMPLSP2:def 2
(1 : ( ( ) ( non zero natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( non zero natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non zero complex V34() V36() ext-real ) Element of RAT : ( ( ) ( non zero V39() V64() V65() V66() V67() V70() ) set ) ) * (z : ( ( Relation-like Function-like real-valued ) ( Relation-like Function-like complex-valued ext-real-valued real-valued ) set ) + (z : ( ( Relation-like Function-like real-valued ) ( Relation-like Function-like complex-valued ext-real-valued real-valued ) set ) *') : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ;
end;

theorem :: COMPLSP2:27
for z being ( ( complex ) ( complex ) number ) holds z : ( ( complex ) ( complex ) number ) - (z : ( ( complex ) ( complex ) number ) *') : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) = (2 : ( ( ) ( non zero natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) * (Im z : ( ( complex ) ( complex ) number ) ) : ( ( ) ( complex V34() ext-real ) Element of REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ) ) : ( ( ) ( complex V34() ext-real ) Element of REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ) * <i> : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ;

definition
let z be ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ;
func Im z -> ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) FinSequence of REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ) equals :: COMPLSP2:def 3
(- ((1 : ( ( ) ( non zero natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( non zero natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non zero complex V34() V36() ext-real ) Element of RAT : ( ( ) ( non zero V39() V64() V65() V66() V67() V70() ) set ) ) * <i> : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) * (z : ( ( Relation-like Function-like real-valued ) ( Relation-like Function-like complex-valued ext-real-valued real-valued ) set ) - (z : ( ( Relation-like Function-like real-valued ) ( Relation-like Function-like complex-valued ext-real-valued real-valued ) set ) *') : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ;
end;

definition
let x, y be ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ;
func |(x,y)| -> ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) equals :: COMPLSP2:def 4
((|((Re x : ( ( Relation-like Function-like real-valued ) ( Relation-like Function-like complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) FinSequence of REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ) ,(Re y : ( ( ) ( ) set ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) FinSequence of REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ) )| : ( ( ) ( complex V34() ext-real ) Element of REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ) - (<i> : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) * |((Re x : ( ( Relation-like Function-like real-valued ) ( Relation-like Function-like complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) FinSequence of REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ) ,(Im y : ( ( ) ( ) set ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) FinSequence of REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ) )| : ( ( ) ( complex V34() ext-real ) Element of REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) + (<i> : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) * |((Im x : ( ( Relation-like Function-like real-valued ) ( Relation-like Function-like complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) FinSequence of REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ) ,(Re y : ( ( ) ( ) set ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) FinSequence of REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ) )| : ( ( ) ( complex V34() ext-real ) Element of REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) + |((Im x : ( ( Relation-like Function-like real-valued ) ( Relation-like Function-like complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) FinSequence of REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ) ,(Im y : ( ( ) ( ) set ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) FinSequence of REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ) )| : ( ( ) ( complex V34() ext-real ) Element of REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ;
end;

theorem :: COMPLSP2:28
for x, y, z being ( ( Relation-like Function-like FinSequence-like complex-valued ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence) st len x : ( ( Relation-like Function-like FinSequence-like complex-valued ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence) : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) = len y : ( ( Relation-like Function-like FinSequence-like complex-valued ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence) : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) & len y : ( ( Relation-like Function-like FinSequence-like complex-valued ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence) : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) = len z : ( ( Relation-like Function-like FinSequence-like complex-valued ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence) : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) holds
x : ( ( Relation-like Function-like FinSequence-like complex-valued ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence) + (y : ( ( Relation-like Function-like FinSequence-like complex-valued ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence) + z : ( ( Relation-like Function-like FinSequence-like complex-valued ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence) ) : ( ( Relation-like Function-like ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) set ) : ( ( Relation-like Function-like ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) set ) = (x : ( ( Relation-like Function-like FinSequence-like complex-valued ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence) + y : ( ( Relation-like Function-like FinSequence-like complex-valued ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence) ) : ( ( Relation-like Function-like ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) set ) + z : ( ( Relation-like Function-like FinSequence-like complex-valued ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence) : ( ( Relation-like Function-like ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) set ) ;

theorem :: COMPLSP2:29
for x, y being ( ( Relation-like Function-like FinSequence-like complex-valued ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence) st len x : ( ( Relation-like Function-like FinSequence-like complex-valued ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence) : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) = len y : ( ( Relation-like Function-like FinSequence-like complex-valued ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence) : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) holds
x : ( ( Relation-like Function-like FinSequence-like complex-valued ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence) + y : ( ( Relation-like Function-like FinSequence-like complex-valued ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence) : ( ( Relation-like Function-like ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) set ) = y : ( ( Relation-like Function-like FinSequence-like complex-valued ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence) + x : ( ( Relation-like Function-like FinSequence-like complex-valued ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence) : ( ( Relation-like Function-like ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) set ) ;

theorem :: COMPLSP2:30
for c being ( ( complex ) ( complex ) number )
for x, y being ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) st len x : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) = len y : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) holds
c : ( ( complex ) ( complex ) number ) * (x : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) + y : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) = (c : ( ( complex ) ( complex ) number ) * x : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) + (c : ( ( complex ) ( complex ) number ) * y : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ;

theorem :: COMPLSP2:31
for x, y being ( ( Relation-like Function-like FinSequence-like complex-valued ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence) st len x : ( ( Relation-like Function-like FinSequence-like complex-valued ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence) : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) = len y : ( ( Relation-like Function-like FinSequence-like complex-valued ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence) : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) holds
x : ( ( Relation-like Function-like FinSequence-like complex-valued ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence) - y : ( ( Relation-like Function-like FinSequence-like complex-valued ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence) : ( ( Relation-like Function-like ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) set ) = x : ( ( Relation-like Function-like FinSequence-like complex-valued ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence) + (- y : ( ( Relation-like Function-like FinSequence-like complex-valued ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence) ) : ( ( Relation-like Function-like complex-valued ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) set ) : ( ( Relation-like Function-like ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) set ) ;

theorem :: COMPLSP2:32
for x, y being ( ( Relation-like Function-like FinSequence-like complex-valued ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence) st len x : ( ( Relation-like Function-like FinSequence-like complex-valued ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence) : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) = len y : ( ( Relation-like Function-like FinSequence-like complex-valued ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence) : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) & x : ( ( Relation-like Function-like FinSequence-like complex-valued ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence) + y : ( ( Relation-like Function-like FinSequence-like complex-valued ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence) : ( ( Relation-like Function-like ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) set ) = 0c (len x : ( ( Relation-like Function-like FinSequence-like complex-valued ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence) ) : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) Element of COMPLEX (len b1 : ( ( Relation-like Function-like FinSequence-like complex-valued ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence) ) : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ) holds
( x : ( ( Relation-like Function-like FinSequence-like complex-valued ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence) = - y : ( ( Relation-like Function-like FinSequence-like complex-valued ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence) : ( ( Relation-like Function-like complex-valued ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) set ) & y : ( ( Relation-like Function-like FinSequence-like complex-valued ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence) = - x : ( ( Relation-like Function-like FinSequence-like complex-valued ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence) : ( ( Relation-like Function-like complex-valued ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) set ) ) ;

theorem :: COMPLSP2:33
for x being ( ( Relation-like Function-like FinSequence-like complex-valued ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence) holds x : ( ( Relation-like Function-like FinSequence-like complex-valued ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence) + (0c (len x : ( ( Relation-like Function-like FinSequence-like complex-valued ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence) ) : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) Element of COMPLEX (len b1 : ( ( Relation-like Function-like FinSequence-like complex-valued ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence) ) : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ) : ( ( Relation-like Function-like ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) set ) = x : ( ( Relation-like Function-like FinSequence-like complex-valued ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence) ;

theorem :: COMPLSP2:34
for x being ( ( Relation-like Function-like FinSequence-like complex-valued ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence) holds x : ( ( Relation-like Function-like FinSequence-like complex-valued ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence) + (- x : ( ( Relation-like Function-like FinSequence-like complex-valued ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence) ) : ( ( Relation-like Function-like complex-valued ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) set ) : ( ( Relation-like Function-like ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) set ) = 0c (len x : ( ( Relation-like Function-like FinSequence-like complex-valued ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence) ) : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) Element of COMPLEX (len b1 : ( ( Relation-like Function-like FinSequence-like complex-valued ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence) ) : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ) ;

theorem :: COMPLSP2:35
for x, y being ( ( Relation-like Function-like FinSequence-like complex-valued ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence) st len x : ( ( Relation-like Function-like FinSequence-like complex-valued ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence) : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) = len y : ( ( Relation-like Function-like FinSequence-like complex-valued ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence) : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) holds
- (x : ( ( Relation-like Function-like FinSequence-like complex-valued ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence) + y : ( ( Relation-like Function-like FinSequence-like complex-valued ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence) ) : ( ( Relation-like Function-like ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) set ) : ( ( Relation-like Function-like complex-valued ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) set ) = (- x : ( ( Relation-like Function-like FinSequence-like complex-valued ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence) ) : ( ( Relation-like Function-like complex-valued ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) set ) + (- y : ( ( Relation-like Function-like FinSequence-like complex-valued ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence) ) : ( ( Relation-like Function-like complex-valued ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) set ) : ( ( Relation-like Function-like ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) set ) ;

theorem :: COMPLSP2:36
for x, y, z being ( ( Relation-like Function-like FinSequence-like complex-valued ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence) st len x : ( ( Relation-like Function-like FinSequence-like complex-valued ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence) : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) = len y : ( ( Relation-like Function-like FinSequence-like complex-valued ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence) : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) & len y : ( ( Relation-like Function-like FinSequence-like complex-valued ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence) : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) = len z : ( ( Relation-like Function-like FinSequence-like complex-valued ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence) : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) holds
(x : ( ( Relation-like Function-like FinSequence-like complex-valued ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence) - y : ( ( Relation-like Function-like FinSequence-like complex-valued ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence) ) : ( ( Relation-like Function-like ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) set ) - z : ( ( Relation-like Function-like FinSequence-like complex-valued ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence) : ( ( Relation-like Function-like ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) set ) = x : ( ( Relation-like Function-like FinSequence-like complex-valued ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence) - (y : ( ( Relation-like Function-like FinSequence-like complex-valued ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence) + z : ( ( Relation-like Function-like FinSequence-like complex-valued ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence) ) : ( ( Relation-like Function-like ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) set ) : ( ( Relation-like Function-like ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) set ) ;

theorem :: COMPLSP2:37
for x, y, z being ( ( Relation-like Function-like FinSequence-like complex-valued ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence) st len x : ( ( Relation-like Function-like FinSequence-like complex-valued ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence) : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) = len y : ( ( Relation-like Function-like FinSequence-like complex-valued ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence) : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) & len y : ( ( Relation-like Function-like FinSequence-like complex-valued ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence) : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) = len z : ( ( Relation-like Function-like FinSequence-like complex-valued ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence) : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) holds
x : ( ( Relation-like Function-like FinSequence-like complex-valued ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence) + (y : ( ( Relation-like Function-like FinSequence-like complex-valued ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence) - z : ( ( Relation-like Function-like FinSequence-like complex-valued ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence) ) : ( ( Relation-like Function-like ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) set ) : ( ( Relation-like Function-like ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) set ) = (x : ( ( Relation-like Function-like FinSequence-like complex-valued ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence) + y : ( ( Relation-like Function-like FinSequence-like complex-valued ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence) ) : ( ( Relation-like Function-like ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) set ) - z : ( ( Relation-like Function-like FinSequence-like complex-valued ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence) : ( ( Relation-like Function-like ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) set ) ;

theorem :: COMPLSP2:38
canceled;

theorem :: COMPLSP2:39
for x, y being ( ( Relation-like Function-like FinSequence-like complex-valued ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence) st len x : ( ( Relation-like Function-like FinSequence-like complex-valued ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence) : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) = len y : ( ( Relation-like Function-like FinSequence-like complex-valued ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence) : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) holds
- (x : ( ( Relation-like Function-like FinSequence-like complex-valued ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence) - y : ( ( Relation-like Function-like FinSequence-like complex-valued ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence) ) : ( ( Relation-like Function-like ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) set ) : ( ( Relation-like Function-like complex-valued ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) set ) = (- x : ( ( Relation-like Function-like FinSequence-like complex-valued ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence) ) : ( ( Relation-like Function-like complex-valued ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) set ) + y : ( ( Relation-like Function-like FinSequence-like complex-valued ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence) : ( ( Relation-like Function-like ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) set ) ;

theorem :: COMPLSP2:40
for x, y, z being ( ( Relation-like Function-like FinSequence-like complex-valued ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence) st len x : ( ( Relation-like Function-like FinSequence-like complex-valued ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence) : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) = len y : ( ( Relation-like Function-like FinSequence-like complex-valued ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence) : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) & len y : ( ( Relation-like Function-like FinSequence-like complex-valued ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence) : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) = len z : ( ( Relation-like Function-like FinSequence-like complex-valued ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence) : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) holds
x : ( ( Relation-like Function-like FinSequence-like complex-valued ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence) - (y : ( ( Relation-like Function-like FinSequence-like complex-valued ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence) - z : ( ( Relation-like Function-like FinSequence-like complex-valued ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence) ) : ( ( Relation-like Function-like ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) set ) : ( ( Relation-like Function-like ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) set ) = (x : ( ( Relation-like Function-like FinSequence-like complex-valued ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence) - y : ( ( Relation-like Function-like FinSequence-like complex-valued ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence) ) : ( ( Relation-like Function-like ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) set ) + z : ( ( Relation-like Function-like FinSequence-like complex-valued ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence) : ( ( Relation-like Function-like ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) set ) ;

theorem :: COMPLSP2:41
for x being ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) )
for c being ( ( complex ) ( complex ) number ) holds c : ( ( complex ) ( complex ) number ) * (0c (len x : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ) : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) Element of COMPLEX (len b1 : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ) : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) = 0c (len x : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ) : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) Element of COMPLEX (len b1 : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ) : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ) ;

theorem :: COMPLSP2:42
for x being ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) )
for c being ( ( complex ) ( complex ) number ) holds - (c : ( ( complex ) ( complex ) number ) * x : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) = c : ( ( complex ) ( complex ) number ) * (- x : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ;

theorem :: COMPLSP2:43
for c being ( ( complex ) ( complex ) number )
for x, y being ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) st len x : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) = len y : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) holds
c : ( ( complex ) ( complex ) number ) * (x : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) - y : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) = (c : ( ( complex ) ( complex ) number ) * x : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) - (c : ( ( complex ) ( complex ) number ) * y : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ;

theorem :: COMPLSP2:44
for x1, y1 being ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) )
for x2, y2 being ( ( ) ( complex V34() ext-real ) Real) st x1 : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) = x2 : ( ( ) ( complex V34() ext-real ) Real) & y1 : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) = y2 : ( ( ) ( complex V34() ext-real ) Real) holds
addcomplex : ( ( Function-like V18([:COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ,COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) :] : ( ( ) ( complex-valued ) set ) , COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ) ( Relation-like [:COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ,COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) :] : ( ( ) ( complex-valued ) set ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V14([:COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ,COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) :] : ( ( ) ( complex-valued ) set ) ) V18([:COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ,COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) :] : ( ( ) ( complex-valued ) set ) , COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) V25( COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) complex-valued V145( COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) V146( COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ) Element of bool [:[:COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ,COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) :] : ( ( ) ( complex-valued ) set ) ,COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) :] : ( ( ) ( complex-valued ) set ) : ( ( ) ( ) set ) ) . (x1 : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ,y1 : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) = addreal : ( ( Function-like V18([:REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ,REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) , REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ) ) ( Relation-like [:REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ,REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) -defined REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) -valued Function-like V14([:REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ,REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) V18([:REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ,REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) , REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ) V25( REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ) complex-valued ext-real-valued real-valued V145( REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ) V146( REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ) ) Element of bool [:[:REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ,REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ,REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) ) . (x2 : ( ( ) ( complex V34() ext-real ) Real) ,y2 : ( ( ) ( complex V34() ext-real ) Real) ) : ( ( ) ( complex V34() ext-real ) Element of REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ) ;

theorem :: COMPLSP2:45
for C being ( ( Function-like V18([:COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ,COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) :] : ( ( ) ( complex-valued ) set ) , COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ) ( Relation-like [:COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ,COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) :] : ( ( ) ( complex-valued ) set ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V14([:COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ,COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) :] : ( ( ) ( complex-valued ) set ) ) V18([:COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ,COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) :] : ( ( ) ( complex-valued ) set ) , COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) complex-valued ) Function of [:COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ,COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) :] : ( ( ) ( complex-valued ) set ) , COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) )
for G being ( ( Function-like V18([:REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ,REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) , REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ) ) ( Relation-like [:REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ,REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) -defined REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) -valued Function-like V14([:REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ,REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) V18([:REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ,REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) , REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ) complex-valued ext-real-valued real-valued ) Function of [:REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ,REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) , REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) )
for x1, y1 being ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) )
for x2, y2 being ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) FinSequence of REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ) st x1 : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) = x2 : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) FinSequence of REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ) & y1 : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) = y2 : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) FinSequence of REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ) & len x1 : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) = len y2 : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) FinSequence of REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ) : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) & ( for i being ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) st i : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) in dom x1 : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) : ( ( ) ( V64() V65() V66() V67() V68() V69() ) Element of bool NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) holds
C : ( ( Function-like V18([:COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ,COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) :] : ( ( ) ( complex-valued ) set ) , COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ) ( Relation-like [:COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ,COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) :] : ( ( ) ( complex-valued ) set ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V14([:COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ,COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) :] : ( ( ) ( complex-valued ) set ) ) V18([:COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ,COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) :] : ( ( ) ( complex-valued ) set ) , COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) complex-valued ) Function of [:COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ,COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) :] : ( ( ) ( complex-valued ) set ) , COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) . ((x1 : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) . i : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ,(y1 : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) . i : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) = G : ( ( Function-like V18([:REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ,REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) , REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ) ) ( Relation-like [:REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ,REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) -defined REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) -valued Function-like V14([:REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ,REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) V18([:REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ,REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) , REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ) complex-valued ext-real-valued real-valued ) Function of [:REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ,REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) , REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ) . ((x2 : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) FinSequence of REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ) . i : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex V34() ext-real ) Element of REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ) ,(y2 : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) FinSequence of REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ) . i : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex V34() ext-real ) Element of REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ) ) : ( ( ) ( complex V34() ext-real ) Element of REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ) ) holds
C : ( ( Function-like V18([:COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ,COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) :] : ( ( ) ( complex-valued ) set ) , COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ) ( Relation-like [:COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ,COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) :] : ( ( ) ( complex-valued ) set ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V14([:COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ,COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) :] : ( ( ) ( complex-valued ) set ) ) V18([:COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ,COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) :] : ( ( ) ( complex-valued ) set ) , COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) complex-valued ) Function of [:COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ,COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) :] : ( ( ) ( complex-valued ) set ) , COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) .: (x1 : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ,y1 : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) = G : ( ( Function-like V18([:REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ,REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) , REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ) ) ( Relation-like [:REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ,REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) -defined REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) -valued Function-like V14([:REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ,REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) V18([:REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ,REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) , REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ) complex-valued ext-real-valued real-valued ) Function of [:REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ,REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) , REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ) .: (x2 : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) FinSequence of REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ) ,y2 : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) FinSequence of REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) FinSequence of REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ) ;

theorem :: COMPLSP2:46
for x1, y1 being ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) )
for x2, y2 being ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) FinSequence of REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ) st x1 : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) = x2 : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) FinSequence of REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ) & y1 : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) = y2 : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) FinSequence of REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ) & len x1 : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) = len y2 : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) FinSequence of REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ) : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) holds
addcomplex : ( ( Function-like V18([:COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ,COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) :] : ( ( ) ( complex-valued ) set ) , COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ) ( Relation-like [:COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ,COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) :] : ( ( ) ( complex-valued ) set ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V14([:COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ,COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) :] : ( ( ) ( complex-valued ) set ) ) V18([:COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ,COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) :] : ( ( ) ( complex-valued ) set ) , COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) V25( COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) complex-valued V145( COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) V146( COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ) Element of bool [:[:COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ,COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) :] : ( ( ) ( complex-valued ) set ) ,COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) :] : ( ( ) ( complex-valued ) set ) : ( ( ) ( ) set ) ) .: (x1 : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ,y1 : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) = addreal : ( ( Function-like V18([:REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ,REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) , REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ) ) ( Relation-like [:REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ,REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) -defined REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) -valued Function-like V14([:REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ,REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) V18([:REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ,REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) , REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ) V25( REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ) complex-valued ext-real-valued real-valued V145( REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ) V146( REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ) ) Element of bool [:[:REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ,REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ,REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) ) .: (x2 : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) FinSequence of REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ) ,y2 : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) FinSequence of REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) FinSequence of REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ) ;

theorem :: COMPLSP2:47
for x1, y1 being ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) )
for x2, y2 being ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) FinSequence of REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ) st x1 : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) = x2 : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) FinSequence of REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ) & y1 : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) = y2 : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) FinSequence of REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ) & len x1 : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) = len y2 : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) FinSequence of REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ) : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) holds
x1 : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) + y1 : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) = x2 : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) FinSequence of REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ) + y2 : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) FinSequence of REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) FinSequence of REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ) ;

theorem :: COMPLSP2:48
for x being ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) holds
( len (Re x : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) FinSequence of REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ) : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) = len x : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) & len (Im x : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) FinSequence of REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ) : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) = len x : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) ) ;

theorem :: COMPLSP2:49
for x, y being ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) st len x : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) = len y : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) holds
( Re (x : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) + y : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) FinSequence of REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ) = (Re x : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) FinSequence of REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ) + (Re y : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) FinSequence of REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) FinSequence of REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ) & Im (x : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) + y : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) FinSequence of REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ) = (Im x : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) FinSequence of REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ) + (Im y : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) FinSequence of REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) FinSequence of REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ) ) ;

theorem :: COMPLSP2:50
for x1, y1 being ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) )
for x2, y2 being ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) FinSequence of REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ) st x1 : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) = x2 : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) FinSequence of REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ) & y1 : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) = y2 : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) FinSequence of REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ) & len x1 : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) = len y2 : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) FinSequence of REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ) : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) holds
diffcomplex : ( ( Function-like V18([:COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ,COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) :] : ( ( ) ( complex-valued ) set ) , COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ) ( Relation-like [:COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ,COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) :] : ( ( ) ( complex-valued ) set ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V14([:COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ,COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) :] : ( ( ) ( complex-valued ) set ) ) V18([:COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ,COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) :] : ( ( ) ( complex-valued ) set ) , COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) complex-valued ) Element of bool [:[:COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ,COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) :] : ( ( ) ( complex-valued ) set ) ,COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) :] : ( ( ) ( complex-valued ) set ) : ( ( ) ( ) set ) ) .: (x1 : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ,y1 : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) = diffreal : ( ( Function-like V18([:REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ,REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) , REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ) ) ( Relation-like [:REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ,REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) -defined REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) -valued Function-like V14([:REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ,REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) V18([:REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ,REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) , REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ) complex-valued ext-real-valued real-valued ) Element of bool [:[:REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ,REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ,REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) ) .: (x2 : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) FinSequence of REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ) ,y2 : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) FinSequence of REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) FinSequence of REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ) ;

theorem :: COMPLSP2:51
for x1, y1 being ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) )
for x2, y2 being ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) FinSequence of REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ) st x1 : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) = x2 : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) FinSequence of REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ) & y1 : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) = y2 : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) FinSequence of REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ) & len x1 : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) = len y2 : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) FinSequence of REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ) : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) holds
x1 : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) - y1 : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) = x2 : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) FinSequence of REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ) - y2 : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) FinSequence of REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) FinSequence of REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ) ;

theorem :: COMPLSP2:52
for x, y being ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) st len x : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) = len y : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) holds
( Re (x : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) - y : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) FinSequence of REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ) = (Re x : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) FinSequence of REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ) - (Re y : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) FinSequence of REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) FinSequence of REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ) & Im (x : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) - y : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) FinSequence of REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ) = (Im x : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) FinSequence of REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ) - (Im y : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) FinSequence of REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) FinSequence of REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ) ) ;

theorem :: COMPLSP2:53
for z being ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) )
for a, b being ( ( complex ) ( complex ) number ) holds a : ( ( complex ) ( complex ) number ) * (b : ( ( complex ) ( complex ) number ) * z : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) = (a : ( ( complex ) ( complex ) number ) * b : ( ( complex ) ( complex ) number ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) * z : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ;

theorem :: COMPLSP2:54
for x being ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) )
for c being ( ( complex ) ( complex ) number ) holds (- c : ( ( complex ) ( complex ) number ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) * x : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) = - (c : ( ( complex ) ( complex ) number ) * x : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ;

theorem :: COMPLSP2:55
for h being ( ( Function-like V18( COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) , COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ) ( Relation-like COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like non zero V14( COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) V18( COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) , COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) complex-valued ) Function of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) , COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) )
for g being ( ( Function-like V18( REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) , REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) -defined REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) -valued Function-like non zero V14( REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ) V18( REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) , REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ) complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) , REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) )
for y1 being ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) )
for y2 being ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) FinSequence of REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ) st len y1 : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) = len y2 : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) FinSequence of REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ) : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) & ( for i being ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) st i : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) in dom y1 : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) : ( ( ) ( V64() V65() V66() V67() V68() V69() ) Element of bool NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) holds
h : ( ( Function-like V18( COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) , COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ) ( Relation-like COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like non zero V14( COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) V18( COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) , COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) complex-valued ) Function of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) , COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) . (y1 : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) . i : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) = g : ( ( Function-like V18( REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) , REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) -defined REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) -valued Function-like non zero V14( REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ) V18( REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) , REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ) complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) , REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ) . (y2 : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) FinSequence of REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ) . i : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex V34() ext-real ) Element of REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ) : ( ( ) ( complex V34() ext-real ) Element of REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ) ) holds
h : ( ( Function-like V18( COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) , COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ) ( Relation-like COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like non zero V14( COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) V18( COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) , COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) complex-valued ) Function of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) , COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) * y1 : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) = g : ( ( Function-like V18( REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) , REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) -defined REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) -valued Function-like non zero V14( REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ) V18( REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) , REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ) complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) , REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ) * y2 : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) FinSequence of REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) FinSequence of REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ) ;

theorem :: COMPLSP2:56
for y1 being ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) )
for y2 being ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) FinSequence of REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ) st y1 : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) = y2 : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) FinSequence of REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ) & len y1 : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) = len y2 : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) FinSequence of REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ) : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) holds
compcomplex : ( ( Function-like V18( COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) , COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ) ( Relation-like COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like non zero V14( COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) V18( COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) , COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) complex-valued ) Element of bool [:COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ,COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) :] : ( ( ) ( complex-valued ) set ) : ( ( ) ( ) set ) ) * y1 : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) = compreal : ( ( Function-like V18( REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) , REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) -defined REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) -valued Function-like non zero V14( REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ) V18( REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) , REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ) complex-valued ext-real-valued real-valued ) Element of bool [:REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ,REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) ) * y2 : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) FinSequence of REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) FinSequence of REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ) ;

theorem :: COMPLSP2:57
for y1 being ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) )
for y2 being ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) FinSequence of REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ) st y1 : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) = y2 : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) FinSequence of REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ) & len y1 : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) = len y2 : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) FinSequence of REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ) : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) holds
- y1 : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) = - y2 : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) FinSequence of REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) FinSequence of REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ) ;

theorem :: COMPLSP2:58
for x being ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) holds
( Re (<i> : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) * x : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) FinSequence of REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ) = - (Im x : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) FinSequence of REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) FinSequence of REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ) & Im (<i> : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) * x : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) FinSequence of REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ) = Re x : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) FinSequence of REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ) ) ;

theorem :: COMPLSP2:59
for x, y being ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) st len x : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) = len y : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) holds
|((<i> : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) * x : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ,y : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) )| : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) = <i> : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) * |(x : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ,y : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) )| : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ;

theorem :: COMPLSP2:60
for x, y being ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) st len x : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) = len y : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) holds
|(x : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ,(<i> : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) * y : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) )| : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) = - (<i> : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) * |(x : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ,y : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) )| : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ;

theorem :: COMPLSP2:61
for a1 being ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) )
for y1 being ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) )
for a2 being ( ( ) ( complex V34() ext-real ) Element of REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) )
for y2 being ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) FinSequence of REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ) st a1 : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) = a2 : ( ( ) ( complex V34() ext-real ) Element of REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ) & y1 : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) = y2 : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) FinSequence of REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ) & len y1 : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) = len y2 : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) FinSequence of REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ) : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) holds
(a1 : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) multcomplex) : ( ( Function-like V18( COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) , COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ) ( Relation-like COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like non zero V14( COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) V18( COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) , COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) complex-valued ) Element of bool [:COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ,COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) :] : ( ( ) ( complex-valued ) set ) : ( ( ) ( ) set ) ) * y1 : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) = (a2 : ( ( ) ( complex V34() ext-real ) Element of REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ) multreal) : ( ( Function-like V18( REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) , REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) -defined REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) -valued Function-like non zero V14( REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ) V18( REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) , REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ) complex-valued ext-real-valued real-valued ) Element of bool [:REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ,REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) :] : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) : ( ( ) ( ) set ) ) * y2 : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) FinSequence of REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) FinSequence of REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ) ;

theorem :: COMPLSP2:62
for a1 being ( ( complex ) ( complex ) number )
for y1 being ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) )
for a2 being ( ( ) ( complex V34() ext-real ) Element of REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) )
for y2 being ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) FinSequence of REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ) st a1 : ( ( complex ) ( complex ) number ) = a2 : ( ( ) ( complex V34() ext-real ) Element of REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ) & y1 : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) = y2 : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) FinSequence of REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ) & len y1 : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) = len y2 : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) FinSequence of REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ) : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) holds
a1 : ( ( complex ) ( complex ) number ) * y1 : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) = a2 : ( ( ) ( complex V34() ext-real ) Element of REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ) * y2 : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) FinSequence of REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) FinSequence of REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ) ;

theorem :: COMPLSP2:63
for z being ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) )
for a, b being ( ( complex ) ( complex ) number ) holds (a : ( ( complex ) ( complex ) number ) + b : ( ( complex ) ( complex ) number ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) * z : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) = (a : ( ( complex ) ( complex ) number ) * z : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) + (b : ( ( complex ) ( complex ) number ) * z : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ;

theorem :: COMPLSP2:64
for z being ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) )
for a, b being ( ( complex ) ( complex ) number ) holds (a : ( ( complex ) ( complex ) number ) - b : ( ( complex ) ( complex ) number ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) * z : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) = (a : ( ( complex ) ( complex ) number ) * z : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) - (b : ( ( complex ) ( complex ) number ) * z : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ;

theorem :: COMPLSP2:65
for a being ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) )
for x being ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) holds
( Re (a : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) * x : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) FinSequence of REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ) = ((Re a : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ) : ( ( ) ( complex V34() ext-real ) Element of REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ) * (Re x : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) FinSequence of REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) FinSequence of REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ) - ((Im a : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ) : ( ( ) ( complex V34() ext-real ) Element of REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ) * (Im x : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) FinSequence of REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) FinSequence of REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) FinSequence of REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ) & Im (a : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) * x : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) FinSequence of REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ) = ((Im a : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ) : ( ( ) ( complex V34() ext-real ) Element of REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ) * (Re x : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) FinSequence of REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) FinSequence of REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ) + ((Re a : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ) : ( ( ) ( complex V34() ext-real ) Element of REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ) * (Im x : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) FinSequence of REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) FinSequence of REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) FinSequence of REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ) ) ;

begin

theorem :: COMPLSP2:66
for x1, x2, y being ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) st len x1 : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) = len x2 : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) & len x2 : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) = len y : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) holds
|((x1 : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) + x2 : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ,y : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) )| : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) = |(x1 : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ,y : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) )| : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) + |(x2 : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ,y : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) )| : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ;

theorem :: COMPLSP2:67
for x1, x2 being ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) st len x1 : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) = len x2 : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) holds
|((- x1 : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ,x2 : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) )| : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) = - |(x1 : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ,x2 : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) )| : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ;

theorem :: COMPLSP2:68
for x1, x2 being ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) st len x1 : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) = len x2 : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) holds
|(x1 : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ,(- x2 : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) )| : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) = - |(x1 : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ,x2 : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) )| : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ;

theorem :: COMPLSP2:69
for x1, x2 being ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) st len x1 : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) = len x2 : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) holds
|((- x1 : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ,(- x2 : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) )| : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) = |(x1 : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ,x2 : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) )| : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ;

theorem :: COMPLSP2:70
for x1, x2, x3 being ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) st len x1 : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) = len x2 : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) & len x2 : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) = len x3 : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) holds
|((x1 : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) - x2 : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ,x3 : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) )| : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) = |(x1 : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ,x3 : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) )| : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) - |(x2 : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ,x3 : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) )| : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ;

theorem :: COMPLSP2:71
for x, y1, y2 being ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) st len x : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) = len y1 : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) & len y1 : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) = len y2 : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) holds
|(x : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ,(y1 : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) + y2 : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) )| : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) = |(x : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ,y1 : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) )| : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) + |(x : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ,y2 : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) )| : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ;

theorem :: COMPLSP2:72
for x, y1, y2 being ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) st len x : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) = len y1 : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) & len y1 : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) = len y2 : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) holds
|(x : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ,(y1 : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) - y2 : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) )| : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) = |(x : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ,y1 : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) )| : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) - |(x : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ,y2 : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) )| : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ;

theorem :: COMPLSP2:73
for x1, x2, y1, y2 being ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) st len x1 : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) = len x2 : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) & len x2 : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) = len y1 : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) & len y1 : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) = len y2 : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) holds
|((x1 : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) + x2 : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ,(y1 : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) + y2 : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) )| : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) = ((|(x1 : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ,y1 : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) )| : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) + |(x1 : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ,y2 : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) )| : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) + |(x2 : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ,y1 : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) )| : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) + |(x2 : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ,y2 : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) )| : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ;

theorem :: COMPLSP2:74
for x1, x2, y1, y2 being ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) st len x1 : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) = len x2 : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) & len x2 : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) = len y1 : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) & len y1 : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) = len y2 : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) holds
|((x1 : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) - x2 : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ,(y1 : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) - y2 : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) )| : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) = ((|(x1 : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ,y1 : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) )| : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) - |(x1 : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ,y2 : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) )| : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) - |(x2 : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ,y1 : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) )| : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) + |(x2 : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ,y2 : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) )| : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ;

theorem :: COMPLSP2:75
for x, y being ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) holds |(x : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ,y : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) )| : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) = |(y : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ,x : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) )| : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) *' : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ;

theorem :: COMPLSP2:76
for x, y being ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) st len x : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) = len y : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) holds
|((x : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) + y : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ,(x : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) + y : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) )| : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) = (|(x : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ,x : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) )| : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) + (2 : ( ( ) ( non zero natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) * (Re |(x : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ,y : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) )| : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ) : ( ( ) ( complex V34() ext-real ) Element of REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ) ) : ( ( ) ( complex V34() ext-real ) Element of REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) + |(y : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ,y : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) )| : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ;

theorem :: COMPLSP2:77
for x, y being ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) st len x : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) = len y : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) holds
|((x : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) - y : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ,(x : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) - y : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) )| : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) = (|(x : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ,x : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) )| : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) - (2 : ( ( ) ( non zero natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) * (Re |(x : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ,y : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) )| : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ) : ( ( ) ( complex V34() ext-real ) Element of REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ) ) : ( ( ) ( complex V34() ext-real ) Element of REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) + |(y : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ,y : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) )| : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ;

theorem :: COMPLSP2:78
for a being ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) )
for x, y being ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) st len x : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) = len y : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) holds
|((a : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) * x : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ,y : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) )| : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) = a : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) * |(x : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ,y : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) )| : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ;

theorem :: COMPLSP2:79
for a being ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) )
for x, y being ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) st len x : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) = len y : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) holds
|(x : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ,(a : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) * y : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) )| : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) = (a : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) *') : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) * |(x : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ,y : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) )| : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ;

theorem :: COMPLSP2:80
for a, b being ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) )
for x, y, z being ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) st len x : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) = len y : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) & len y : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) = len z : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) holds
|(((a : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) * x : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) + (b : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) * y : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ,z : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) )| : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) = (a : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) * |(x : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ,z : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) )| : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) + (b : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) * |(y : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ,z : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) )| : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ;

theorem :: COMPLSP2:81
for a, b being ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) )
for x, y, z being ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) st len x : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) = len y : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) & len y : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) = len z : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) : ( ( ) ( natural complex V34() V35() V36() ext-real V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) ) holds
|(x : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ,((a : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) * y : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) + (b : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) * z : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) )| : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) = ((a : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) *') : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) * |(x : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ,y : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) )| : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) + ((b : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) *') : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) * |(x : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ,z : ( ( ) ( Relation-like NAT : ( ( ) ( V64() V65() V66() V67() V68() V69() V70() ) Element of bool REAL : ( ( ) ( non zero V39() V64() V65() V66() V70() ) set ) : ( ( ) ( ) set ) ) -defined COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) -valued Function-like V39() FinSequence-like FinSubsequence-like complex-valued ) FinSequence of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) )| : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non zero V39() V64() V70() ) set ) ) ;