begin
definition
let C be ( ( non
empty ) ( non
empty )
set ) ;
let f1,
f2 be ( (
Function-like ) (
V1()
V4(
C : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
PartFunc of ,) ;
func f1 / f2 -> ( (
Function-like ) (
V1()
V4(
C : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
PartFunc of ,)
means
(
dom it : ( (
Function-like ) (
V1()
V4(
C : ( ( non
empty ) ( non
empty )
set ) )
V5(
f1 : ( (
Function-like V30(
C : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) ) (
V1()
V4(
C : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V30(
C : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
V38() )
Element of
K19(
K20(
C : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) ) )
V5(
RAT : ( ( ) ( non
empty V48()
V49()
V50()
V51()
V54()
V55() )
set ) )
Function-like V38()
V39()
V40() )
Element of
K19(
K20(
C : ( ( non
empty ) ( non
empty )
set ) ,
f1 : ( (
Function-like V30(
C : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) ) (
V1()
V4(
C : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V30(
C : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
V38() )
Element of
K19(
K20(
C : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
Element of
K19(
C : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) )
= (dom f1 : ( ( Function-like V30(C : ( ( non empty ) ( non empty ) set ) , COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) ) ( V1() V4(C : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V30(C : ( ( non empty ) ( non empty ) set ) , COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) V38() ) Element of K19(K20(C : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) : ( ( ) ( V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( )
Element of
K19(
C : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) )
/\ ((dom f2 : ( ( Function-like ) ( V1() V4(C : ( ( non empty ) ( non empty ) set ) ) V5(f1 : ( ( Function-like V30(C : ( ( non empty ) ( non empty ) set ) , COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) ) ( V1() V4(C : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V30(C : ( ( non empty ) ( non empty ) set ) , COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) V38() ) Element of K19(K20(C : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) : ( ( ) ( V38() ) set ) ) : ( ( ) ( ) set ) ) ) V5( INT : ( ( ) ( non empty V48() V49() V50() V51() V52() V54() V55() ) set ) ) Function-like V38() V39() V40() ) Element of K19(K20(C : ( ( non empty ) ( non empty ) set ) ,f1 : ( ( Function-like V30(C : ( ( non empty ) ( non empty ) set ) , COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) ) ( V1() V4(C : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V30(C : ( ( non empty ) ( non empty ) set ) , COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) V38() ) Element of K19(K20(C : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) : ( ( ) ( V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) Element of K19(C : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) \ (f2 : ( ( Function-like ) ( V1() V4(C : ( ( non empty ) ( non empty ) set ) ) V5(f1 : ( ( Function-like V30(C : ( ( non empty ) ( non empty ) set ) , COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) ) ( V1() V4(C : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V30(C : ( ( non empty ) ( non empty ) set ) , COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) V38() ) Element of K19(K20(C : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) : ( ( ) ( V38() ) set ) ) : ( ( ) ( ) set ) ) ) V5( INT : ( ( ) ( non empty V48() V49() V50() V51() V52() V54() V55() ) set ) ) Function-like V38() V39() V40() ) Element of K19(K20(C : ( ( non empty ) ( non empty ) set ) ,f1 : ( ( Function-like V30(C : ( ( non empty ) ( non empty ) set ) , COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) ) ( V1() V4(C : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V30(C : ( ( non empty ) ( non empty ) set ) , COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) V38() ) Element of K19(K20(C : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) : ( ( ) ( V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) " {0 : ( ( ) ( ext-real natural complex real V36() V37() V48() V49() V50() V51() V52() V53() ) Element of NAT : ( ( ) ( V48() V49() V50() V51() V52() V53() V54() ) Element of K19(REAL : ( ( ) ( non empty V48() V49() V50() V54() V55() ) set ) ) : ( ( ) ( ) set ) ) ) } : ( ( ) ( V48() V49() V50() V51() V52() V53() ) set ) ) : ( ( ) ( ) Element of K19(C : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( )
Element of
K19(
C : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
Element of
K19(
C : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) & ( for
c being ( ( ) ( )
Element of
C : ( ( non
empty ) ( non
empty )
set ) ) st
c : ( ( ) ( )
Element of
C : ( ( non
empty ) ( non
empty )
set ) )
in dom it : ( (
Function-like ) (
V1()
V4(
C : ( ( non
empty ) ( non
empty )
set ) )
V5(
f1 : ( (
Function-like V30(
C : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) ) (
V1()
V4(
C : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V30(
C : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
V38() )
Element of
K19(
K20(
C : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) ) )
V5(
RAT : ( ( ) ( non
empty V48()
V49()
V50()
V51()
V54()
V55() )
set ) )
Function-like V38()
V39()
V40() )
Element of
K19(
K20(
C : ( ( non
empty ) ( non
empty )
set ) ,
f1 : ( (
Function-like V30(
C : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) ) (
V1()
V4(
C : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V30(
C : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
V38() )
Element of
K19(
K20(
C : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
Element of
K19(
C : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) holds
it : ( (
Function-like ) (
V1()
V4(
C : ( ( non
empty ) ( non
empty )
set ) )
V5(
f1 : ( (
Function-like V30(
C : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) ) (
V1()
V4(
C : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V30(
C : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
V38() )
Element of
K19(
K20(
C : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) ) )
V5(
RAT : ( ( ) ( non
empty V48()
V49()
V50()
V51()
V54()
V55() )
set ) )
Function-like V38()
V39()
V40() )
Element of
K19(
K20(
C : ( ( non
empty ) ( non
empty )
set ) ,
f1 : ( (
Function-like V30(
C : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) ) (
V1()
V4(
C : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V30(
C : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
V38() )
Element of
K19(
K20(
C : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
/. c : ( ( ) ( )
Element of
C : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) (
complex )
Element of
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
= (f1 : ( ( Function-like V30(C : ( ( non empty ) ( non empty ) set ) , COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) ) ( V1() V4(C : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V30(C : ( ( non empty ) ( non empty ) set ) , COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) V38() ) Element of K19(K20(C : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) : ( ( ) ( V38() ) set ) ) : ( ( ) ( ) set ) ) /. c : ( ( ) ( ) Element of C : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) (
complex )
Element of
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
* ((f2 : ( ( Function-like ) ( V1() V4(C : ( ( non empty ) ( non empty ) set ) ) V5(f1 : ( ( Function-like V30(C : ( ( non empty ) ( non empty ) set ) , COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) ) ( V1() V4(C : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V30(C : ( ( non empty ) ( non empty ) set ) , COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) V38() ) Element of K19(K20(C : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) : ( ( ) ( V38() ) set ) ) : ( ( ) ( ) set ) ) ) V5( INT : ( ( ) ( non empty V48() V49() V50() V51() V52() V54() V55() ) set ) ) Function-like V38() V39() V40() ) Element of K19(K20(C : ( ( non empty ) ( non empty ) set ) ,f1 : ( ( Function-like V30(C : ( ( non empty ) ( non empty ) set ) , COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) ) ( V1() V4(C : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V30(C : ( ( non empty ) ( non empty ) set ) , COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) V38() ) Element of K19(K20(C : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) : ( ( ) ( V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) /. c : ( ( ) ( ) Element of C : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) ") : ( ( ) (
complex )
Element of
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
complex )
Element of
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) ) );
end;
definition
let C be ( ( non
empty ) ( non
empty )
set ) ;
let f be ( (
Function-like ) (
V1()
V4(
C : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
PartFunc of ,) ;
func f ^ -> ( (
Function-like ) (
V1()
V4(
C : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
PartFunc of ,)
means
(
dom it : ( (
Function-like ) (
V1()
V4(
C : ( ( non
empty ) ( non
empty )
set ) )
V5(
f : ( (
Function-like V30(
C : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) ) (
V1()
V4(
C : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V30(
C : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
V38() )
Element of
K19(
K20(
C : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) ) )
V5(
INT : ( ( ) ( non
empty V48()
V49()
V50()
V51()
V52()
V54()
V55() )
set ) )
Function-like V38()
V39()
V40() )
Element of
K19(
K20(
C : ( ( non
empty ) ( non
empty )
set ) ,
f : ( (
Function-like V30(
C : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) ) (
V1()
V4(
C : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V30(
C : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
V38() )
Element of
K19(
K20(
C : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
Element of
K19(
C : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) )
= (dom f : ( ( Function-like V30(C : ( ( non empty ) ( non empty ) set ) , COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) ) ( V1() V4(C : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V30(C : ( ( non empty ) ( non empty ) set ) , COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) V38() ) Element of K19(K20(C : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) : ( ( ) ( V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( )
Element of
K19(
C : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) )
\ (f : ( ( Function-like V30(C : ( ( non empty ) ( non empty ) set ) , COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) ) ( V1() V4(C : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V30(C : ( ( non empty ) ( non empty ) set ) , COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) V38() ) Element of K19(K20(C : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) : ( ( ) ( V38() ) set ) ) : ( ( ) ( ) set ) ) " {0 : ( ( ) ( ext-real natural complex real V36() V37() V48() V49() V50() V51() V52() V53() ) Element of NAT : ( ( ) ( V48() V49() V50() V51() V52() V53() V54() ) Element of K19(REAL : ( ( ) ( non empty V48() V49() V50() V54() V55() ) set ) ) : ( ( ) ( ) set ) ) ) } : ( ( ) ( V48() V49() V50() V51() V52() V53() ) set ) ) : ( ( ) ( )
Element of
K19(
C : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
Element of
K19(
C : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) & ( for
c being ( ( ) ( )
Element of
C : ( ( non
empty ) ( non
empty )
set ) ) st
c : ( ( ) ( )
Element of
C : ( ( non
empty ) ( non
empty )
set ) )
in dom it : ( (
Function-like ) (
V1()
V4(
C : ( ( non
empty ) ( non
empty )
set ) )
V5(
f : ( (
Function-like V30(
C : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) ) (
V1()
V4(
C : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V30(
C : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
V38() )
Element of
K19(
K20(
C : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) ) )
V5(
INT : ( ( ) ( non
empty V48()
V49()
V50()
V51()
V52()
V54()
V55() )
set ) )
Function-like V38()
V39()
V40() )
Element of
K19(
K20(
C : ( ( non
empty ) ( non
empty )
set ) ,
f : ( (
Function-like V30(
C : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) ) (
V1()
V4(
C : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V30(
C : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
V38() )
Element of
K19(
K20(
C : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
Element of
K19(
C : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) holds
it : ( (
Function-like ) (
V1()
V4(
C : ( ( non
empty ) ( non
empty )
set ) )
V5(
f : ( (
Function-like V30(
C : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) ) (
V1()
V4(
C : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V30(
C : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
V38() )
Element of
K19(
K20(
C : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) ) )
V5(
INT : ( ( ) ( non
empty V48()
V49()
V50()
V51()
V52()
V54()
V55() )
set ) )
Function-like V38()
V39()
V40() )
Element of
K19(
K20(
C : ( ( non
empty ) ( non
empty )
set ) ,
f : ( (
Function-like V30(
C : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) ) (
V1()
V4(
C : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V30(
C : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
V38() )
Element of
K19(
K20(
C : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
/. c : ( ( ) ( )
Element of
C : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) (
complex )
Element of
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
= (f : ( ( Function-like V30(C : ( ( non empty ) ( non empty ) set ) , COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) ) ( V1() V4(C : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V30(C : ( ( non empty ) ( non empty ) set ) , COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) V38() ) Element of K19(K20(C : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) : ( ( ) ( V38() ) set ) ) : ( ( ) ( ) set ) ) /. c : ( ( ) ( ) Element of C : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) (
complex )
Element of
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
" : ( ( ) (
complex )
Element of
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) ) );
end;
theorem
for
C being ( ( non
empty ) ( non
empty )
set )
for
f1,
f2 being ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
PartFunc of ,) holds
(
dom (f1 : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) + f2 : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) ) : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
Element of
K19(
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) )
= (dom f1 : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) ) : ( ( ) ( )
Element of
K19(
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) )
/\ (dom f2 : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) ) : ( ( ) ( )
Element of
K19(
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
Element of
K19(
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) & ( for
c being ( ( ) ( )
Element of
C : ( ( non
empty ) ( non
empty )
set ) ) st
c : ( ( ) ( )
Element of
b1 : ( ( non
empty ) ( non
empty )
set ) )
in dom (f1 : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) + f2 : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) ) : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
Element of
K19(
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) holds
(f1 : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) + f2 : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) ) : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) )
/. c : ( ( ) ( )
Element of
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) (
complex )
Element of
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
= (f1 : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) /. c : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) (
complex )
Element of
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
+ (f2 : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) /. c : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) (
complex )
Element of
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
complex )
Element of
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) ) ) ;
theorem
for
C being ( ( non
empty ) ( non
empty )
set )
for
f1,
f2 being ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
PartFunc of ,) holds
(
dom (f1 : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) - f2 : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) ) : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
Element of
K19(
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) )
= (dom f1 : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) ) : ( ( ) ( )
Element of
K19(
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) )
/\ (dom f2 : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) ) : ( ( ) ( )
Element of
K19(
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
Element of
K19(
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) & ( for
c being ( ( ) ( )
Element of
C : ( ( non
empty ) ( non
empty )
set ) ) st
c : ( ( ) ( )
Element of
b1 : ( ( non
empty ) ( non
empty )
set ) )
in dom (f1 : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) - f2 : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) ) : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
Element of
K19(
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) holds
(f1 : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) - f2 : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) ) : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) )
/. c : ( ( ) ( )
Element of
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) (
complex )
Element of
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
= (f1 : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) /. c : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) (
complex )
Element of
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
- (f2 : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) /. c : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) (
complex )
Element of
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
complex )
Element of
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) ) ) ;
theorem
for
C being ( ( non
empty ) ( non
empty )
set )
for
f1,
f2 being ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
PartFunc of ,) holds
(
dom (f1 : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) (#) f2 : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) ) : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
Element of
K19(
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) )
= (dom f1 : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) ) : ( ( ) ( )
Element of
K19(
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) )
/\ (dom f2 : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) ) : ( ( ) ( )
Element of
K19(
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
Element of
K19(
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) & ( for
c being ( ( ) ( )
Element of
C : ( ( non
empty ) ( non
empty )
set ) ) st
c : ( ( ) ( )
Element of
b1 : ( ( non
empty ) ( non
empty )
set ) )
in dom (f1 : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) (#) f2 : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) ) : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
Element of
K19(
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) holds
(f1 : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) (#) f2 : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) ) : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) )
/. c : ( ( ) ( )
Element of
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) (
complex )
Element of
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
= (f1 : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) /. c : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) (
complex )
Element of
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
* (f2 : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) /. c : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) (
complex )
Element of
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
complex )
Element of
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) ) ) ;
theorem
for
C being ( ( non
empty ) ( non
empty )
set )
for
f1,
f2 being ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
PartFunc of ,) holds
(dom (f1 : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) (#) f2 : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) ) : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) Element of K19(K20(b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) : ( ( ) ( V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( )
Element of
K19(
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) )
\ ((f1 : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) (#) f2 : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) ) : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) Element of K19(K20(b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) : ( ( ) ( V38() ) set ) ) : ( ( ) ( ) set ) ) " {0 : ( ( ) ( ext-real natural complex real V36() V37() V48() V49() V50() V51() V52() V53() ) Element of NAT : ( ( ) ( V48() V49() V50() V51() V52() V53() V54() ) Element of K19(REAL : ( ( ) ( non empty V48() V49() V50() V54() V55() ) set ) ) : ( ( ) ( ) set ) ) ) } : ( ( ) ( V48() V49() V50() V51() V52() V53() ) set ) ) : ( ( ) ( )
Element of
K19(
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
Element of
K19(
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) )
= ((dom f1 : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) ) : ( ( ) ( ) Element of K19(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) \ (f1 : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) " {0 : ( ( ) ( ext-real natural complex real V36() V37() V48() V49() V50() V51() V52() V53() ) Element of NAT : ( ( ) ( V48() V49() V50() V51() V52() V53() V54() ) Element of K19(REAL : ( ( ) ( non empty V48() V49() V50() V54() V55() ) set ) ) : ( ( ) ( ) set ) ) ) } : ( ( ) ( V48() V49() V50() V51() V52() V53() ) set ) ) : ( ( ) ( ) Element of K19(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( )
Element of
K19(
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) )
/\ ((dom f2 : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) ) : ( ( ) ( ) Element of K19(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) \ (f2 : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) " {0 : ( ( ) ( ext-real natural complex real V36() V37() V48() V49() V50() V51() V52() V53() ) Element of NAT : ( ( ) ( V48() V49() V50() V51() V52() V53() V54() ) Element of K19(REAL : ( ( ) ( non empty V48() V49() V50() V54() V55() ) set ) ) : ( ( ) ( ) set ) ) ) } : ( ( ) ( V48() V49() V50() V51() V52() V53() ) set ) ) : ( ( ) ( ) Element of K19(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( )
Element of
K19(
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
Element of
K19(
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) ;
begin
theorem
for
C being ( ( non
empty ) ( non
empty )
set )
for
f1,
f2,
f3 being ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
PartFunc of ,) holds
(f1 : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) + f2 : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) ) : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) )
+ f3 : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
PartFunc of ,) : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) )
= f1 : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
PartFunc of ,)
+ (f2 : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) + f3 : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) ) : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) ) ;
theorem
for
C being ( ( non
empty ) ( non
empty )
set )
for
f1,
f2,
f3 being ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
PartFunc of ,) holds
(f1 : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) (#) f2 : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) ) : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) )
(#) f3 : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
PartFunc of ,) : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) )
= f1 : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
PartFunc of ,)
(#) (f2 : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) (#) f3 : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) ) : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) ) ;
theorem
for
C being ( ( non
empty ) ( non
empty )
set )
for
f1,
f2,
f3 being ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
PartFunc of ,) holds
(f1 : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) + f2 : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) ) : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) )
(#) f3 : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
PartFunc of ,) : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) )
= (f1 : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) (#) f3 : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) ) : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) )
+ (f2 : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) (#) f3 : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) ) : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) ) ;
theorem
for
C being ( ( non
empty ) ( non
empty )
set )
for
f3,
f1,
f2 being ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
PartFunc of ,) holds
f3 : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
PartFunc of ,)
(#) (f1 : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) + f2 : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) ) : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) )
= (f3 : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) (#) f1 : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) ) : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) )
+ (f3 : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) (#) f2 : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) ) : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) ) ;
theorem
for
C being ( ( non
empty ) ( non
empty )
set )
for
f1,
f2 being ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
PartFunc of ,)
for
r being ( ( ) (
complex )
Element of
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) holds
r : ( ( ) (
complex )
Element of
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
(#) (f1 : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) (#) f2 : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) ) : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) )
= (r : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) (#) f1 : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) ) : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) )
(#) f2 : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
PartFunc of ,) : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) ) ;
theorem
for
C being ( ( non
empty ) ( non
empty )
set )
for
f1,
f2 being ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
PartFunc of ,)
for
r being ( ( ) (
complex )
Element of
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) holds
r : ( ( ) (
complex )
Element of
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
(#) (f1 : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) (#) f2 : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) ) : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) )
= f1 : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
PartFunc of ,)
(#) (r : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) (#) f2 : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) ) : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) ) ;
theorem
for
C being ( ( non
empty ) ( non
empty )
set )
for
f1,
f2,
f3 being ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
PartFunc of ,) holds
(f1 : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) - f2 : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) ) : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) )
(#) f3 : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
PartFunc of ,) : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) )
= (f1 : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) (#) f3 : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) ) : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) )
- (f2 : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) (#) f3 : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) ) : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) ) ;
theorem
for
C being ( ( non
empty ) ( non
empty )
set )
for
f3,
f1,
f2 being ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
PartFunc of ,) holds
(f3 : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) (#) f1 : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) ) : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) )
- (f3 : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) (#) f2 : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) ) : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) )
= f3 : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
PartFunc of ,)
(#) (f1 : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) - f2 : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) ) : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) ) ;
theorem
for
C being ( ( non
empty ) ( non
empty )
set )
for
f1,
f2 being ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
PartFunc of ,)
for
r being ( ( ) (
complex )
Element of
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) holds
r : ( ( ) (
complex )
Element of
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
(#) (f1 : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) + f2 : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) ) : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) )
= (r : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) (#) f1 : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) ) : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) )
+ (r : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) (#) f2 : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) ) : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) ) ;
theorem
for
C being ( ( non
empty ) ( non
empty )
set )
for
f1,
f2 being ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
PartFunc of ,)
for
r being ( ( ) (
complex )
Element of
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) holds
r : ( ( ) (
complex )
Element of
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
(#) (f1 : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) - f2 : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) ) : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) )
= (r : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) (#) f1 : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) ) : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) )
- (r : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) (#) f2 : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) ) : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) ) ;
theorem
for
C being ( ( non
empty ) ( non
empty )
set )
for
f1,
f2,
f3 being ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
PartFunc of ,) holds
f1 : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
PartFunc of ,)
- (f2 : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) + f3 : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) ) : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) )
= (f1 : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) - f2 : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) ) : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) )
- f3 : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
PartFunc of ,) : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) ) ;
theorem
for
C being ( ( non
empty ) ( non
empty )
set )
for
f1,
f2,
f3 being ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
PartFunc of ,) holds
f1 : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
PartFunc of ,)
- (f2 : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) - f3 : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) ) : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) )
= (f1 : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) - f2 : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) ) : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) )
+ f3 : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
PartFunc of ,) : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) ) ;
theorem
for
C being ( ( non
empty ) ( non
empty )
set )
for
f1,
f2,
f3 being ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
PartFunc of ,) holds
f1 : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
PartFunc of ,)
+ (f2 : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) - f3 : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) ) : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) )
= (f1 : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) + f2 : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) ) : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) )
- f3 : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
PartFunc of ,) : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) ) ;
theorem
for
C being ( ( non
empty ) ( non
empty )
set )
for
f1,
f2 being ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
PartFunc of ,) holds
|.(f1 : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) (#) f2 : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) ) : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) Element of K19(K20(b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) : ( ( ) ( V38() ) set ) ) : ( ( ) ( ) set ) ) .| : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
REAL : ( ( ) ( non
empty V48()
V49()
V50()
V54()
V55() )
set ) )
Function-like V38()
V39()
V40() )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V48()
V49()
V50()
V54()
V55() )
set ) ) : ( ( ) (
V38()
V39()
V40() )
set ) ) : ( ( ) ( )
set ) )
= |.f1 : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) .| : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
REAL : ( ( ) ( non
empty V48()
V49()
V50()
V54()
V55() )
set ) )
Function-like V38()
V39()
V40() )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V48()
V49()
V50()
V54()
V55() )
set ) ) : ( ( ) (
V38()
V39()
V40() )
set ) ) : ( ( ) ( )
set ) )
(#) |.f2 : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) .| : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
REAL : ( ( ) ( non
empty V48()
V49()
V50()
V54()
V55() )
set ) )
Function-like V38()
V39()
V40() )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V48()
V49()
V50()
V54()
V55() )
set ) ) : ( ( ) (
V38()
V39()
V40() )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
REAL : ( ( ) ( non
empty V48()
V49()
V50()
V54()
V55() )
set ) )
Function-like V38()
V39()
V40() )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V48()
V49()
V50()
V54()
V55() )
set ) ) : ( ( ) (
V38()
V39()
V40() )
set ) ) : ( ( ) ( )
set ) ) ;
theorem
for
C being ( ( non
empty ) ( non
empty )
set )
for
f1,
f2 being ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
PartFunc of ,) holds
(f1 : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) (#) f2 : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) ) : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) )
^ : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
PartFunc of ,)
= (f1 : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) ^) : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
PartFunc of ,)
(#) (f2 : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) ^) : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
PartFunc of ,) : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) ) ;
theorem
for
C being ( ( non
empty ) ( non
empty )
set )
for
g,
f being ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
PartFunc of ,)
for
r being ( ( ) (
complex )
Element of
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) holds
r : ( ( ) (
complex )
Element of
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
(#) (g : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) / f : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) ) : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
PartFunc of ,) : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) )
= (r : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) (#) g : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) ) : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) )
/ f : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
PartFunc of ,) : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
PartFunc of ,) ;
theorem
for
C being ( ( non
empty ) ( non
empty )
set )
for
f,
g being ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
PartFunc of ,) holds
(f : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) / g : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) ) : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
PartFunc of ,)
(#) g : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
PartFunc of ,) : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) )
= f : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
PartFunc of ,)
| (dom (g : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) ^) : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) ) : ( ( ) ( )
Element of
K19(
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) ) ;
theorem
for
C being ( ( non
empty ) ( non
empty )
set )
for
f,
g,
f1,
g1 being ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
PartFunc of ,) holds
(f : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) / g : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) ) : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
PartFunc of ,)
(#) (f1 : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) / g1 : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) ) : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
PartFunc of ,) : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) )
= (f : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) (#) f1 : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) ) : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) )
/ (g : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) (#) g1 : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) ) : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
PartFunc of ,) ;
theorem
for
C being ( ( non
empty ) ( non
empty )
set )
for
f1,
f2 being ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
PartFunc of ,) holds
(f1 : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) / f2 : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) ) : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
PartFunc of ,)
^ : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
PartFunc of ,)
= (f2 : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) | (dom (f2 : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) ^) : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) ) : ( ( ) ( ) Element of K19(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) )
/ f1 : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
PartFunc of ,) : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
PartFunc of ,) ;
theorem
for
C being ( ( non
empty ) ( non
empty )
set )
for
g,
f1,
f2 being ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
PartFunc of ,) holds
g : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
PartFunc of ,)
(#) (f1 : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) / f2 : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) ) : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
PartFunc of ,) : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) )
= (g : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) (#) f1 : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) ) : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) )
/ f2 : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
PartFunc of ,) : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
PartFunc of ,) ;
theorem
for
C being ( ( non
empty ) ( non
empty )
set )
for
g,
f1,
f2 being ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
PartFunc of ,) holds
g : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
PartFunc of ,)
/ (f1 : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) / f2 : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) ) : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
PartFunc of ,) : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
PartFunc of ,)
= (g : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) (#) (f2 : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) | (dom (f2 : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) ^) : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) ) : ( ( ) ( ) Element of K19(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) Element of K19(K20(b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) : ( ( ) ( V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) )
/ f1 : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
PartFunc of ,) : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
PartFunc of ,) ;
theorem
for
C being ( ( non
empty ) ( non
empty )
set )
for
f,
g being ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
PartFunc of ,) holds
(
- (f : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) / g : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) ) : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
PartFunc of ,) : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) )
= (- f : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) ) : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) )
/ g : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
PartFunc of ,) : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
PartFunc of ,) &
f : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
PartFunc of ,)
/ (- g : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) ) : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
PartFunc of ,)
= - (f : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) / g : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) ) : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
PartFunc of ,) : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) ) ) ;
theorem
for
C being ( ( non
empty ) ( non
empty )
set )
for
f1,
f,
f2 being ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
PartFunc of ,) holds
(
(f1 : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) / f : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) ) : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
PartFunc of ,)
+ (f2 : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) / f : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) ) : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
PartFunc of ,) : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) )
= (f1 : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) + f2 : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) ) : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) )
/ f : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
PartFunc of ,) : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
PartFunc of ,) &
(f1 : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) / f : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) ) : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
PartFunc of ,)
- (f2 : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) / f : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) ) : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
PartFunc of ,) : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) )
= (f1 : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) - f2 : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) ) : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) )
/ f : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
PartFunc of ,) : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
PartFunc of ,) ) ;
theorem
for
C being ( ( non
empty ) ( non
empty )
set )
for
f1,
f,
g1,
g being ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
PartFunc of ,) holds
(f1 : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) / f : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) ) : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
PartFunc of ,)
+ (g1 : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) / g : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) ) : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
PartFunc of ,) : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) )
= ((f1 : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) (#) g : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) ) : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) Element of K19(K20(b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) : ( ( ) ( V38() ) set ) ) : ( ( ) ( ) set ) ) + (g1 : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) (#) f : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) ) : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) Element of K19(K20(b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) : ( ( ) ( V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) )
/ (f : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) (#) g : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) ) : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
PartFunc of ,) ;
theorem
for
C being ( ( non
empty ) ( non
empty )
set )
for
f,
g,
f1,
g1 being ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
PartFunc of ,) holds
(f : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) / g : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) ) : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
PartFunc of ,)
/ (f1 : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) / g1 : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) ) : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
PartFunc of ,) : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
PartFunc of ,)
= (f : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) (#) (g1 : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) | (dom (g1 : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) ^) : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) ) : ( ( ) ( ) Element of K19(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) Element of K19(K20(b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) : ( ( ) ( V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) )
/ (g : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) (#) f1 : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) ) : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
PartFunc of ,) ;
theorem
for
C being ( ( non
empty ) ( non
empty )
set )
for
f1,
f,
g1,
g being ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
PartFunc of ,) holds
(f1 : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) / f : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) ) : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
PartFunc of ,)
- (g1 : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) / g : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) ) : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
PartFunc of ,) : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) )
= ((f1 : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) (#) g : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) ) : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) Element of K19(K20(b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) : ( ( ) ( V38() ) set ) ) : ( ( ) ( ) set ) ) - (g1 : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) (#) f : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) ) : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) Element of K19(K20(b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) : ( ( ) ( V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) )
/ (f : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) (#) g : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) ) : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
PartFunc of ,) ;
theorem
for
C being ( ( non
empty ) ( non
empty )
set )
for
f1,
f2 being ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
PartFunc of ,) holds
|.(f1 : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) / f2 : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) ) : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) .| : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
REAL : ( ( ) ( non
empty V48()
V49()
V50()
V54()
V55() )
set ) )
Function-like V38()
V39()
V40() )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V48()
V49()
V50()
V54()
V55() )
set ) ) : ( ( ) (
V38()
V39()
V40() )
set ) ) : ( ( ) ( )
set ) )
= |.f1 : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) .| : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
REAL : ( ( ) ( non
empty V48()
V49()
V50()
V54()
V55() )
set ) )
Function-like V38()
V39()
V40() )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V48()
V49()
V50()
V54()
V55() )
set ) ) : ( ( ) (
V38()
V39()
V40() )
set ) ) : ( ( ) ( )
set ) )
/ |.f2 : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) .| : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
REAL : ( ( ) ( non
empty V48()
V49()
V50()
V54()
V55() )
set ) )
Function-like V38()
V39()
V40() )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V48()
V49()
V50()
V54()
V55() )
set ) ) : ( ( ) (
V38()
V39()
V40() )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
REAL : ( ( ) ( non
empty V48()
V49()
V50()
V54()
V55() )
set ) )
Function-like V38()
V39()
V40() )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V48()
V49()
V50()
V54()
V55() )
set ) ) : ( ( ) (
V38()
V39()
V40() )
set ) ) : ( ( ) ( )
set ) ) ;
theorem
for
X being ( ( ) ( )
set )
for
C being ( ( non
empty ) ( non
empty )
set )
for
f1,
f2 being ( (
Function-like ) (
V1()
V4(
b2 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
PartFunc of ,) holds
(
(f1 : ( ( Function-like ) ( V1() V4(b2 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) + f2 : ( ( Function-like ) ( V1() V4(b2 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) ) : ( (
Function-like ) (
V1()
V4(
b2 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
Element of
K19(
K20(
b2 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) )
| X : ( ( ) ( )
set ) : ( (
Function-like ) (
V1()
V4(
b2 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
Element of
K19(
K20(
b2 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) )
= (f1 : ( ( Function-like ) ( V1() V4(b2 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) | X : ( ( ) ( ) set ) ) : ( (
Function-like ) (
V1()
V4(
b2 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
Element of
K19(
K20(
b2 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) )
+ (f2 : ( ( Function-like ) ( V1() V4(b2 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) | X : ( ( ) ( ) set ) ) : ( (
Function-like ) (
V1()
V4(
b2 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
Element of
K19(
K20(
b2 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like ) (
V1()
V4(
b2 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
Element of
K19(
K20(
b2 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) ) &
(f1 : ( ( Function-like ) ( V1() V4(b2 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) + f2 : ( ( Function-like ) ( V1() V4(b2 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) ) : ( (
Function-like ) (
V1()
V4(
b2 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
Element of
K19(
K20(
b2 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) )
| X : ( ( ) ( )
set ) : ( (
Function-like ) (
V1()
V4(
b2 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
Element of
K19(
K20(
b2 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) )
= (f1 : ( ( Function-like ) ( V1() V4(b2 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) | X : ( ( ) ( ) set ) ) : ( (
Function-like ) (
V1()
V4(
b2 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
Element of
K19(
K20(
b2 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) )
+ f2 : ( (
Function-like ) (
V1()
V4(
b2 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
PartFunc of ,) : ( (
Function-like ) (
V1()
V4(
b2 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
Element of
K19(
K20(
b2 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) ) &
(f1 : ( ( Function-like ) ( V1() V4(b2 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) + f2 : ( ( Function-like ) ( V1() V4(b2 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) ) : ( (
Function-like ) (
V1()
V4(
b2 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
Element of
K19(
K20(
b2 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) )
| X : ( ( ) ( )
set ) : ( (
Function-like ) (
V1()
V4(
b2 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
Element of
K19(
K20(
b2 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) )
= f1 : ( (
Function-like ) (
V1()
V4(
b2 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
PartFunc of ,)
+ (f2 : ( ( Function-like ) ( V1() V4(b2 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) | X : ( ( ) ( ) set ) ) : ( (
Function-like ) (
V1()
V4(
b2 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
Element of
K19(
K20(
b2 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like ) (
V1()
V4(
b2 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
Element of
K19(
K20(
b2 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) ) ) ;
theorem
for
X being ( ( ) ( )
set )
for
C being ( ( non
empty ) ( non
empty )
set )
for
f1,
f2 being ( (
Function-like ) (
V1()
V4(
b2 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
PartFunc of ,) holds
(
(f1 : ( ( Function-like ) ( V1() V4(b2 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) (#) f2 : ( ( Function-like ) ( V1() V4(b2 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) ) : ( (
Function-like ) (
V1()
V4(
b2 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
Element of
K19(
K20(
b2 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) )
| X : ( ( ) ( )
set ) : ( (
Function-like ) (
V1()
V4(
b2 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
Element of
K19(
K20(
b2 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) )
= (f1 : ( ( Function-like ) ( V1() V4(b2 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) | X : ( ( ) ( ) set ) ) : ( (
Function-like ) (
V1()
V4(
b2 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
Element of
K19(
K20(
b2 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) )
(#) (f2 : ( ( Function-like ) ( V1() V4(b2 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) | X : ( ( ) ( ) set ) ) : ( (
Function-like ) (
V1()
V4(
b2 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
Element of
K19(
K20(
b2 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like ) (
V1()
V4(
b2 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
Element of
K19(
K20(
b2 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) ) &
(f1 : ( ( Function-like ) ( V1() V4(b2 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) (#) f2 : ( ( Function-like ) ( V1() V4(b2 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) ) : ( (
Function-like ) (
V1()
V4(
b2 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
Element of
K19(
K20(
b2 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) )
| X : ( ( ) ( )
set ) : ( (
Function-like ) (
V1()
V4(
b2 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
Element of
K19(
K20(
b2 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) )
= (f1 : ( ( Function-like ) ( V1() V4(b2 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) | X : ( ( ) ( ) set ) ) : ( (
Function-like ) (
V1()
V4(
b2 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
Element of
K19(
K20(
b2 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) )
(#) f2 : ( (
Function-like ) (
V1()
V4(
b2 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
PartFunc of ,) : ( (
Function-like ) (
V1()
V4(
b2 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
Element of
K19(
K20(
b2 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) ) &
(f1 : ( ( Function-like ) ( V1() V4(b2 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) (#) f2 : ( ( Function-like ) ( V1() V4(b2 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) ) : ( (
Function-like ) (
V1()
V4(
b2 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
Element of
K19(
K20(
b2 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) )
| X : ( ( ) ( )
set ) : ( (
Function-like ) (
V1()
V4(
b2 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
Element of
K19(
K20(
b2 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) )
= f1 : ( (
Function-like ) (
V1()
V4(
b2 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
PartFunc of ,)
(#) (f2 : ( ( Function-like ) ( V1() V4(b2 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) | X : ( ( ) ( ) set ) ) : ( (
Function-like ) (
V1()
V4(
b2 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
Element of
K19(
K20(
b2 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like ) (
V1()
V4(
b2 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
Element of
K19(
K20(
b2 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) ) ) ;
theorem
for
X being ( ( ) ( )
set )
for
C being ( ( non
empty ) ( non
empty )
set )
for
f being ( (
Function-like ) (
V1()
V4(
b2 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
PartFunc of ,) holds
(
(- f : ( ( Function-like ) ( V1() V4(b2 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) ) : ( (
Function-like ) (
V1()
V4(
b2 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
Element of
K19(
K20(
b2 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) )
| X : ( ( ) ( )
set ) : ( (
Function-like ) (
V1()
V4(
b2 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
Element of
K19(
K20(
b2 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) )
= - (f : ( ( Function-like ) ( V1() V4(b2 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) | X : ( ( ) ( ) set ) ) : ( (
Function-like ) (
V1()
V4(
b2 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
Element of
K19(
K20(
b2 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like ) (
V1()
V4(
b2 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
Element of
K19(
K20(
b2 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) ) &
(f : ( ( Function-like ) ( V1() V4(b2 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) ^) : ( (
Function-like ) (
V1()
V4(
b2 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
PartFunc of ,)
| X : ( ( ) ( )
set ) : ( (
Function-like ) (
V1()
V4(
b2 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
Element of
K19(
K20(
b2 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) )
= (f : ( ( Function-like ) ( V1() V4(b2 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) | X : ( ( ) ( ) set ) ) : ( (
Function-like ) (
V1()
V4(
b2 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
Element of
K19(
K20(
b2 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) )
^ : ( (
Function-like ) (
V1()
V4(
b2 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
PartFunc of ,) &
|.f : ( ( Function-like ) ( V1() V4(b2 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) .| : ( (
Function-like ) (
V1()
V4(
b2 : ( ( non
empty ) ( non
empty )
set ) )
V5(
REAL : ( ( ) ( non
empty V48()
V49()
V50()
V54()
V55() )
set ) )
Function-like V38()
V39()
V40() )
Element of
K19(
K20(
b2 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V48()
V49()
V50()
V54()
V55() )
set ) ) : ( ( ) (
V38()
V39()
V40() )
set ) ) : ( ( ) ( )
set ) )
| X : ( ( ) ( )
set ) : ( (
Function-like ) (
V1()
V4(
b2 : ( ( non
empty ) ( non
empty )
set ) )
V5(
REAL : ( ( ) ( non
empty V48()
V49()
V50()
V54()
V55() )
set ) )
Function-like V38()
V39()
V40() )
Element of
K19(
K20(
b2 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V48()
V49()
V50()
V54()
V55() )
set ) ) : ( ( ) (
V38()
V39()
V40() )
set ) ) : ( ( ) ( )
set ) )
= |.(f : ( ( Function-like ) ( V1() V4(b2 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) | X : ( ( ) ( ) set ) ) : ( ( Function-like ) ( V1() V4(b2 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) Element of K19(K20(b2 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) : ( ( ) ( V38() ) set ) ) : ( ( ) ( ) set ) ) .| : ( (
Function-like ) (
V1()
V4(
b2 : ( ( non
empty ) ( non
empty )
set ) )
V5(
REAL : ( ( ) ( non
empty V48()
V49()
V50()
V54()
V55() )
set ) )
Function-like V38()
V39()
V40() )
Element of
K19(
K20(
b2 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V48()
V49()
V50()
V54()
V55() )
set ) ) : ( ( ) (
V38()
V39()
V40() )
set ) ) : ( ( ) ( )
set ) ) ) ;
theorem
for
X being ( ( ) ( )
set )
for
C being ( ( non
empty ) ( non
empty )
set )
for
f1,
f2 being ( (
Function-like ) (
V1()
V4(
b2 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
PartFunc of ,) holds
(
(f1 : ( ( Function-like ) ( V1() V4(b2 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) - f2 : ( ( Function-like ) ( V1() V4(b2 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) ) : ( (
Function-like ) (
V1()
V4(
b2 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
Element of
K19(
K20(
b2 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) )
| X : ( ( ) ( )
set ) : ( (
Function-like ) (
V1()
V4(
b2 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
Element of
K19(
K20(
b2 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) )
= (f1 : ( ( Function-like ) ( V1() V4(b2 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) | X : ( ( ) ( ) set ) ) : ( (
Function-like ) (
V1()
V4(
b2 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
Element of
K19(
K20(
b2 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) )
- (f2 : ( ( Function-like ) ( V1() V4(b2 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) | X : ( ( ) ( ) set ) ) : ( (
Function-like ) (
V1()
V4(
b2 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
Element of
K19(
K20(
b2 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like ) (
V1()
V4(
b2 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
Element of
K19(
K20(
b2 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) ) &
(f1 : ( ( Function-like ) ( V1() V4(b2 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) - f2 : ( ( Function-like ) ( V1() V4(b2 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) ) : ( (
Function-like ) (
V1()
V4(
b2 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
Element of
K19(
K20(
b2 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) )
| X : ( ( ) ( )
set ) : ( (
Function-like ) (
V1()
V4(
b2 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
Element of
K19(
K20(
b2 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) )
= (f1 : ( ( Function-like ) ( V1() V4(b2 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) | X : ( ( ) ( ) set ) ) : ( (
Function-like ) (
V1()
V4(
b2 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
Element of
K19(
K20(
b2 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) )
- f2 : ( (
Function-like ) (
V1()
V4(
b2 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
PartFunc of ,) : ( (
Function-like ) (
V1()
V4(
b2 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
Element of
K19(
K20(
b2 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) ) &
(f1 : ( ( Function-like ) ( V1() V4(b2 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) - f2 : ( ( Function-like ) ( V1() V4(b2 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) ) : ( (
Function-like ) (
V1()
V4(
b2 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
Element of
K19(
K20(
b2 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) )
| X : ( ( ) ( )
set ) : ( (
Function-like ) (
V1()
V4(
b2 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
Element of
K19(
K20(
b2 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) )
= f1 : ( (
Function-like ) (
V1()
V4(
b2 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
PartFunc of ,)
- (f2 : ( ( Function-like ) ( V1() V4(b2 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) | X : ( ( ) ( ) set ) ) : ( (
Function-like ) (
V1()
V4(
b2 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
Element of
K19(
K20(
b2 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like ) (
V1()
V4(
b2 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
Element of
K19(
K20(
b2 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) ) ) ;
theorem
for
X being ( ( ) ( )
set )
for
C being ( ( non
empty ) ( non
empty )
set )
for
f1,
f2 being ( (
Function-like ) (
V1()
V4(
b2 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
PartFunc of ,) holds
(
(f1 : ( ( Function-like ) ( V1() V4(b2 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) / f2 : ( ( Function-like ) ( V1() V4(b2 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) ) : ( (
Function-like ) (
V1()
V4(
b2 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
PartFunc of ,)
| X : ( ( ) ( )
set ) : ( (
Function-like ) (
V1()
V4(
b2 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
Element of
K19(
K20(
b2 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) )
= (f1 : ( ( Function-like ) ( V1() V4(b2 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) | X : ( ( ) ( ) set ) ) : ( (
Function-like ) (
V1()
V4(
b2 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
Element of
K19(
K20(
b2 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) )
/ (f2 : ( ( Function-like ) ( V1() V4(b2 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) | X : ( ( ) ( ) set ) ) : ( (
Function-like ) (
V1()
V4(
b2 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
Element of
K19(
K20(
b2 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like ) (
V1()
V4(
b2 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
PartFunc of ,) &
(f1 : ( ( Function-like ) ( V1() V4(b2 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) / f2 : ( ( Function-like ) ( V1() V4(b2 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) ) : ( (
Function-like ) (
V1()
V4(
b2 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
PartFunc of ,)
| X : ( ( ) ( )
set ) : ( (
Function-like ) (
V1()
V4(
b2 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
Element of
K19(
K20(
b2 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) )
= (f1 : ( ( Function-like ) ( V1() V4(b2 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) | X : ( ( ) ( ) set ) ) : ( (
Function-like ) (
V1()
V4(
b2 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
Element of
K19(
K20(
b2 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) )
/ f2 : ( (
Function-like ) (
V1()
V4(
b2 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
PartFunc of ,) : ( (
Function-like ) (
V1()
V4(
b2 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
PartFunc of ,) &
(f1 : ( ( Function-like ) ( V1() V4(b2 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) / f2 : ( ( Function-like ) ( V1() V4(b2 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) ) : ( (
Function-like ) (
V1()
V4(
b2 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
PartFunc of ,)
| X : ( ( ) ( )
set ) : ( (
Function-like ) (
V1()
V4(
b2 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
Element of
K19(
K20(
b2 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) )
= f1 : ( (
Function-like ) (
V1()
V4(
b2 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
PartFunc of ,)
/ (f2 : ( ( Function-like ) ( V1() V4(b2 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) | X : ( ( ) ( ) set ) ) : ( (
Function-like ) (
V1()
V4(
b2 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
Element of
K19(
K20(
b2 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) ) : ( (
Function-like ) (
V1()
V4(
b2 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
PartFunc of ,) ) ;
begin
theorem
for
C being ( ( non
empty ) ( non
empty )
set )
for
f1,
f2 being ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
PartFunc of ,) holds
( (
f1 : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
PartFunc of ,) is
total &
f2 : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
PartFunc of ,) is
total implies
f1 : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
PartFunc of ,)
+ f2 : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
PartFunc of ,) : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) ) is
total ) & (
f1 : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
PartFunc of ,)
+ f2 : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
PartFunc of ,) : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) ) is
total implies (
f1 : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
PartFunc of ,) is
total &
f2 : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
PartFunc of ,) is
total ) ) & (
f1 : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
PartFunc of ,) is
total &
f2 : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
PartFunc of ,) is
total implies
f1 : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
PartFunc of ,)
- f2 : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
PartFunc of ,) : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) ) is
total ) & (
f1 : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
PartFunc of ,)
- f2 : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
PartFunc of ,) : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) ) is
total implies (
f1 : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
PartFunc of ,) is
total &
f2 : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
PartFunc of ,) is
total ) ) & (
f1 : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
PartFunc of ,) is
total &
f2 : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
PartFunc of ,) is
total implies
f1 : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
PartFunc of ,)
(#) f2 : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
PartFunc of ,) : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) ) is
total ) & (
f1 : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
PartFunc of ,)
(#) f2 : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
PartFunc of ,) : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) ) is
total implies (
f1 : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
PartFunc of ,) is
total &
f2 : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
PartFunc of ,) is
total ) ) ) ;
theorem
for
C being ( ( non
empty ) ( non
empty )
set )
for
c being ( ( ) ( )
Element of
C : ( ( non
empty ) ( non
empty )
set ) )
for
f1,
f2 being ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
PartFunc of ,) st
f1 : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
PartFunc of ,) is
total &
f2 : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
PartFunc of ,) is
total holds
(
(f1 : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) + f2 : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) ) : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) )
/. c : ( ( ) ( )
Element of
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) (
complex )
Element of
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
= (f1 : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) /. c : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) (
complex )
Element of
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
+ (f2 : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) /. c : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) (
complex )
Element of
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
complex )
Element of
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) &
(f1 : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) - f2 : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) ) : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) )
/. c : ( ( ) ( )
Element of
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) (
complex )
Element of
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
= (f1 : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) /. c : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) (
complex )
Element of
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
- (f2 : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) /. c : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) (
complex )
Element of
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
complex )
Element of
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) &
(f1 : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) (#) f2 : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) ) : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
Element of
K19(
K20(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) )
/. c : ( ( ) ( )
Element of
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) (
complex )
Element of
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
= (f1 : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) /. c : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) (
complex )
Element of
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
* (f2 : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) /. c : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) (
complex )
Element of
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
complex )
Element of
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) ) ;
theorem
for
C being ( ( non
empty ) ( non
empty )
set )
for
c being ( ( ) ( )
Element of
C : ( ( non
empty ) ( non
empty )
set ) )
for
f1,
f2 being ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
PartFunc of ,) st
f1 : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
PartFunc of ,) is
total &
f2 : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
PartFunc of ,)
^ : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
PartFunc of ,) is
total holds
(f1 : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) / f2 : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) ) : ( (
Function-like ) (
V1()
V4(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
PartFunc of ,)
/. c : ( ( ) ( )
Element of
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) (
complex )
Element of
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
= (f1 : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) /. c : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) (
complex )
Element of
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
* ((f2 : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) /. c : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) ") : ( ( ) (
complex )
Element of
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
complex )
Element of
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) ;
begin
theorem
for
X,
Y being ( ( ) ( )
set )
for
C being ( ( non
empty ) ( non
empty )
set )
for
f1,
f2 being ( (
Function-like ) (
V1()
V4(
b3 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
PartFunc of ,) st
f1 : ( (
Function-like ) (
V1()
V4(
b3 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
PartFunc of ,)
| X : ( ( ) ( )
set ) : ( (
Function-like ) (
V1()
V4(
b3 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
Element of
K19(
K20(
b3 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) ) is
bounded &
f2 : ( (
Function-like ) (
V1()
V4(
b3 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
PartFunc of ,)
| Y : ( ( ) ( )
set ) : ( (
Function-like ) (
V1()
V4(
b3 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
Element of
K19(
K20(
b3 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) ) is
bounded holds
(f1 : ( ( Function-like ) ( V1() V4(b3 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) + f2 : ( ( Function-like ) ( V1() V4(b3 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) ) : ( (
Function-like ) (
V1()
V4(
b3 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
Element of
K19(
K20(
b3 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) )
| (X : ( ( ) ( ) set ) /\ Y : ( ( ) ( ) set ) ) : ( ( ) ( )
set ) : ( (
Function-like ) (
V1()
V4(
b3 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
Element of
K19(
K20(
b3 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) ) is
bounded ;
theorem
for
X,
Y being ( ( ) ( )
set )
for
C being ( ( non
empty ) ( non
empty )
set )
for
f1,
f2 being ( (
Function-like ) (
V1()
V4(
b3 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
PartFunc of ,) st
f1 : ( (
Function-like ) (
V1()
V4(
b3 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
PartFunc of ,)
| X : ( ( ) ( )
set ) : ( (
Function-like ) (
V1()
V4(
b3 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
Element of
K19(
K20(
b3 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) ) is
bounded &
f2 : ( (
Function-like ) (
V1()
V4(
b3 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
PartFunc of ,)
| Y : ( ( ) ( )
set ) : ( (
Function-like ) (
V1()
V4(
b3 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
Element of
K19(
K20(
b3 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) ) is
bounded holds
(
(f1 : ( ( Function-like ) ( V1() V4(b3 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) (#) f2 : ( ( Function-like ) ( V1() V4(b3 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) ) : ( (
Function-like ) (
V1()
V4(
b3 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
Element of
K19(
K20(
b3 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) )
| (X : ( ( ) ( ) set ) /\ Y : ( ( ) ( ) set ) ) : ( ( ) ( )
set ) : ( (
Function-like ) (
V1()
V4(
b3 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
Element of
K19(
K20(
b3 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) ) is
bounded &
(f1 : ( ( Function-like ) ( V1() V4(b3 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) - f2 : ( ( Function-like ) ( V1() V4(b3 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) ) : ( (
Function-like ) (
V1()
V4(
b3 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
Element of
K19(
K20(
b3 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) )
| (X : ( ( ) ( ) set ) /\ Y : ( ( ) ( ) set ) ) : ( ( ) ( )
set ) : ( (
Function-like ) (
V1()
V4(
b3 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
Element of
K19(
K20(
b3 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) ) is
bounded ) ;
theorem
for
X,
Y being ( ( ) ( )
set )
for
C being ( ( non
empty ) ( non
empty )
set )
for
f1,
f2 being ( (
Function-like ) (
V1()
V4(
b3 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
PartFunc of ,) st
f1 : ( (
Function-like ) (
V1()
V4(
b3 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
PartFunc of ,)
| X : ( ( ) ( )
set ) : ( (
Function-like ) (
V1()
V4(
b3 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
Element of
K19(
K20(
b3 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) ) is
constant &
f2 : ( (
Function-like ) (
V1()
V4(
b3 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
PartFunc of ,)
| Y : ( ( ) ( )
set ) : ( (
Function-like ) (
V1()
V4(
b3 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
Element of
K19(
K20(
b3 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) ) is
constant holds
(
(f1 : ( ( Function-like ) ( V1() V4(b3 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) + f2 : ( ( Function-like ) ( V1() V4(b3 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) ) : ( (
Function-like ) (
V1()
V4(
b3 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
Element of
K19(
K20(
b3 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) )
| (X : ( ( ) ( ) set ) /\ Y : ( ( ) ( ) set ) ) : ( ( ) ( )
set ) : ( (
Function-like ) (
V1()
V4(
b3 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
Element of
K19(
K20(
b3 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) ) is
constant &
(f1 : ( ( Function-like ) ( V1() V4(b3 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) - f2 : ( ( Function-like ) ( V1() V4(b3 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) ) : ( (
Function-like ) (
V1()
V4(
b3 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
Element of
K19(
K20(
b3 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) )
| (X : ( ( ) ( ) set ) /\ Y : ( ( ) ( ) set ) ) : ( ( ) ( )
set ) : ( (
Function-like ) (
V1()
V4(
b3 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
Element of
K19(
K20(
b3 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) ) is
constant &
(f1 : ( ( Function-like ) ( V1() V4(b3 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) (#) f2 : ( ( Function-like ) ( V1() V4(b3 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) ) : ( (
Function-like ) (
V1()
V4(
b3 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
Element of
K19(
K20(
b3 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) )
| (X : ( ( ) ( ) set ) /\ Y : ( ( ) ( ) set ) ) : ( ( ) ( )
set ) : ( (
Function-like ) (
V1()
V4(
b3 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
Element of
K19(
K20(
b3 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) ) is
constant ) ;
theorem
for
Y being ( ( ) ( )
set )
for
C being ( ( non
empty ) ( non
empty )
set )
for
f being ( (
Function-like ) (
V1()
V4(
b2 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
PartFunc of ,) st
f : ( (
Function-like ) (
V1()
V4(
b2 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
PartFunc of ,)
| Y : ( ( ) ( )
set ) : ( (
Function-like ) (
V1()
V4(
b2 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
Element of
K19(
K20(
b2 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) ) is
constant holds
( ( for
r being ( ( ) (
complex )
Element of
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) holds
(r : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) (#) f : ( ( Function-like ) ( V1() V4(b2 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) ) : ( (
Function-like ) (
V1()
V4(
b2 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
Element of
K19(
K20(
b2 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) )
| Y : ( ( ) ( )
set ) : ( (
Function-like ) (
V1()
V4(
b2 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
Element of
K19(
K20(
b2 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) ) is
bounded ) &
(- f : ( ( Function-like ) ( V1() V4(b2 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) ) : ( (
Function-like ) (
V1()
V4(
b2 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
Element of
K19(
K20(
b2 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) )
| Y : ( ( ) ( )
set ) : ( (
Function-like ) (
V1()
V4(
b2 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
Element of
K19(
K20(
b2 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) ) is
bounded &
|.f : ( ( Function-like ) ( V1() V4(b2 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) .| : ( (
Function-like ) (
V1()
V4(
b2 : ( ( non
empty ) ( non
empty )
set ) )
V5(
REAL : ( ( ) ( non
empty V48()
V49()
V50()
V54()
V55() )
set ) )
Function-like V38()
V39()
V40() )
Element of
K19(
K20(
b2 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V48()
V49()
V50()
V54()
V55() )
set ) ) : ( ( ) (
V38()
V39()
V40() )
set ) ) : ( ( ) ( )
set ) )
| Y : ( ( ) ( )
set ) : ( (
Function-like ) (
V1()
V4(
b2 : ( ( non
empty ) ( non
empty )
set ) )
V5(
REAL : ( ( ) ( non
empty V48()
V49()
V50()
V54()
V55() )
set ) )
Function-like V38()
V39()
V40() )
Element of
K19(
K20(
b2 : ( ( non
empty ) ( non
empty )
set ) ,
REAL : ( ( ) ( non
empty V48()
V49()
V50()
V54()
V55() )
set ) ) : ( ( ) (
V38()
V39()
V40() )
set ) ) : ( ( ) ( )
set ) ) is
bounded ) ;
theorem
for
X,
Y being ( ( ) ( )
set )
for
C being ( ( non
empty ) ( non
empty )
set )
for
f1,
f2 being ( (
Function-like ) (
V1()
V4(
b3 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
PartFunc of ,) st
f1 : ( (
Function-like ) (
V1()
V4(
b3 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
PartFunc of ,)
| X : ( ( ) ( )
set ) : ( (
Function-like ) (
V1()
V4(
b3 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
Element of
K19(
K20(
b3 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) ) is
bounded &
f2 : ( (
Function-like ) (
V1()
V4(
b3 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
PartFunc of ,)
| Y : ( ( ) ( )
set ) : ( (
Function-like ) (
V1()
V4(
b3 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
Element of
K19(
K20(
b3 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) ) is
constant holds
(f1 : ( ( Function-like ) ( V1() V4(b3 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) + f2 : ( ( Function-like ) ( V1() V4(b3 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) ) : ( (
Function-like ) (
V1()
V4(
b3 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
Element of
K19(
K20(
b3 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) )
| (X : ( ( ) ( ) set ) /\ Y : ( ( ) ( ) set ) ) : ( ( ) ( )
set ) : ( (
Function-like ) (
V1()
V4(
b3 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
Element of
K19(
K20(
b3 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) ) is
bounded ;
theorem
for
X,
Y being ( ( ) ( )
set )
for
C being ( ( non
empty ) ( non
empty )
set )
for
f1,
f2 being ( (
Function-like ) (
V1()
V4(
b3 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
PartFunc of ,) st
f1 : ( (
Function-like ) (
V1()
V4(
b3 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
PartFunc of ,)
| X : ( ( ) ( )
set ) : ( (
Function-like ) (
V1()
V4(
b3 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
Element of
K19(
K20(
b3 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) ) is
bounded &
f2 : ( (
Function-like ) (
V1()
V4(
b3 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
PartFunc of ,)
| Y : ( ( ) ( )
set ) : ( (
Function-like ) (
V1()
V4(
b3 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
Element of
K19(
K20(
b3 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) ) is
constant holds
(
(f1 : ( ( Function-like ) ( V1() V4(b3 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) - f2 : ( ( Function-like ) ( V1() V4(b3 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) ) : ( (
Function-like ) (
V1()
V4(
b3 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
Element of
K19(
K20(
b3 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) )
| (X : ( ( ) ( ) set ) /\ Y : ( ( ) ( ) set ) ) : ( ( ) ( )
set ) : ( (
Function-like ) (
V1()
V4(
b3 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
Element of
K19(
K20(
b3 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) ) is
bounded &
(f2 : ( ( Function-like ) ( V1() V4(b3 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) - f1 : ( ( Function-like ) ( V1() V4(b3 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) ) : ( (
Function-like ) (
V1()
V4(
b3 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
Element of
K19(
K20(
b3 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) )
| (X : ( ( ) ( ) set ) /\ Y : ( ( ) ( ) set ) ) : ( ( ) ( )
set ) : ( (
Function-like ) (
V1()
V4(
b3 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
Element of
K19(
K20(
b3 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) ) is
bounded &
(f1 : ( ( Function-like ) ( V1() V4(b3 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) (#) f2 : ( ( Function-like ) ( V1() V4(b3 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) ) : ( (
Function-like ) (
V1()
V4(
b3 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
Element of
K19(
K20(
b3 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) )
| (X : ( ( ) ( ) set ) /\ Y : ( ( ) ( ) set ) ) : ( ( ) ( )
set ) : ( (
Function-like ) (
V1()
V4(
b3 : ( ( non
empty ) ( non
empty )
set ) )
V5(
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) )
Function-like V38() )
Element of
K19(
K20(
b3 : ( ( non
empty ) ( non
empty )
set ) ,
COMPLEX : ( ( ) ( non
empty V48()
V54()
V55() )
set ) ) : ( ( ) (
V38() )
set ) ) : ( ( ) ( )
set ) ) is
bounded ) ;