:: CFUNCT_1 semantic presentation

begin

definition
mode Complex is ( ( complex ) ( complex ) number ) ;
end;

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 :: CFUNCT_1:def 1
( 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 :: CFUNCT_1:def 2
( 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 :: CFUNCT_1:1
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 :: CFUNCT_1:2
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 :: CFUNCT_1:3
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 :: CFUNCT_1:4
for C being ( ( non empty ) ( non empty ) set )
for 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
( dom (r : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non empty V48() V54() V55() ) 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 ) ) : ( ( ) ( ) Element of K19(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) = dom f : ( ( 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 ) ) & ( for c being ( ( ) ( ) Element of C : ( ( non empty ) ( non empty ) set ) ) st c : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) in dom (r : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non empty V48() V54() V55() ) 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 ) ) : ( ( ) ( ) Element of K19(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) holds
(r : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non empty V48() V54() V55() ) 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 ) ) /. c : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) = r : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non empty V48() V54() V55() ) 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 ,) /. 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 :: CFUNCT_1:5
for C being ( ( non empty ) ( non empty ) set )
for 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 ,) holds
( dom (- 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 ) ) : ( ( ) ( ) Element of K19(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) = dom f : ( ( 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 ) ) & ( for c being ( ( ) ( ) Element of C : ( ( non empty ) ( non empty ) set ) ) st c : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) in dom (- 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 ) ) : ( ( ) ( ) Element of K19(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) 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 ,) ) : ( ( 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 ) ) = - (f : ( ( 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 :: CFUNCT_1:6
for C being ( ( non empty ) ( non empty ) set )
for 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
( 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 ) ) c= 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 ,) : ( ( ) ( ) Element of K19(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & (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 ,) ) : ( ( ) ( ) Element of K19(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) /\ ((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 ,) ) : ( ( ) ( ) Element of K19(b1 : ( ( non empty ) ( non empty ) 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 ,) " {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 ) ) = (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 ,) ) : ( ( ) ( ) Element of K19(b1 : ( ( non empty ) ( non empty ) 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 ,) " {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 ) ) ) ;

theorem :: CFUNCT_1:7
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 ) ) ;

theorem :: CFUNCT_1:8
for C being ( ( non empty ) ( non empty ) set )
for c being ( ( ) ( ) Element of C : ( ( non empty ) ( non empty ) set ) )
for 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 ,) st c : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) in dom (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 ,) : ( ( ) ( ) Element of K19(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) 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 ,) /. c : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non empty V48() V54() V55() ) 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 ) ) ) ;

theorem :: CFUNCT_1:9
for C being ( ( non empty ) ( non empty ) set )
for 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 ,) 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 ,) ^) : ( ( 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 ) ) = {} : ( ( ) ( ) set ) ;

theorem :: CFUNCT_1:10
for C being ( ( non empty ) ( non empty ) set )
for 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 ,) 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 ,) .| : ( ( 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 ) ) " {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 ) ) = f : ( ( 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 ) ) & (- 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 ) ) " {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 ) ) = f : ( ( 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 ) ) ) ;

theorem :: CFUNCT_1:11
for C being ( ( non empty ) ( non empty ) set )
for 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 ,) holds dom ((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() ) PartFunc of ,) : ( ( ) ( ) Element of K19(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) = dom (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 (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 ,) ) : ( ( ) ( ) 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 ) ) : ( ( ) ( ) Element of K19(b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: CFUNCT_1:12
for C being ( ( non empty ) ( non empty ) set )
for 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 ) ) st r : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non empty V48() V54() V55() ) 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 ) ) ) holds
(r : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non empty V48() V54() V55() ) 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 ) ) " {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 ) ) = f : ( ( 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 ) ) ;

begin

theorem :: CFUNCT_1:13
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 :: CFUNCT_1:14
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 :: CFUNCT_1:15
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 :: CFUNCT_1:16
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 :: CFUNCT_1:17
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 :: CFUNCT_1:18
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 :: CFUNCT_1:19
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 :: CFUNCT_1:20
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 :: CFUNCT_1:21
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 :: CFUNCT_1:22
for C being ( ( non empty ) ( non empty ) set )
for 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, q being ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) holds (r : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) * q : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non empty V48() V54() V55() ) 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 ) ) = r : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) (#) (q : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non empty V48() V54() V55() ) 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 ) ) : ( ( 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 :: CFUNCT_1:23
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 :: CFUNCT_1:24
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 ) ) = (- 1r : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non empty V48() V54() V55() ) 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 ,) - 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() ) Element of K19(K20(b1 : ( ( non empty ) ( non empty ) set ) ,COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) : ( ( ) ( V38() ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: CFUNCT_1:25
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 :: CFUNCT_1:26
for C being ( ( non empty ) ( non empty ) set )
for 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 ,) holds 1r : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non empty V48() V54() V55() ) 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 ) ) = f : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) ;

theorem :: CFUNCT_1:27
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 :: CFUNCT_1:28
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 :: CFUNCT_1:29
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 :: CFUNCT_1:30
for C being ( ( non empty ) ( non empty ) set )
for 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 ) ) (#) 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( 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 ) ) = |.r : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) .| : ( ( ) ( ext-real complex real ) Element of REAL : ( ( ) ( non empty V48() V49() V50() V54() V55() ) 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( 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 :: CFUNCT_1:31
for C being ( ( non empty ) ( non empty ) set )
for 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 ,) 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 ,) : ( ( 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 ) ) = (- 1r : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non empty V48() V54() V55() ) 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 ) ) ;

theorem :: CFUNCT_1:32
canceled;

theorem :: CFUNCT_1:33
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() ) 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 ) ) ;

theorem :: CFUNCT_1:34
for C being ( ( non empty ) ( non empty ) set )
for 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 ,) 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 ,) ^) : ( ( 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 ,) | (dom (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 ,) ) : ( ( ) ( ) 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 :: CFUNCT_1:35
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 :: CFUNCT_1:36
for C being ( ( non empty ) ( non empty ) set )
for 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 ) ) st r : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non empty V48() V54() V55() ) 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 ) ) ) holds
(r : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non empty V48() V54() V55() ) 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 ) ) ^ : ( ( 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 ) ) ") : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non empty V48() V54() V55() ) 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 ,) : ( ( 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 :: CFUNCT_1:37
for C being ( ( non empty ) ( non empty ) set )
for 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 ,) 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 ,) ) : ( ( 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 ,) = (- 1r : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non empty V48() V54() V55() ) 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 ,) : ( ( 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 :: CFUNCT_1:38
for C being ( ( non empty ) ( non empty ) set )
for 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 ,) 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 ,) .| : ( ( 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 ) ) = |.(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( 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 :: CFUNCT_1:39
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 ,) = 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 :: CFUNCT_1:40
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 :: CFUNCT_1:41
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 :: CFUNCT_1:42
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 :: CFUNCT_1:43
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 :: CFUNCT_1:44
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 :: CFUNCT_1:45
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 :: CFUNCT_1:46
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 :: CFUNCT_1:47
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 :: CFUNCT_1:48
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 :: CFUNCT_1:49
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 :: CFUNCT_1:50
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 :: CFUNCT_1:51
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 :: CFUNCT_1:52
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 :: CFUNCT_1:53
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 :: CFUNCT_1:54
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 :: CFUNCT_1:55
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 :: CFUNCT_1:56
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 ,) ) ;

theorem :: CFUNCT_1:57
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 ,)
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 ) ) | 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 ) ) = 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 ,) | 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 ) ) ;

begin

theorem :: CFUNCT_1:58
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 :: CFUNCT_1:59
for C being ( ( non empty ) ( non empty ) set )
for 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
( f : ( ( 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 iff r : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non empty V48() V54() V55() ) 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 ) ) is total ) ;

theorem :: CFUNCT_1:60
for C being ( ( non empty ) ( non empty ) set )
for 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 ,) 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 ,) is total iff - 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 ) ) is total ) ;

theorem :: CFUNCT_1:61
for C being ( ( non empty ) ( non empty ) set )
for 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 ,) 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 ,) is total iff |.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( 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 ) ) is total ) ;

theorem :: CFUNCT_1:62
for C being ( ( non empty ) ( non empty ) set )
for 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 ,) 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 ,) ^ : ( ( 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 iff ( f : ( ( 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 ) ) = {} : ( ( ) ( ) 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 ,) is total ) ) ;

theorem :: CFUNCT_1:63
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 ,) " {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 ) ) = {} : ( ( ) ( ) 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 ,) is total ) iff 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 ,) is total ) ;

theorem :: CFUNCT_1:64
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 :: CFUNCT_1:65
for C being ( ( non empty ) ( non empty ) set )
for c being ( ( ) ( ) Element of C : ( ( non empty ) ( non empty ) set ) )
for 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 ) ) st f : ( ( 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
(r : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non empty V48() V54() V55() ) 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 ) ) /. c : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) = r : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non empty V48() V54() V55() ) 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 ,) /. 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 :: CFUNCT_1:66
for C being ( ( non empty ) ( non empty ) set )
for c being ( ( ) ( ) Element of C : ( ( non empty ) ( non empty ) set ) )
for 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 ,) st f : ( ( 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
( (- 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 ) ) /. c : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non empty V48() V54() V55() ) 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 ,) /. 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 ) ) & |.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( 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 ) ) . c : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ext-real complex real ) 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 ,) /. c : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) .| : ( ( ) ( ext-real complex real ) Element of REAL : ( ( ) ( non empty V48() V49() V50() V54() V55() ) set ) ) ) ;

theorem :: CFUNCT_1:67
for C being ( ( non empty ) ( non empty ) set )
for c being ( ( ) ( ) Element of C : ( ( non empty ) ( non empty ) set ) )
for 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 ,) st 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 ,) is total 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 ,) ^) : ( ( 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 ) ) = (f : ( ( 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 :: CFUNCT_1:68
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 :: CFUNCT_1:69
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 ,) 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 ,) | 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 iff ex p being ( ( real ) ( ext-real complex real ) number ) st
for c being ( ( ) ( ) Element of C : ( ( non empty ) ( non empty ) set ) ) st c : ( ( ) ( ) Element of b2 : ( ( non empty ) ( non empty ) set ) ) in Y : ( ( ) ( ) set ) /\ (dom f : ( ( Function-like ) ( V1() V4(b2 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) ) : ( ( ) ( ) Element of K19(b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) Element of K19(b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) 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 ,) /. c : ( ( ) ( ) Element of b2 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) .| : ( ( ) ( ext-real complex real ) Element of REAL : ( ( ) ( non empty V48() V49() V50() V54() V55() ) set ) ) <= p : ( ( real ) ( ext-real complex real ) number ) ) ;

theorem :: CFUNCT_1:70
for Y, X being ( ( ) ( ) set )
for C being ( ( non empty ) ( non empty ) set )
for f 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 Y : ( ( ) ( ) set ) c= X : ( ( ) ( ) set ) & f : ( ( 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 holds
f : ( ( 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 ;

theorem :: CFUNCT_1:71
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 ,) st X : ( ( ) ( ) set ) misses dom f : ( ( Function-like ) ( V1() V4(b2 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) : ( ( ) ( ) Element of K19(b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) 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 ,) | 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 ) ) is bounded ;

theorem :: CFUNCT_1:72
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 ,)
for r being ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) 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 bounded 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 ;

theorem :: CFUNCT_1:73
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( 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 ) ) is bounded_below ;

theorem :: CFUNCT_1:74
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 bounded 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( 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 & (- 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 ) ;

theorem :: CFUNCT_1:75
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 :: CFUNCT_1:76
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 :: CFUNCT_1:77
for X, Y being ( ( ) ( ) set )
for C being ( ( non empty ) ( non empty ) set )
for f 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 f : ( ( 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 & f : ( ( 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
f : ( ( 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 ) \/ 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 :: CFUNCT_1:78
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 :: CFUNCT_1:79
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 ,)
for q being ( ( ) ( complex ) Element of COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) 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
(q : ( ( ) ( 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 constant ;

theorem :: CFUNCT_1:80
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
( |.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 constant & (- 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 constant ) ;

theorem :: CFUNCT_1:81
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
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 bounded ;

theorem :: CFUNCT_1:82
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 :: CFUNCT_1:83
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 :: CFUNCT_1:84
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 ) ;

theorem :: CFUNCT_1:85
for C being ( ( non empty ) ( non empty ) set )
for 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 ,) 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 ,) .| : ( ( 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 ) ) is bounded iff f : ( ( Function-like ) ( V1() V4(b1 : ( ( non empty ) ( non empty ) set ) ) V5( COMPLEX : ( ( ) ( non empty V48() V54() V55() ) set ) ) Function-like V38() ) PartFunc of ,) is bounded ) ;