let A be non empty set ; :: thesis: for f, g, h being Element of Funcs (A,COMPLEX) holds
( h = (ComplexFuncAdd A) . (f,g) iff for x being Element of A holds h . x = (f . x) + (g . x) )
let f, g, h be Element of Funcs (A,COMPLEX); :: thesis: ( h = (ComplexFuncAdd A) . (f,g) iff for x being Element of A holds h . x = (f . x) + (g . x) )
A1: now :: thesis: ( ( for x being Element of A holds h . x = (f . x) + (g . x) ) implies h = (ComplexFuncAdd A) . (f,g) )
assume A2: for x being Element of A holds h . x = (f . x) + (g . x) ; :: thesis: h = (ComplexFuncAdd A) . (f,g)
now :: thesis: for x being Element of A holds ((ComplexFuncAdd A) . (f,g)) . x = h . x
let x be Element of A; :: thesis: ((ComplexFuncAdd A) . (f,g)) . x = h . x
A3: x in dom (addcomplex .: (f,g)) by Lm1;
thus ((ComplexFuncAdd A) . (f,g)) . x = (addcomplex .: (f,g)) . x by Def1
.= addcomplex . ((f . x),(g . x)) by A3, FUNCOP_1:22
.= (f . x) + (g . x) by BINOP_2:def 3
.= h . x by A2 ; :: thesis: verum
end;
hence h = (ComplexFuncAdd A) . (f,g) by FUNCT_2:63; :: thesis: verum
end;
now :: thesis: ( h = (ComplexFuncAdd A) . (f,g) implies for x being Element of A holds h . x = (f . x) + (g . x) )
assume A4: h = (ComplexFuncAdd A) . (f,g) ; :: thesis: for x being Element of A holds h . x = (f . x) + (g . x)
let x be Element of A; :: thesis: h . x = (f . x) + (g . x)
A5: x in dom (addcomplex .: (f,g)) by Lm1;
thus h . x = (addcomplex .: (f,g)) . x by A4, Def1
.= addcomplex . ((f . x),(g . x)) by A5, FUNCOP_1:22
.= (f . x) + (g . x) by BINOP_2:def 3 ; :: thesis: verum
end;
hence ( h = (ComplexFuncAdd A) . (f,g) iff for x being Element of A holds h . x = (f . x) + (g . x) ) by A1; :: thesis: verum