let A be non empty set ; :: thesis: for h, f, g 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 h, f, g 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

assume A2: 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)
A3: x in dom (addcomplex .: f,g) by Lm1;
thus h . x = (addcomplex .: f,g) . x by A2, Def1
.= addcomplex . (f . x),(g . x) by A3, FUNCOP_1:28
.= (f . x) + (g . x) by BINOP_2:def 3 ; :: thesis: verum

end;

now

assume A4: for x being Element of A holds h . x = (f . x) + (g . x) ; :: thesis: h = (ComplexFuncAdd A) . f,g

now

let x be Element of A; :: thesis: ((ComplexFuncAdd A) . f,g) . x = h . x
A5: 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 A5, FUNCOP_1:28
.= (f . x) + (g . x) by BINOP_2:def 3
.= h . x by A4 ; :: thesis: verum

end;

hence h = (ComplexFuncAdd A) . f,g by FUNCT_2:113; :: 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