let A be non empty set ; :: thesis: for h, f, g being Element of Funcs A,COMPLEX holds
( h = (ComplexFuncMult 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 = (ComplexFuncMult A) . f,g iff for x being Element of A holds h . x = (f . x) * (g . x) )

A1: now

assume A2: for x being Element of A holds h . x = (f . x) * (g . x) ; :: thesis: h = (ComplexFuncMult A) . f,g

now

let x be Element of A; :: thesis: ((ComplexFuncMult A) . f,g) . x = h . x
A3: x in dom (multcomplex .: f,g) by Lm1;
thus ((ComplexFuncMult A) . f,g) . x = (multcomplex .: f,g) . x by Def2
.= multcomplex . (f . x),(g . x) by A3, FUNCOP_1:28
.= (f . x) * (g . x) by BINOP_2:def 5
.= h . x by A2 ; :: thesis: verum

end;

hence h = (ComplexFuncMult A) . f,g by FUNCT_2:113; :: thesis: verum

end;

now

assume A4: h = (ComplexFuncMult 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 (multcomplex .: f,g) by Lm1;
thus h . x = (multcomplex .: f,g) . x by A4, Def2
.= multcomplex . (f . x),(g . x) by A5, FUNCOP_1:28
.= (f . x) * (g . x) by BINOP_2:def 5 ; :: thesis: verum

end;

hence ( h = (ComplexFuncMult A) . f,g iff for x being Element of A holds h . x = (f . x) * (g . x) ) by A1; :: thesis: verum