let A be non empty set ; :: thesis: for f being Element of Funcs (A,COMPLEX) holds (ComplexFuncMult A) . ((ComplexFuncUnit A),f) = f
let f be Element of Funcs (A,COMPLEX); :: thesis: (ComplexFuncMult A) . ((ComplexFuncUnit A),f) = f

now :: thesis: for x being Element of A holds ((ComplexFuncMult A) . ((ComplexFuncUnit A),f)) . x = f . x

let x be Element of A; :: thesis: ((ComplexFuncMult A) . ((ComplexFuncUnit A),f)) . x = f . x
thus ((ComplexFuncMult A) . ((ComplexFuncUnit A),f)) . x = ((ComplexFuncUnit A) . x) * (f . x) by Th2
.= 1r * (f . x) by FUNCOP_1:7
.= f . x by COMPLEX1:def 4 ; :: thesis: verum

end;

hence (ComplexFuncMult A) . ((ComplexFuncUnit A),f) = f by FUNCT_2:63; :: thesis: verum