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

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

let x be Element of A; :: thesis: ((ComplexFuncMult A) . (f,((ComplexFuncAdd A) . (g,h)))) . x = ((ComplexFuncAdd A) . (((ComplexFuncMult A) . (f,g)),((ComplexFuncMult A) . (f,h)))) . x
thus ((ComplexFuncMult A) . (f,((ComplexFuncAdd A) . (g,h)))) . x = (f . x) * (((ComplexFuncAdd A) . (g,h)) . x) by Th2
.= (f . x) * ((g . x) + (h . x)) by Th1
.= ((f . x) * (g . x)) + ((f . x) * (h . x))
.= (((ComplexFuncMult A) . (f,g)) . x) + ((f . x) * (h . x)) by Th2
.= (((ComplexFuncMult A) . (f,g)) . x) + (((ComplexFuncMult A) . (f,h)) . x) by Th2
.= ((ComplexFuncAdd A) . (((ComplexFuncMult A) . (f,g)),((ComplexFuncMult A) . (f,h)))) . x by Th1 ; :: thesis: verum

end;

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