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

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_2:113; :: thesis: verum