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

now

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

end;

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