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

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_2:63; :: thesis: verum