let X be non empty set ; :: thesis: for Y being RealLinearSpace
for f, g, h being Element of Funcs X,the carrier of Y holds (FuncAdd X,Y) . f,((FuncAdd X,Y) . g,h) = (FuncAdd X,Y) . ((FuncAdd X,Y) . f,g),h
let Y be RealLinearSpace; :: thesis: for f, g, h being Element of Funcs X,the carrier of Y holds (FuncAdd X,Y) . f,((FuncAdd X,Y) . g,h) = (FuncAdd X,Y) . ((FuncAdd X,Y) . f,g),h
let f, g, h be Element of Funcs X,the carrier of Y; :: thesis: (FuncAdd X,Y) . f,((FuncAdd X,Y) . g,h) = (FuncAdd X,Y) . ((FuncAdd X,Y) . f,g),h

now

let x be Element of X; :: thesis: ((FuncAdd X,Y) . f,((FuncAdd X,Y) . g,h)) . x = ((FuncAdd X,Y) . ((FuncAdd X,Y) . f,g),h) . x
thus ((FuncAdd X,Y) . f,((FuncAdd X,Y) . g,h)) . x = (f . x) + (((FuncAdd X,Y) . g,h) . x) by Th3
.= (f . x) + ((g . x) + (h . x)) by Th3
.= ((f . x) + (g . x)) + (h . x) by RLVECT_1:def 6
.= (((FuncAdd X,Y) . f,g) . x) + (h . x) by Th3
.= ((FuncAdd X,Y) . ((FuncAdd X,Y) . f,g),h) . x by Th3 ; :: thesis: verum

end;

hence (FuncAdd X,Y) . f,((FuncAdd X,Y) . g,h) = (FuncAdd X,Y) . ((FuncAdd X,Y) . f,g),h by FUNCT_2:113; :: thesis: verum