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

now

let x be Element of A; :: thesis: ((RealFuncMult A) . f,((RealFuncMult A) . g,h)) . x = ((RealFuncMult A) . ((RealFuncMult A) . f,g),h) . x
thus ((RealFuncMult A) . f,((RealFuncMult A) . g,h)) . x = (f . x) * (((RealFuncMult A) . g,h) . x) by Th11
.= (f . x) * ((g . x) * (h . x)) by Th11
.= ((f . x) * (g . x)) * (h . x)
.= (((RealFuncMult A) . f,g) . x) * (h . x) by Th11
.= ((RealFuncMult A) . ((RealFuncMult A) . f,g),h) . x by Th11 ; :: thesis: verum

end;

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