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

now

let x be Element of A; :: thesis: ((RealFuncMult A) . f,((RealFuncAdd A) . g,h)) . x = ((RealFuncAdd A) . ((RealFuncMult A) . f,g),((RealFuncMult A) . f,h)) . x
thus ((RealFuncMult A) . f,((RealFuncAdd A) . g,h)) . x = (f . x) * (((RealFuncAdd A) . g,h) . x) by Th11
.= (f . x) * ((g . x) + (h . x)) by Th10
.= ((f . x) * (g . x)) + ((f . x) * (h . x))
.= (((RealFuncMult A) . f,g) . x) + ((f . x) * (h . x)) by Th11
.= (((RealFuncMult A) . f,g) . x) + (((RealFuncMult A) . f,h) . x) by Th11
.= ((RealFuncAdd A) . ((RealFuncMult A) . f,g),((RealFuncMult A) . f,h)) . x by Th10 ; :: thesis: verum

end;

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