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

now

let x be Element of A; :: thesis: ((RealFuncMult A) . ((RealFuncExtMult A) . [a,f]),g) . x = ((RealFuncExtMult A) . [a,((RealFuncMult A) . f,g)]) . x
thus ((RealFuncMult A) . ((RealFuncExtMult A) . [a,f]),g) . x = (((RealFuncExtMult A) . [a,f]) . x) * (g . x) by Th11
.= (a * (f . x)) * (g . x) by Th15
.= a * ((f . x) * (g . x))
.= a * (((RealFuncMult A) . f,g) . x) by Th11
.= ((RealFuncExtMult A) . [a,((RealFuncMult A) . f,g)]) . x by Th15 ; :: thesis: verum

end;

hence (RealFuncMult A) . ((RealFuncExtMult A) . [a,f]),g = (RealFuncExtMult A) . [a,((RealFuncMult A) . f,g)] by FUNCT_2:113; :: thesis: verum