let A be non empty set ; :: thesis: for f, g being Element of Funcs A,COMPLEX
for a being Complex holds (ComplexFuncMult A) . ((ComplexFuncExtMult A) . [a,f]),g = (ComplexFuncExtMult A) . [a,((ComplexFuncMult A) . f,g)]
let f, g be Element of Funcs A,COMPLEX ; :: thesis: for a being Complex holds (ComplexFuncMult A) . ((ComplexFuncExtMult A) . [a,f]),g = (ComplexFuncExtMult A) . [a,((ComplexFuncMult A) . f,g)]
let a be Complex; :: thesis: (ComplexFuncMult A) . ((ComplexFuncExtMult A) . [a,f]),g = (ComplexFuncExtMult A) . [a,((ComplexFuncMult A) . f,g)]

now

let x be Element of A; :: thesis: ((ComplexFuncMult A) . ((ComplexFuncExtMult A) . [a,f]),g) . x = ((ComplexFuncExtMult A) . [a,((ComplexFuncMult A) . f,g)]) . x
thus ((ComplexFuncMult A) . ((ComplexFuncExtMult A) . [a,f]),g) . x = (((ComplexFuncExtMult A) . [a,f]) . x) * (g . x) by Th2
.= (a * (f . x)) * (g . x) by Th6
.= a * ((f . x) * (g . x))
.= a * (((ComplexFuncMult A) . f,g) . x) by Th2
.= ((ComplexFuncExtMult A) . [a,((ComplexFuncMult A) . f,g)]) . x by Th6 ; :: thesis: verum

end;

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