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

now

let x be Element of A; :: thesis: ((ComplexFuncAdd A) . ((ComplexFuncExtMult A) . [a,f]),((ComplexFuncExtMult A) . [a,g])) . x = ((ComplexFuncExtMult A) . [a,((ComplexFuncAdd A) . f,g)]) . x
thus ((ComplexFuncAdd A) . ((ComplexFuncExtMult A) . [a,f]),((ComplexFuncExtMult A) . [a,g])) . x = (((ComplexFuncExtMult A) . [a,f]) . x) + (((ComplexFuncExtMult A) . [a,g]) . x) by Th1
.= (a * (f . x)) + (((ComplexFuncExtMult A) . [a,g]) . x) by Th6
.= (a * (f . x)) + (a * (g . x)) by Th6
.= a * ((f . x) + (g . x))
.= a * (((ComplexFuncAdd A) . f,g) . x) by Th1
.= ((ComplexFuncExtMult A) . [a,((ComplexFuncAdd A) . f,g)]) . x by Th6 ; :: thesis: verum

end;

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