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))]
reconsider a = a as Element of COMPLEX by XCMPLX_0:def 2;

now :: thesis: for x being Element of A holds ((ComplexFuncAdd A) . (((ComplexFuncExtMult A) . [a,f]),((ComplexFuncExtMult A) . [a,g]))) . x = ((ComplexFuncExtMult A) . [a,((ComplexFuncAdd A) . (f,g))]) . x

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 Th4
.= (a * (f . x)) + (a * (g . x)) by Th4
.= a * ((f . x) + (g . x))
.= a * (((ComplexFuncAdd A) . (f,g)) . x) by Th1
.= ((ComplexFuncExtMult A) . [a,((ComplexFuncAdd A) . (f,g))]) . x by Th4 ; :: thesis: verum

end;

hence (ComplexFuncAdd A) . (((ComplexFuncExtMult A) . [a,f]),((ComplexFuncExtMult A) . [a,g])) = (ComplexFuncExtMult A) . [a,((ComplexFuncAdd A) . (f,g))] by FUNCT_2:63; :: thesis: verum