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

let f be Element of Funcs (A,COMPLEX); :: thesis: for a, b being Complex holds () . (( . [a,f]),( . [b,f])) = . [(a + b),f]
let a, b be Complex; :: thesis: () . (( . [a,f]),( . [b,f])) = . [(a + b),f]
reconsider a = a, b = b as Element of COMPLEX by XCMPLX_0:def 2;
reconsider ab = a + b as Element of COMPLEX by XCMPLX_0:def 2;
now :: thesis: for x being Element of A holds (() . (( . [a,f]),( . [b,f]))) . x = ( . [ab,f]) . x
let x be Element of A; :: thesis: (() . (( . [a,f]),( . [b,f]))) . x = ( . [ab,f]) . x
thus (() . (( . [a,f]),( . [b,f]))) . x = (( . [a,f]) . x) + (( . [b,f]) . x) by Th1
.= (a * (f . x)) + (( . [b,f]) . x) by Th4
.= (a * (f . x)) + (b * (f . x)) by Th4
.= (a + b) * (f . x)
.= ( . [ab,f]) . x by Th4 ; :: thesis: verum
end;
hence (ComplexFuncAdd A) . (( . [a,f]),( . [b,f])) = . [(a + b),f] by FUNCT_2:63; :: thesis: verum