let F be FinSequence of COMPLEX ; :: thesis: for F1, F2 being complex-valued FinSequence st F = addcomplex .: F1,F2 holds
F = addcomplex .: F2,F1
let F1, F2 be complex-valued FinSequence; :: thesis: ( F = addcomplex .: F1,F2 implies F = addcomplex .: F2,F1 )
assume A5: F = addcomplex .: F1,F2 ; :: thesis: F = addcomplex .: F2,F1
reconsider F1 = F1, F2 = F2 as FinSequence of COMPLEX by Lm4;
A6: dom addcomplex = [:COMPLEX ,COMPLEX :] by FUNCT_2:def 1;
then A7: [:(rng F2),(rng F1):] c= dom addcomplex by ZFMISC_1:119;
[:(rng F1),(rng F2):] c= dom addcomplex by A6, ZFMISC_1:119;
then A8: dom (addcomplex .: F1,F2) = (dom F1) /\ (dom F2) by FUNCOP_1:84
.= dom (addcomplex .: F2,F1) by A7, FUNCOP_1:84 ;
for z being set st z in dom (addcomplex .: F2,F1) holds
F . z = addcomplex . (F2 . z),(F1 . z)

proof

let z be set ; :: thesis: ( z in dom (addcomplex .: F2,F1) implies F . z = addcomplex . (F2 . z),(F1 . z) )
assume z in dom (addcomplex .: F2,F1) ; :: thesis: F . z = addcomplex . (F2 . z),(F1 . z)
hence F . z = addcomplex . (F1 . z),(F2 . z) by A5, A8, FUNCOP_1:28
.= (F1 . z) + (F2 . z) by BINOP_2:def 3
.= addcomplex . (F2 . z),(F1 . z) by BINOP_2:def 3 ;
:: thesis: verum

end;

hence F = addcomplex .: F2,F1 by A5, A8, FUNCOP_1:27; :: thesis: verum