let F be FinSequence of REAL ; :: thesis: for F1, F2 being real-valued FinSequence st F = addreal .: (F1,F2) holds
F = addreal .: (F2,F1)
let F1, F2 be real-valued FinSequence; :: thesis: ( F = addreal .: (F1,F2) implies F = addreal .: (F2,F1) )
assume A5: F = addreal .: (F1,F2) ; :: thesis: F = addreal .: (F2,F1)
reconsider F1 = F1, F2 = F2 as FinSequence of REAL by Lm2;
A6: dom addreal = [:REAL,REAL:] by FUNCT_2:def 1;
then A7: [:(rng F2),(rng F1):] c= dom addreal by ZFMISC_1:96;
[:(rng F1),(rng F2):] c= dom addreal by A6, ZFMISC_1:96;
then A8: dom (addreal .: (F1,F2)) = (dom F1) /\ (dom F2) by FUNCOP_1:69
.= dom (addreal .: (F2,F1)) by A7, FUNCOP_1:69 ;
for z being set st z in dom (addreal .: (F2,F1)) holds
F . z = addreal . ((F2 . z),(F1 . z))

proof

let z be set ; :: thesis: ( z in dom (addreal .: (F2,F1)) implies F . z = addreal . ((F2 . z),(F1 . z)) )
assume z in dom (addreal .: (F2,F1)) ; :: thesis: F . z = addreal . ((F2 . z),(F1 . z))
hence F . z = addreal . ((F1 . z),(F2 . z)) by A5, A8, FUNCOP_1:22
.= (F1 . z) + (F2 . z) by BINOP_2:def 9
.= addreal . ((F2 . z),(F1 . z)) by BINOP_2:def 9 ;
:: thesis: verum

end;

hence F = addreal .: (F2,F1) by A5, A8, FUNCOP_1:21; :: thesis: verum