let F be FinSequence of REAL ; :: thesis: for F1, F2 being real-valued FinSequence st F = multreal .: F1,F2 holds
F = multreal .: F2,F1
let F1, F2 be real-valued FinSequence; :: thesis: ( F = multreal .: F1,F2 implies F = multreal .: F2,F1 )
assume A5: F = multreal .: F1,F2 ; :: thesis: F = multreal .: F2,F1
A6: dom multreal = [:REAL ,REAL :] by FUNCT_2:def 1;
reconsider F3 = F1, F4 = F2 as FinSequence of REAL by Lm4;
A7: [:(rng F4),(rng F3):] c= dom multreal by A6, ZFMISC_1:119;
[:(rng F3),(rng F4):] c= dom multreal by A6, ZFMISC_1:119;
then A8: dom (multreal .: F1,F2) = (dom F1) /\ (dom F2) by FUNCOP_1:84
.= dom (multreal .: F2,F1) by A7, FUNCOP_1:84 ;
for z being set st z in dom (multreal .: F2,F1) holds
F . z = multreal . (F2 . z),(F1 . z)

proof

let z be set ; :: thesis: ( z in dom (multreal .: F2,F1) implies F . z = multreal . (F2 . z),(F1 . z) )
assume A9: z in dom (multreal .: F2,F1) ; :: thesis: F . z = multreal . (F2 . z),(F1 . z)
set F1z = F1 . z;
set F2z = F2 . z;
thus F . z = multreal . (F1 . z),(F2 . z) by A5, A8, A9, FUNCOP_1:28
.= (F1 . z) * (F2 . z) by BINOP_2:def 11
.= multreal . (F2 . z),(F1 . z) by BINOP_2:def 11 ; :: thesis: verum

end;

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