reconsider F3 = F1, F4 = F2 as FinSequence of COMPLEX by Lm2;
let F be FinSequence of COMPLEX ; :: thesis: ( F = F1 - F2 iff F = diffcomplex .: (F1,F2) )
A1: dom (F1 - F2) = (dom F1) /\ (dom F2) by VALUED_1:12;
dom diffcomplex = [:COMPLEX,COMPLEX:] by FUNCT_2:def 1;
then A2: [:(rng F3),(rng F4):] c= dom diffcomplex by ZFMISC_1:96;
then A3: dom (diffcomplex .: (F1,F2)) = (dom F1) /\ (dom F2) by FUNCOP_1:69;
thus ( F = F1 - F2 implies F = diffcomplex .: (F1,F2) ) :: thesis: ( F = diffcomplex .: (F1,F2) implies F = F1 - F2 )
proof
assume A4: F = F1 - F2 ; :: thesis: F = diffcomplex .: (F1,F2)
for z being set st z in dom (diffcomplex .: (F1,F2)) holds
F . z = diffcomplex . ((F1 . z),(F2 . z))
proof
let z be set ; :: thesis: ( z in dom (diffcomplex .: (F1,F2)) implies F . z = diffcomplex . ((F1 . z),(F2 . z)) )
assume z in dom (diffcomplex .: (F1,F2)) ; :: thesis: F . z = diffcomplex . ((F1 . z),(F2 . z))
hence F . z = (F1 . z) - (F2 . z) by A1, A3, A4, VALUED_1:13
.= diffcomplex . ((F1 . z),(F2 . z)) by BINOP_2:def 4 ;
:: thesis: verum
end;
hence F = diffcomplex .: (F1,F2) by A1, A3, A4, FUNCOP_1:21; :: thesis: verum
end;
assume A5: F = diffcomplex .: (F1,F2) ; :: thesis: F = F1 - F2
now :: thesis: for c being object st c in dom F holds
F . c = (F1 . c) - (F2 . c)
let c be object ; :: thesis: ( c in dom F implies F . c = (F1 . c) - (F2 . c) )
assume c in dom F ; :: thesis: F . c = (F1 . c) - (F2 . c)
hence F . c = diffcomplex . ((F1 . c),(F2 . c)) by A5, FUNCOP_1:22
.= (F1 . c) - (F2 . c) by BINOP_2:def 4 ;
:: thesis: verum
end;
hence F = F1 - F2 by A1, A2, A5, FUNCOP_1:69, VALUED_1:14; :: thesis: verum