let i be Element of NAT ; :: thesis: for x, y being complex-valued FinSequence st len x = len y holds
(x + y) . i = (x . i) + (y . i)
let x, y be complex-valued FinSequence; :: thesis: ( len x = len y implies (x + y) . i = (x . i) + (y . i) )
A: ( x is FinSequence of COMPLEX & y is FinSequence of COMPLEX ) by Lm4;
then reconsider x2 = x as Element of (len x) -tuples_on COMPLEX by FINSEQ_2:110;
reconsider y2 = y as Element of (len y) -tuples_on COMPLEX by A, FINSEQ_2:110;
reconsider y29 = x2 + y2 as Element of (len (x2 + y2)) -tuples_on COMPLEX by FINSEQ_2:110;
assume A1: len x = len y ; :: thesis: (x + y) . i = (x . i) + (y . i)
then A2: len (x + y) = len x by Th6;

per cases ( not i in Seg (len x) or i in Seg (len x) ) ;

suppose A3: not i in Seg (len x) ; :: thesis: (x + y) . i = (x . i) + (y . i)

then A4: not i in dom x2 by FINSEQ_2:144;
A5: not i in dom y2 by A1, A3, FINSEQ_2:144;
not i in dom y29 by A2, A3, FINSEQ_2:144;
then (x2 + y2) . i = 0 + 0 by FUNCT_1:def 4
.= (x2 . i) + 0 by A4, FUNCT_1:def 4
.= (x2 . i) + (y2 . i) by A5, FUNCT_1:def 4 ;
hence (x + y) . i = (x . i) + (y . i) ; :: thesis: verum

end;

suppose i in Seg (len x) ; :: thesis: (x + y) . i = (x . i) + (y . i)

then i in dom y29 by A2, FINSEQ_2:144;
hence (x + y) . i = (x . i) + (y . i) by VALUED_1:def 1; :: thesis: verum

end;

end;