set g = multreal ;
set h = multcomplex ;
let F be FinSequence of REAL ; :: thesis: Product F = multreal $$ F
rng F c= COMPLEX by NUMBERS:11, XBOOLE_1:1;
then reconsider f = F as FinSequence of COMPLEX by FINSEQ_1:def 4;
defpred S₁[ Nat] means multreal $$ ((finSeg $1),([#] (F,(the_unity_wrt multreal)))) = multcomplex $$ ((finSeg $1),([#] (f,(the_unity_wrt multcomplex))));
consider n being Nat such that
A1: dom f = Seg n by FINSEQ_1:def 2;
A2: ( multreal $$ F = multreal $$ ((finSeg n),([#] (F,(the_unity_wrt multreal)))) & multcomplex $$ f = multcomplex $$ ((finSeg n),([#] (f,(the_unity_wrt multcomplex)))) ) by A1, SETWOP_2:def 2;
A3: for k being Nat st S₁[k] holds
S₁[k + 1]

proof

set j = [#] (f,(the_unity_wrt multcomplex));
set i = [#] (F,(the_unity_wrt multreal));
let k be Nat; :: thesis: ( S₁[k] implies S₁[k + 1] )
assume A4: S₁[k] ; :: thesis: S₁[k + 1]
A5: ([#] (F,(the_unity_wrt multreal))) . (k + 1) = ([#] (f,(the_unity_wrt multcomplex))) . (k + 1)

proof

per cases ( k + 1 in dom f or not k + 1 in dom f ) ;

suppose A6: k + 1 in dom f ; :: thesis: ([#] (F,(the_unity_wrt multreal))) . (k + 1) = ([#] (f,(the_unity_wrt multcomplex))) . (k + 1)

then ([#] (f,(the_unity_wrt multcomplex))) . (k + 1) = F . (k + 1) by SETWOP_2:20
.= ([#] (F,(the_unity_wrt multreal))) . (k + 1) by A6, SETWOP_2:20 ;
hence ([#] (F,(the_unity_wrt multreal))) . (k + 1) = ([#] (f,(the_unity_wrt multcomplex))) . (k + 1) ; :: thesis: verum

end;

suppose A7: not k + 1 in dom f ; :: thesis: ([#] (F,(the_unity_wrt multreal))) . (k + 1) = ([#] (f,(the_unity_wrt multcomplex))) . (k + 1)

then ([#] (f,(the_unity_wrt multcomplex))) . (k + 1) = the_unity_wrt multcomplex by SETWOP_2:20
.= ([#] (F,(the_unity_wrt multreal))) . (k + 1) by A7, BINOP_2:6, BINOP_2:7, SETWOP_2:20 ;
hence ([#] (F,(the_unity_wrt multreal))) . (k + 1) = ([#] (f,(the_unity_wrt multcomplex))) . (k + 1) ; :: thesis: verum

end;

end;

end;

A8: not k + 1 in Seg k by FINSEQ_3:8;
reconsider k = k as Element of NAT by ORDINAL1:def 12;
multreal $$ ((finSeg (k + 1)),([#] (F,(the_unity_wrt multreal)))) = multreal $$ (((finSeg k) \/ {.(In ((k + 1),NAT)).}),([#] (F,(the_unity_wrt multreal)))) by FINSEQ_1:9
.= multreal . ((multreal $$ ((finSeg k),([#] (F,(the_unity_wrt multreal))))),(([#] (F,(the_unity_wrt multreal))) . (k + 1))) by A8, SETWOP_2:2
.= (multreal $$ ((finSeg k),([#] (F,(the_unity_wrt multreal))))) * (([#] (F,(the_unity_wrt multreal))) . (k + 1)) by BINOP_2:def 11
.= multcomplex . ((multcomplex $$ ((finSeg k),([#] (f,(the_unity_wrt multcomplex))))),(([#] (f,(the_unity_wrt multcomplex))) . (k + 1))) by A4, A5, BINOP_2:def 5
.= multcomplex $$ (((finSeg k) \/ {.(In ((k + 1),NAT)).}),([#] (f,(the_unity_wrt multcomplex)))) by A8, SETWOP_2:2
.= multcomplex $$ ((finSeg (k + 1)),([#] (f,(the_unity_wrt multcomplex)))) by FINSEQ_1:9 ;
hence S₁[k + 1] ; :: thesis: verum

end;

A9: Seg 0 = {}. NAT ;
then multreal $$ ((finSeg 0),([#] (F,(the_unity_wrt multreal)))) = the_unity_wrt multcomplex by BINOP_2:6, BINOP_2:7, SETWISEO:31
.= multcomplex $$ ((finSeg 0),([#] (f,(the_unity_wrt multcomplex)))) by A9, SETWISEO:31 ;
then A10: S₁[ 0 ] ;
for k being Nat holds S₁[k] from NAT_1:sch 2(A10, A3);
then multreal $$ F = multcomplex $$ f by A2;
hence Product F = multreal $$ F by Def13; :: thesis: verum