let cF be complex-valued XFinSequence; :: thesis: ( cF is natural-valued implies Product cF = multnat "**" cF )
assume cF is natural-valued ; :: thesis: Product cF = multnat "**" cF
then rng cF c= NAT by VALUED_0:def 6;
then A1: cF is NAT -valued by RELAT_1:def 19;

per cases ( len cF = 0 or len cF >= 1 ) by NAT_1:14;

suppose A2: len cF = 0 ; :: thesis: Product cF = multnat "**" cF

hence multnat "**" cF = the_unity_wrt multcomplex by BINOP_2:6, BINOP_2:10, AFINSQ_2:def 8, A1
.= Product cF by AFINSQ_2:def 8, A2 ;
:: thesis: verum

end;

suppose A3: len cF >= 1 ; :: thesis: Product cF = multnat "**" cF

A4: NAT = NAT /\ COMPLEX by MEMBERED:1, XBOOLE_1:28;

now :: thesis: for x, y being object st x in NAT & y in NAT holds
( multnat . (x,y) = multcomplex . (x,y) & multnat . (x,y) in NAT )

let x, y be object ; :: thesis: ( x in NAT & y in NAT implies ( multnat . (x,y) = multcomplex . (x,y) & multnat . (x,y) in NAT ) )
assume ( x in NAT & y in NAT ) ; :: thesis: ( multnat . (x,y) = multcomplex . (x,y) & multnat . (x,y) in NAT )
then reconsider X = x, Y = y as Element of NAT ;
multnat . (x,y) = X * Y by BINOP_2:def 24;
hence ( multnat . (x,y) = multcomplex . (x,y) & multnat . (x,y) in NAT ) by BINOP_2:def 5, ORDINAL1:def 12; :: thesis: verum

end;

hence Product cF = multnat "**" cF by AFINSQ_2:47, A3, A4, A1; :: thesis: verum

end;

end;