reserve i,j,n,n1,n2,m,k,l,u for Nat,
        r,r1,r2 for Real,
        x,y for Integer,
        a,b for non trivial Nat,
        F for XFinSequence,
        cF,cF1,cF2 for complex-valued XFinSequence,
        c,c1,c2 for Complex;

theorem Th3:
  cF is natural-valued implies Product cF = multnat "**" cF
proof
  assume cF is natural-valued;
  then rng cF c= NAT by VALUED_0:def 6;
  then
A1:  cF is NAT-valued by RELAT_1:def 19;
  per cases by NAT_1:14;
  suppose
A2:   len cF=0;
    hence multnat "**" cF = the_unity_wrt multcomplex
        by BINOP_2:6,10,AFINSQ_2:def 8,A1
      .= Product cF by AFINSQ_2:def 8,A2;
    end;
    suppose
A3:   len cF>=1;
A4:   NAT = NAT /\ COMPLEX by MEMBERED:1,XBOOLE_1:28;
      now let x,y be object;
        assume x in NAT & 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;
      end;
      hence thesis by AFINSQ_2:47,A3,A4,A1;
    end;
end;
