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
  cF is INT-valued implies Product cF = multint "**" cF
proof
  assume
A1: cF is INT-valued;
  per cases by NAT_1:14;
    suppose
A2:   len cF=0;
      hence multint "**" cF = the_unity_wrt multcomplex
          by BINOP_2:6,9,AFINSQ_2:def 8,A1
        .= Product cF by AFINSQ_2:def 8,A2;
    end;
    suppose
A3:      len cF>=1;
A4:      INT = INT /\ COMPLEX by MEMBERED:1,XBOOLE_1:28;
       now let x,y be object;
         assume x in INT & y in INT;
         then reconsider X=x,Y=y as Element of INT;
         multint.(x,y) = X*Y by BINOP_2:def 22;
         hence multint.(x,y) = multcomplex.(x,y) & multint.(x,y) in INT
           by BINOP_2:def 5,INT_1:def 2;
       end;
       hence thesis by AFINSQ_2:47,A3,A4,A1;
     end;
end;
