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 Th1:
  cF is real-valued implies Product cF = multreal "**" cF
proof
  assume
A1: cF is real-valued;
  per cases by NAT_1:14;
  suppose
A2: len cF=0;
    hence multreal "**" cF = the_unity_wrt multcomplex
        by BINOP_2:6,7,AFINSQ_2:def 8,A1
      .= Product cF by AFINSQ_2:def 8,A2;
  end;
  suppose
A3: len cF>=1;
A4: REAL = REAL /\ COMPLEX by MEMBERED:1,XBOOLE_1:28;
    now let x,y be object; assume x in REAL & y in REAL;
      then reconsider X=x,Y=y as Element of REAL;
      multreal.(x,y) = X*Y by BINOP_2:def 11;
      hence multreal.(x,y) =multcomplex.(x,y) & multreal.(x,y) in REAL
        by BINOP_2:def 5,XREAL_0:def 1;
    end;
    hence thesis by AFINSQ_2:47,A3,A4,A1;
  end;
end;
