
theorem
  for m being INT-valued FinSequence, i,j being Nat st i <> j holds
  Product(m) / (m.i * m.j) is Integer
proof
  let m be INT-valued FinSequence, i9,j9 be Nat;
  assume that
A3: j9 <> i9;
    per cases;
    suppose that
A1:   i9 in dom m;
      reconsider i = i9, j = j9 as Element of NAT by ORDINAL1:def 12;
A5:   ex z being Integer st z * m.i = Product(m) / m.j by A1,A3,Lm6;
      per cases;
      suppose m.i = 0;
        hence thesis;
      end;
      suppose
A6:     m.i <> 0;
        reconsider u = Product(m) / m.j as Integer;
A7:     u / m.i = Product(m) * ((m.j)" * (m.i)")
        .= Product(m) / (m.i * m.j) by XCMPLX_1:204;
        m.i divides u by A5,INT_1:def 3;
        hence thesis by A6,A7,WSIERP_1:17;
      end;
    end;
    suppose not i9 in dom m;
      then m.i9 = 0 by FUNCT_1:def 2;
      hence thesis;
    end;
  end;
