
theorem Th10:
  for m being INT-valued FinSequence, i being Nat
  st i in dom m ex z being Integer st z * m.i = Product(m)
proof
  let m be INT-valued FinSequence, i be Nat;
  assume
A1: i in dom m;
  per cases;
  suppose
A2: m.i <> 0;
    take z = Product(m) / m.i;
    thus z * m.i = Product(m) * ((m.i)" * m.i)
      .= Product(m) * 1 by A2,XCMPLX_0:def 7
      .= Product(m);
  end;
  suppose
A3: m.i = 0;
    take 1;
    thus thesis by A1,A3,Th6;
  end;
end;
