reserve a, b, k, n, m for Nat,
  i for Integer,
  r for Real,
  p for Rational,
  c for Complex,
  x for object,
  f for Function;

theorem Th4:
  m mod n = 0 implies m / n = m div n
proof
  reconsider i = m as Integer;
  assume
A1: m mod n = 0;
  per cases;
  suppose
A2: n = 0;
    hence m / n = 0 .= m div n by A2;
  end;
  suppose
A3: n <> 0;
    then i - (i div n) * n = 0 by A1,INT_1:def 10;
    hence thesis by A3,XCMPLX_1:89;
  end;
end;
