
theorem
  for x,y,n being Integer
  holds x,y are_congruent_mod n iff ex k being Integer st x = k * n + y
proof
  let x,y,n be Integer;
  now assume x,y are_congruent_mod n;
    then consider k being Integer such that
 A1: n * k = x - y;
    x = n * k + y by A1;
    hence ex k being Integer st x = k * n + y;
  end;
  hence thesis;
end;
