theorem
  for i being Integer holds i mod i = 0
proof
  let i be Integer;
  per cases;
  suppose
    i = 0;
    hence thesis by Def10;
  end;
  suppose
A1: i <> 0;
    hence i mod i = i - (i div i) * i by Def10
      .= i - 1 * i by A1,Th45
      .= 0;
  end;
end;
