reserve X for set, x,y,z for object,
  k,l,n for Nat,
  r for Real;
reserve i,i0,i1,i2,i3,i4,i5,i8,i9,j for Integer;
reserve r1,p,p1,g,g1,g2 for Real,
  Y for Subset of REAL;

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;
