
theorem
  for i being Integer holds i mod 2 = 0 or i mod 2 = 1
proof
  let i be Integer;
  set A = (i div 2)*2;
  set M = i mod 2;
A1: i div 2 = [\ i / 2 /] by INT_1:def 9;
  then i/2-1<i div 2 by INT_1:def 6;
  then (i/2-1)*2 < A by XREAL_1:68;
  then -(i-2) > -A by XREAL_1:24;
  then i+(2-i) > i+-A by XREAL_1:6;
  then 2 > i-A;
  then
A2: 2 > M by INT_1:def 10;
  i div 2<=i/2 by A1,INT_1:def 6;
  then A <= (i/2)*2 by XREAL_1:64;
  then -i <= -A by XREAL_1:24;
  then i+-i <= i+-A by XREAL_1:6;
  then 0 <= i-A;
  then 0 <= M by INT_1:def 10;
  then reconsider M as Element of NAT by INT_1:3;
  M in { k where k is Nat : k < 2 } by A2;
  then M in 2 by AXIOMS:4;
  hence thesis by CARD_1:50,TARSKI:def 2;
end;
