
theorem
  for a,b be non zero Integer holds
    Parity a > Parity b iff (Parity a) div (Parity b) is non zero even
  proof
    let a,b be non zero Integer;
    thus Parity a > Parity b implies (Parity a) div (Parity b) is non zero even
    proof
      assume Parity a > Parity b; then
      2*(Parity b) divides Parity a by P2P; then
      consider c be Integer such that
      A1: Parity a = 2*(Parity b)*c;
      (0 + (2*c)*(Parity b)) div (Parity b) = (0 div (Parity b)) + 2*c
        by NAT_D:61;
      hence thesis by A1;
    end;
    assume (Parity a) div (Parity b) is non zero even; then
B1: (Parity b)*((Parity a) div Parity b) >= 2*Parity b
    by NAT_D:7,NEWTON02:2;
B2: 2*(Parity b) > 1*(Parity b) by XREAL_1:68;
    (Parity b)*((Parity a) div Parity b) + ((Parity a) mod Parity b) >=
    (Parity b)*((Parity a) div Parity b) + 0 by XREAL_1:6; then
    Parity a >= (Parity b)*((Parity a) div Parity b) + 0 by INT_1:59; then
    Parity a >= 2*(Parity b) by B1,XXREAL_0:2;
    hence thesis by B2,XXREAL_0:2;
  end;
