reserve i, j, k, l, m, n, t for Nat;

theorem
  n > 0 implies (n mod 2 = 0 iff (n -' 1) mod 2 = 1)
proof
  assume
A1: n > 0;
  thus n mod 2 = 0 implies (n -' 1) mod 2 = 1
  proof
    consider t be Nat such that
A2: n = 2 * t + (n mod 2) and
    (n mod 2) < 2 by NAT_D:def 2;
    assume
A3: n mod 2 = 0;
    then t > 0 by A1,A2;
    then
A4: t >= 0 + 1 by NAT_1:13;
    n >= 0 + 1 by A1,NAT_1:13;
    then n -' 1 = 2 * (t - 1 + 1) - 1 by A3,A2,XREAL_1:233
      .= 2 * (t - 1) + (1 + 1 - 1)
      .= 2 * (t-'1) + 1 by A4,XREAL_1:233;
    hence thesis by NAT_D:def 2;
  end;
  assume (n -' 1) mod 2 = 1;
  then consider t be Nat such that
A5: n -' 1 = 2 * t + 1 and
  1 < 2 by NAT_D:def 2;
  n >= 0 + 1 by A1,NAT_1:13;
  then n = 2 * t + 1 + 1 by A5,XREAL_1:235
    .= 2 * (t + 1) + 0;
  hence thesis by NAT_D:def 2;
end;
