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

theorem
  for n being Integer holds n is odd iff n mod 2 = 1
proof
  let n be Integer;
  thus n is odd implies n mod 2 = 1
  proof
    assume n is odd;
    then consider k being Integer such that
A1: n = 2*k+1 by ABIAN:1;
    2*k+1 mod 2 = ((2*k mod 2) + (1 mod 2)) mod 2 by NAT_D:66
    .= (0+1) mod 2 by A5,NAT_D:71
    .= 1 by NAT_D:24;
    hence thesis by A1;
  end;
  assume
A2: n mod 2 = 1;
  assume n is even;
  then ex k being Integer st n = 2*k by ABIAN:11;
  hence thesis by A2,NAT_D:71;
end;
