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

theorem Th21:
  for n being Integer holds n is even iff n mod 2 = 0
proof
  let n be Integer;
  thus n is even implies n mod 2 = 0
  proof
    assume n is even;
    then ex k being Integer st n = 2*k by ABIAN:11;
    hence thesis by NAT_D:71;
  end;
  assume
A2: n mod 2 = 0;
  assume n is odd;
  then consider k being Integer such that
A3: n = 2*k+1 by ABIAN:1;
A4: 2*k mod 2 = 0 by NAT_D:71;
  2*k+1 mod 2 = ((2*k mod 2) + (1 mod 2)) mod 2 by NAT_D:66
  .= 1 by A4,A5;
  hence thesis by A2,A3;
end;
