
theorem PGC:
  for a,b be non zero Integer holds
    Parity (a gcd b) = min (Parity a,Parity b)
  proof
    let a,b be non zero Integer;
    reconsider c = a gcd b as non zero Nat;
    A1: Parity c = 2|^(2|-count (a gcd b)) & Parity b = 2|^(2|-count b)
     & Parity a = 2|^(2|-count a) by Def1;
     2 is non trivial; then
     2|^(2|-count (a gcd b)) = 2|^(min (2|-count a, 2|-count b)) by CCM
     .= min (2|^(2|-count a), 2|^(2|-count b)) by MIN1;
     hence thesis by A1;
   end;
