theorem Th33:
  for r being Nat st r = numberR(a,b,c) holds
  two_or_more_are_odd_among a+r, b+r, c+r
  proof
    let r be Nat such that
A1: r = numberR(a,b,c);
    per cases;
    suppose
A2:   two_or_more_are_even_among a,b,c;
      then r = 1 by A1,Def9;
      hence thesis by A2;
    end;
    suppose
A3:   not two_or_more_are_even_among a,b,c;
      then r = 0 by A1,Def9;
      hence thesis by A3;
    end;
  end;
