
theorem
  for n being Element of NAT holds PFArt (1,n) = { [1,n], [2,n] }
proof
  let n be Element of NAT;
  thus PFArt (1, n) c= { [1,n], [2,n] }
  proof
    let x be object;
    assume
A1: x in PFArt (1, n);
    per cases by A1,Def2;
    suppose ex m being odd Element of NAT st m <= 2*1 & [m,n] = x;
     then consider m being odd Element of NAT such that
A2:     m <= 2*1 & [m,n] = x;
      m = 0 or ... or m = 2 by A2;
      then x = [1,n] by A2;
      hence thesis by TARSKI:def 2;
    end;
    suppose
      [2*1,n] = x;
      hence thesis by TARSKI:def 2;
    end;
  end;
  let x be object;
  assume
A3: x in { [1,n], [2,n] };
  per cases by A3,TARSKI:def 2;
  suppose
A4: x = [1,n];
    1 <= 2 * 1;
    hence thesis by A4,Def2,Lm2,Lm3;
  end;
  suppose
    x = [2,n];
    then x = [2*1,n];
    hence thesis by Def2;
  end;
end;
