
theorem
  for X being finite Subset of NAT holds ex k being odd Element of NAT
  st not k in X
proof
  let X be finite Subset of NAT;
  per cases;
  suppose
    X is non empty;
    then consider n being Element of NAT such that
A1: X c= Seg n \/ {0} by Th1;
A2: not 2*n+1 in X
    proof
A3:   not 2*n+1 in {0} by TARSKI:def 1;
      assume 2*n+1 in X;
      then 2*n+1 in Seg n by A1,A3,XBOOLE_0:def 3;
      then
A4:   2*n+1 <= n by FINSEQ_1:1;
      1*n <= 2*n by NAT_1:4;
      hence thesis by A4,NAT_1:13;
    end;
    assume for k being odd Element of NAT holds k in X;
    hence contradiction by A2;
  end;
  suppose
    X is empty;
    hence thesis;
  end;
end;
