theorem Th14:
  Seg k misses {k + 1}
proof
  set x = the Element of Seg k /\ {k + 1};
  assume not thesis;
  then
A1: Seg k /\ {k + 1} <> {};
  then
A2: x in Seg k by XBOOLE_0:def 4;
  then reconsider x as Element of NAT;
  x in {k + 1} by A1,XBOOLE_0:def 4;
  then
A3: x = k + 1 by TARSKI:def 1;
  x <= k by A2,FINSEQ_1:1;
  hence thesis by A3,XREAL_1:29;
end;
