reserve n,k for Element of NAT;

theorem Th7:
  for L be non empty finite LATTICE for x be Element of L holds height (x) >= 1
proof
  let L be non empty finite LATTICE;
  let x be Element of L;
A1: {Bottom L,x} is Chain of Bottom L,x by Th1,YELLOW_0:44;
  then
A2: card {Bottom L,x} <= height (x) by Def3;
  per cases;
  suppose
    x<>Bottom L;
    then card {Bottom L,x} = 2 by CARD_2:57;
    hence thesis by A2,XXREAL_0:2;
  end;
  suppose
A3: x=Bottom L;
A4: {Bottom L}c={Bottom L,Bottom L}
    proof
      let d be Element of L;
      assume d in{Bottom L};
      then d =Bottom L by TARSKI:def 1;
      hence thesis by TARSKI:def 2;
    end;
    {Bottom L,Bottom L}c={Bottom L}
    proof
      let d be Element of L;
      assume d in {Bottom L,Bottom L};
      then d =Bottom L or d =Bottom L by TARSKI:def 2;
      hence thesis by TARSKI:def 1;
    end;
    then
A5: {Bottom L,Bottom L}={Bottom L} by A4,XBOOLE_0:def 10;
    card {Bottom L,Bottom L} <= height (Bottom L) by A1,A3,Def3;
    hence thesis by A3,A5,CARD_1:30;
  end;
end;
