reserve n,k for Element of NAT;

theorem Th2:
  for L be finite LATTICE for a,b be Element of L holds a < b
  implies height(a) < height(b)
proof
  let L be finite LATTICE;
  let a,b be Element of L;
  ( ex A be Chain of Bottom L,a st height(a) = card A)& for C be Chain of
  Bottom L,a holds card C <= height(a) by Def3;
  then consider A be Chain of Bottom L,a such that
A1: height(a) = card A and
  for C be Chain of Bottom L,a holds card C <= height(a);
  set C=A \/ {b};
  b in {b} by TARSKI:def 1;
  then
A2: b in C by XBOOLE_0:def 3;
A3: Bottom L<=a by YELLOW_0:44;
  then Bottom L in A by Def2;
  then
A4: Bottom L in C by XBOOLE_0:def 3;
  assume
A5: a < b;
  not b in A
  proof
    assume b in A;
    then b <= a by A3,Def2;
    hence contradiction by A5,ORDERS_2:6;
  end;
  then
A6: card C = (card A)+1 by CARD_2:41;
  the InternalRel of L is_strongly_connected_in C
  proof
    let x,y be object;
    x in C & y in C implies [x,y] in the InternalRel of L or [y,x] in the
    InternalRel of L
    proof
      assume
A7:   x in C & y in C;
      per cases by A7,XBOOLE_0:def 3;
      suppose
A8:     x in A & y in A;
        then reconsider x,y as Element of L;
        x <= y or y <= x by A8,ORDERS_2:11;
        hence thesis by ORDERS_2:def 5;
      end;
      suppose
A9:     x in A & y in {b};
        then reconsider x as Element of L;
        Bottom L<=a by YELLOW_0:44;
        then x <= a by A9,Def2;
        then x < b by A5,ORDERS_2:7;
        then
A10:    x <= b by ORDERS_2:def 6;
        y=b by A9,TARSKI:def 1;
        hence thesis by A10,ORDERS_2:def 5;
      end;
      suppose
A11:    x in {b} & y in A;
        then reconsider y as Element of L;
        Bottom L<=a by YELLOW_0:44;
        then y <= a by A11,Def2;
        then y < b by A5,ORDERS_2:7;
        then
A12:    y <= b by ORDERS_2:def 6;
        x=b by A11,TARSKI:def 1;
        hence thesis by A12,ORDERS_2:def 5;
      end;
      suppose
A13:    x in {b} & y in {b};
        then reconsider x,y as Element of L;
        x=b by A13,TARSKI:def 1;
        then x <= y by A13,TARSKI:def 1;
        hence thesis by ORDERS_2:def 5;
      end;
    end;
    hence thesis;
  end;
  then
A14: C is strongly_connected Subset of L by ORDERS_2:def 7;
A15: for z be Element of L st z in C holds Bottom L <= z & z <= b
  proof
    let z be Element of L;
    assume
A16: z in C;
    per cases by A16,XBOOLE_0:def 3;
    suppose
A17:  z in A;
      thus Bottom L<=z by YELLOW_0:44;
      z<=a by A3,A17,Def2;
      then z<b by A5,ORDERS_2:7;
      hence thesis by ORDERS_2:def 6;
    end;
    suppose
      z in {b};
      hence thesis by TARSKI:def 1,YELLOW_0:44;
    end;
  end;
  Bottom L <= b by YELLOW_0:44;
  then C is Chain of Bottom L,b by A4,A2,A15,A14,Def2;
  then height(a) + 1 <= height(b) by A1,A6,Def3;
  hence thesis by NAT_1:13;
end;
