reserve n,k for Element of NAT;

theorem Th9:
  for L be finite LATTICE for y be Element of L st y <> Bottom L
  holds ex x be Element of L st x <(1) y
proof
  let L be finite LATTICE;
  let y be Element of L;
  ( ex A be Chain of Bottom L,y st height(y)= card A)& for A be Chain of
  Bottom L,y holds card A <= height(y) by Def3;
  then consider A be Chain of Bottom L,y such that
A1: height(y)= card A & for A be Chain of Bottom L,y holds card A <= height(y);
  set B=A\{y};
A2: the InternalRel of L is_strongly_connected_in B
  proof
    let p,q be object;
    p in B & q in B implies [p,q] in the InternalRel of L or [q,p] in the
    InternalRel of L
    proof
      assume
A3:   p in B & q in B;
      then
A4:   p in A & q in A by XBOOLE_0:def 5;
      reconsider p,q as Element of L by A3;
      p <= q or q <= p by A4,ORDERS_2:11;
      hence thesis by ORDERS_2:def 5;
    end;
    hence thesis;
  end;
  assume
A5: y <> Bottom L;
  B is non empty
  proof
    Bottom L<=y by YELLOW_0:44;
    then
A6: Bottom L in A by Def2;
    assume
A7: B is empty;
    not Bottom L in {y} by A5,TARSKI:def 1;
    hence contradiction by A7,A6,XBOOLE_0:def 5;
  end;
  then reconsider B as non empty Chain of L by A2,ORDERS_2:def 7;
  take x=max(B);
A8: not ex z be Element of L st x < z & z < y
  proof
    given z be Element of L such that
A9: x < z and
A10: z < y;
A11: Bottom L<=y by YELLOW_0:44;
    then y in A by Def2;
    then
A12: y in A \/ {z} by XBOOLE_0:def 3;
    set C=A \/ {z};
    {y} c= A
    proof
      let h be Element of L;
      assume h in {y};
      then
A13:  h=y by TARSKI:def 1;
      Bottom L<=y by YELLOW_0:44;
      hence thesis by A13,Def2;
    end;
    then
A14: A=(A\{y})\/{y} by XBOOLE_1:45;
    the InternalRel of L is_strongly_connected_in C
    proof
      let x1,y1 be object;
      x1 in C & y1 in C implies [x1,y1] in the InternalRel of L or [y1,x1
      ] in the InternalRel of L
      proof
        assume
A15:    x1 in C & y1 in C;
        per cases by A15,XBOOLE_0:def 3;
        suppose
A16:      x1 in A & y1 in A;
          then reconsider x1,y1 as Element of L;
          x1 <= y1 or y1 <= x1 by A16,ORDERS_2:11;
          hence thesis by ORDERS_2:def 5;
        end;
        suppose
A17:      x1 in A & y1 in {z};
          then
A18:      y1=z by TARSKI:def 1;
          reconsider x1,y1 as Element of L by A17;
          x1 in A\{y} or x1 in {y} by A14,A17,XBOOLE_0:def 3;
          then x1 <= x or x1 = y by Def5,TARSKI:def 1;
          then x1 < y1 or x1 = y by A9,A18,ORDERS_2:7;
          then x1 <= y1 or y1 < x1 by A10,A17,ORDERS_2:def 6,TARSKI:def 1;
          then x1 <= y1 or y1 <= x1 by ORDERS_2:def 6;
          hence thesis by ORDERS_2:def 5;
        end;
        suppose
A19:      y1 in A & x1 in {z};
          then
A20:      x1=z by TARSKI:def 1;
          reconsider x1,y1 as Element of L by A19;
          y1 in A\{y} or y1 in {y} by A14,A19,XBOOLE_0:def 3;
          then y1 <= x or y1 = y by Def5,TARSKI:def 1;
          then y1 < x1 or y1 = y by A9,A20,ORDERS_2:7;
          then y1 <= x1 or x1 <= y1 by A10,A20,ORDERS_2:def 6;
          hence thesis by ORDERS_2:def 5;
        end;
        suppose
A21:      x1 in {z} & y1 in {z};
          then reconsider x1,y1 as Element of L;
          x1=z by A21,TARSKI:def 1;
          then x1<=y1 by A21,TARSKI:def 1;
          hence thesis by ORDERS_2:def 5;
        end;
      end;
      hence thesis;
    end;
    then
A22: C is strongly_connected Subset of L by ORDERS_2:def 7;
A23: z <= y by A10,ORDERS_2:def 6;
A24: for v be Element of L st v in A \/ {z} holds Bottom L <= v & v <= y
    proof
      let v be Element of L;
      assume
A25:  v in A \/ {z};
      per cases by A25,XBOOLE_0:def 3;
      suppose
A26:    v in A;
        thus Bottom L<=v by YELLOW_0:44;
        thus thesis by A11,A26,Def2;
      end;
      suppose
        v in {z};
        hence thesis by A23,TARSKI:def 1,YELLOW_0:44;
      end;
    end;
    not z in A
    proof
      assume
A27:  z in A;
      not z in {y} by A10,TARSKI:def 1;
      then z in B by A27,XBOOLE_0:def 5;
      then z <= x by Def5;
      hence contradiction by A9,ORDERS_2:7;
    end;
    then
A28: card (A \/ {z})=card A + 1 by CARD_2:41;
    Bottom L in A by A11,Def2;
    then Bottom L in A \/ {z} by XBOOLE_0:def 3;
    then A \/ {z} is Chain of Bottom L,y by A22,A11,A12,A24,Def2;
    then card A + 1 <= card A by A1,A28;
    hence contradiction by NAT_1:13;
  end;
A29: x in B by Def5;
  then not x in {y} by XBOOLE_0:def 5;
  then
A30: not x=y by TARSKI:def 1;
  Bottom L<=y & x in A by A29,XBOOLE_0:def 5,YELLOW_0:44;
  then x<=y by Def2;
  then x < y by A30,ORDERS_2:def 6;
  hence thesis by A8;
end;
