 reserve a,Z1,Z2,Z3 for set,
         x,y,z for object,
         k for Nat;
 reserve S for RelStr;
 reserve P,Q for non empty flat Poset;
 reserve p,p1,p2 for Element of P;
 reserve K for non empty Chain of P;
 reserve X,Y for non empty set;

theorem
  for L being non empty flat lower-bounded LATTICE,
      x being Element of L,
      A being Chain of Bottom L, x holds
    card A <= 2
  proof
    let L be non empty flat lower-bounded LATTICE,
        x be Element of L,
        A be Chain of Bottom L, x;
S0: Bottom L <= x by YELLOW_0:44;
    per cases;
    suppose
KK:   Bottom L <> x;
      consider b being Element of L such that
H1:   A = {b} or A = {Bottom L, b} by Thflat01;
H2:   x in A & Bottom L in A by LATTICE7:def 2,S0;
      A <> {b}
      proof
        assume A = {b}; then
        x = b & b = Bottom L by TARSKI:def 1,H2;
        hence thesis by KK;
      end;
      hence card A <= 2 by CARD_2:50,H1;
    end;
    suppose Bottom L = x; then
      card A = 1 by CardA1;
      hence thesis;
    end;
  end;
