 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 CardA1:
  for L being non empty LATTICE,
      x being Element of L,
      A being Chain of x,x holds
    card A = 1
  proof
    let L be non empty LATTICE,
        x be Element of L,
        A be Chain of x,x;
    for z being Element of L st z in A holds z in {x}
    proof
      let z be Element of L;
      assume z in A; then
      x <= z & z <= x by LATTICE7:def 2; then
      z = x by ORDERS_2:2;
      hence thesis by TARSKI:def 1;
    end; then
    A c= {x} by LATTICE7:def 1;
    hence thesis by CARD_2:42,ZFMISC_1:33;
  end;
