reserve L for Lattice,
  p,p1,q,q1,r,r1 for Element of L;
reserve x,y,z,X,Y,Z,X1,X2 for set;
reserve H,F for Filter of L;
reserve D for non empty Subset of L;
reserve D1,D2 for non empty Subset of L;
reserve I for I_Lattice,
  i,j,k for Element of I;
reserve B for B_Lattice,
  FB,HB for Filter of B;
reserve I for I_Lattice,
  i,j,k for Element of I,
  DI for non empty Subset of I,
  FI for Filter of I;
reserve F1,F2 for Filter of I;
reserve a,b,c for Element of B;
reserve o1,o2 for BinOp of F;

theorem
  for L being meet-absorbing meet-commutative meet-associative
join-absorbing join-commutative non empty LattStr for x,y,z being Element of
L st z [= x & z [= y & for z9 being Element of L st z9 [= x & z9 [= y holds z9
  [= z holds z = x "/\" y
proof
  let L be meet-absorbing meet-commutative meet-associative join-absorbing
  join-commutative non empty LattStr;
  let x,y,z being Element of L such that
A1: z [= x & z [= y and
A2: for z9 being Element of L st z9 [= x & z9 [= y holds z9 [= z;
  x "/\" y [= x & x "/\" y [= y by LATTICES:6;
  then
A3: x "/\" y [= z by A2;
  z [= x "/\" y by A1,Th7;
  hence thesis by A3,LATTICES:8;
end;
