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 join-associative
join-absorbing join-commutative non empty LattStr for x,y,z being Element of
  L st x [= z & y [= z & for z9 being Element of L st x [= z9 & y [= z9 holds z
  [= z9 holds z = x "\/" y
proof
  let L be meet-absorbing meet-commutative join-associative join-absorbing
  join-commutative non empty LattStr;
  let x,y,z being Element of L such that
A1: x [= z & y [= z and
A2: for z9 being Element of L st x [= z9 & y [= z9 holds z [= z9;
  x [= x "\/" y & y [= x "\/" y by LATTICES:5;
  then
A3: z [= x "\/" y by A2;
  x "\/" y [= z by A1,Th6;
  hence thesis by A3,LATTICES:8;
end;
