reserve X for non empty set;

theorem Th9:
  for L being Lattice, a,b being Element of LattPOSet L holds a
  "\/" b = %a "\/" %b
proof
  let L be Lattice, a,b be Element of LattPOSet L;
  reconsider x = a, y = b as Element of L;
  set c = x "\/" y;
A1: c% = c;
A2: y [= c & y% = y by LATTICES:5;
A3: x [= c & x% = x by LATTICES:5;
  reconsider c as Element of LattPOSet L;
A4: b <= c by A1,A2,LATTICE3:7;
A5: for d being Element of LattPOSet L st a <= d & b <= d holds c <= d
  proof
    let d be Element of LattPOSet L;
    assume that
A6: a <= d and
A7: b <= d;
    reconsider z = d as Element of L;
    y% <= z% by A7;
    then
A8: y [= z by LATTICE3:7;
    x% <= z% by A6;
    then x [= z by LATTICE3:7;
    then x "\/" y [= z by A8,FILTER_0:6;
    then (x "\/" y)% <= z% by LATTICE3:7;
    hence thesis;
  end;
  a <= c by A1,A3,LATTICE3:7;
  hence thesis by A4,A5,YELLOW_0:22;
end;
