reserve X for non empty set;

theorem Th7:
  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 [= x by LATTICES:6;
A2: c [= y by LATTICES:6;
A3: c% = c;
  reconsider c as Element of LattPOSet L;
A4: y% = y;
  then
A5: c <= b by A2,A3,LATTICE3:7;
A6: x% = x;
A7: for d being Element of LattPOSet L st d <= a & d <= b holds d <= c
  proof
    let d be Element of LattPOSet L;
    reconsider z = d as Element of L;
A8: z% = z;
    assume d <= a & d <= b;
    then z [= x & z [= y by A6,A4,A8,LATTICE3:7;
    then z [= x "/\" y by FILTER_0:7;
    hence thesis by A3,A8,LATTICE3:7;
  end;
  c <= a by A1,A3,A6,LATTICE3:7;
  hence thesis by A5,A7,YELLOW_0:23;
end;
