
theorem Th26:
  for L being with_infima with_suprema naturally_sup-generated
Lattice-like non empty OrthoLattRelStr, x, y being Element of L holds x "|^|"
  y = x |^| y
proof
  let L be with_infima with_suprema naturally_sup-generated Lattice-like non
  empty OrthoLattRelStr, x, y be Element of L;
  x "|^|" y <= x by YELLOW_0:23;
  then
A1: x "|^|" y [= x by Th22;
  x "|^|" y <= y by YELLOW_0:23;
  then
A2: x "|^|" y [= y by Th22;
  x |^| y [= x by LATTICES:6;
  then
A3: x |^| y <= x by Th22;
  x |^| y [= y by LATTICES:6;
  then
A4: x |^| y <= y by Th22;
  (x |^| y) "|^|" (x "|^|" y) = (x |^| y) "|^|" x "|^|" y by LATTICE3:16
    .= (x |^| y) "|^|" y by A3,YELLOW_0:25
    .= x |^| y by A4,YELLOW_0:25;
  then x |^| y <= x "|^|" y by YELLOW_0:25;
  then
A5: x |^| y [= x "|^|" y by Th22;
  (x "|^|" y) |^| (x |^| y) = (x "|^|" y) |^| x |^| y by LATTICES:def 7
    .= (x "|^|" y) |^| y by A1,LATTICES:4
    .= x "|^|" y by A2,LATTICES:4;
  hence thesis by A5,LATTICES:4;
end;
