
theorem Th25:
  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 <= x "|_|" y by YELLOW_0:22;
  then
A1: x [= x "|_|" y by Th22;
  y <= x "|_|" y by YELLOW_0:22;
  then
A2: y [= x "|_|" y by Th22;
  x [= x |_| y by LATTICES:5;
  then
A3: x <= x |_| y by Th22;
  y [= x |_| y by LATTICES:5;
  then
A4: y <= x |_| y by Th22;
  (x |_| y) "|_|" (x "|_|" y) = (x |_| y) "|_|" x "|_|" y by LATTICE3:14
    .= (x |_| y) "|_|" y by A3,YELLOW_0:24
    .= x |_| y by A4,YELLOW_0:24;
  then x "|_|" y <= x |_| y by YELLOW_0:24;
  then
A5: x "|_|" y [= x |_| y by Th22;
  (x "|_|" y) |_| (x |_| y) = (x "|_|" y) |_| x |_| y by LATTICES:def 5
    .= (x "|_|" y) |_| y by A1
    .= x "|_|" y by A2;
  hence thesis by A5;
end;
