reserve L for Ortholattice,
  a, b, c for Element of L;

theorem Th36:
  L is orthomodular iff for a,b being Element of L holds a "\/" b
  = ((a "\/" b) "/\" a) "\/" ((a "\/" b) "/\" a`)
proof
  thus L is orthomodular implies for a,b being Element of L holds a "\/" b = (
  (a "\/" b) "/\" a) "\/" ((a "\/" b) "/\" a`)
  proof
    assume
A1: L is orthomodular;
    let a,b be Element of L;
    a "\/" b = a "\/" ((a "\/" b) "/\" a`) by A1,Th6;
    hence thesis by LATTICES:def 9;
  end;
  assume
A2: for a,b being Element of L holds a "\/" b = ((a "\/" b) "/\" a) "\/"
  ((a "\/" b) "/\" a`);
  let x,y be Element of L;
  assume
A3: x [= y;
  hence y = x "\/" y
    .= ((x "\/" y) "/\" x) "\/" ((x "\/" y) "/\" x`) by A2
    .= (y "/\" x) "\/" ((x "\/" y) "/\" x`) by A3
    .= (y "/\" x) "\/" (y "/\" x`) by A3
    .= x "\/" (x` "/\" y) by A3,LATTICES:4;
end;
