
theorem Th28:
  for L being non empty OrthoLattRelStr st L is involutive
  with_Top de_Morgan Lattice-like naturally_sup-generated holds L is
  Orthocomplemented PartialOrdered
proof
  let L be non empty OrthoLattRelStr;
  assume
  L is involutive with_Top de_Morgan Lattice-like naturally_sup-generated;
  then reconsider L9 = L as involutive with_Top de_Morgan Lattice-like
  naturally_sup-generated non empty OrthoLattRelStr;
  reconsider f = the Compl of L9 as Function of L9, L9;
  for x, y being Element of L9 st x <= y holds f.x >= f.y
  proof
    let x, y be Element of L9;
    assume x <= y;
    then x [= y by Th22;
    then
A1: x` = (x |^| y)` by LATTICES:4
      .= (x` |_| y`)`` by ROBBINS1:def 23
      .= x` |_| y` by Def6;
    f.x = x` & f.y = y` by ROBBINS1:def 3;
    hence thesis by A1,Def10;
  end;
  then
A2: f is antitone by WAYBEL_9:def 1;
A3: for y being Element of L9 holds ex_sup_of {y,f.y},L9 & ex_inf_of {y,f.y
  },L9 & "\/"({y,f.y},L9) is_maximum_of the carrier of L9 & "/\"({y,f.y},L9)
  is_minimum_of the carrier of L9
  proof
    set xx = "\/"(the carrier of L9,L9);
    let y be Element of L9;
    thus ex_sup_of {y,f.y},L9 by YELLOW_0:20;
    thus ex_inf_of {y,f.y},L9 by YELLOW_0:21;
    set t = y |_| y`;
    for b being Element of L9 st b in the carrier of L9 holds b <= t
    proof
      let b be Element of L9;
      assume b in the carrier of L9;
      b |_| (y |_| y`) = b |_| (b |_| b`) by Def7
        .= b |_| b |_| b` by LATTICES:def 5
        .= b |_| b`
        .= y |_| y` by Def7;
      then b [= t;
      hence thesis by Th22;
    end;
    then
A4: t is_>=_than the carrier of L9 by LATTICE3:def 9;
    then L9 is upper-bounded by YELLOW_0:def 5;
    then
A5: ex_sup_of the carrier of L9,L9 by YELLOW_0:43;
    reconsider t as Element of L9;
A6: for a being Element of L9 st the carrier of L9 is_<=_than a holds t
    <= a by LATTICE3:def 9;
    "\/"({y,f.y},L9) = "\/"({y,y`},L9) by ROBBINS1:def 3
      .= y "|_|" y` by YELLOW_0:41
      .= y |_| y` by Th25
      .= xx by A4,A5,A6,YELLOW_0:def 9;
    hence "\/"({y,f.y},L9) is_maximum_of the carrier of L9 by A5,WAYBEL_1:def 7
;
    set xx = "/\"(the carrier of L9,L9);
    set t = y |^| y`;
A7: for a, b being Element of L9 holds a |^| a` = b |^| b`
    proof
      let a, b be Element of L9;
      a |^| a` = (a` |_| a``)` by ROBBINS1:def 23
        .= (b` |_| b``)` by Def7
        .= b |^| b` by ROBBINS1:def 23;
      hence thesis;
    end;
    for b being Element of L9 st b in the carrier of L9 holds b >= t
    proof
      let b be Element of L9;
      assume b in the carrier of L9;
      b |^| (y |^| y`) = b |^| (b |^| b`) by A7
        .= b |^| b |^| b` by LATTICES:def 7
        .= b |^| b`
        .= y |^| y` by A7;
      then t [= b by LATTICES:4;
      hence thesis by Th22;
    end;
    then
A8: t is_<=_than the carrier of L9 by LATTICE3:def 8;
    then L9 is lower-bounded by YELLOW_0:def 4;
    then
A9: ex_inf_of the carrier of L9,L9 by YELLOW_0:42;
    reconsider t as Element of L9;
A10: for a being Element of L9 st the carrier of L9 is_>=_than a holds t
    >= a by LATTICE3:def 8;
    "/\"({y,f.y},L9) = "/\"({y,y`},L9) by ROBBINS1:def 3
      .= y "|^|" y` by YELLOW_0:40
      .= y |^| y` by Th26
      .= xx by A8,A9,A10,YELLOW_0:def 10;
    hence thesis by A9,WAYBEL_1:def 6;
  end;
  for x being Element of L9 holds f.(f.x) = x
  proof
    let x be Element of L9;
    f.(f.x) = f.(x`) by ROBBINS1:def 3
      .= x`` by ROBBINS1:def 3
      .= x by Def6;
    hence thesis;
  end;
  then f is involutive by PARTIT_2:def 3;
  then f is Orderinvolutive by A2,OPOSET_1:def 17;
  then f OrthoComplement_on L9 by A3,OPOSET_1:def 21;
  hence thesis by OPOSET_1:def 22;
end;
