
theorem
  for L being satisfying_Sheffer_1 satisfying_Sheffer_2
  satisfying_Sheffer_3 properly_defined non empty ShefferOrthoLattStr holds L
  is Boolean Lattice
proof
  let L be satisfying_Sheffer_1 satisfying_Sheffer_2 satisfying_Sheffer_3
  properly_defined non empty ShefferOrthoLattStr;
A1: L is distributive & L is distributive' by Th35,Th36;
  ex c being Element of L st for a being Element of L holds c "/\" a = a &
  a "/\" c = a
  proof
    set b = the Element of L;
    take c = (b|b) | ((b|b)|(b|b));
    let a be Element of L;
    thus c "/\" a = a
    proof
      set X = b";
      c "/\" a = (((b | b) | (X | X)) | a)" by Def12
        .= (a | (X | (X | X)))" by Th31
        .= (a | a)" by Def14
        .= a by Def13;
      hence thesis;
    end;
    thus a "/\" c = a
    proof
      set X = b";
      a "/\" c = (a | ((b | b) | (X | X)))" by Def12
        .= (a | a)" by Def14
        .= a by Def13;
      hence thesis;
    end;
  end;
  then
A2: L is upper-bounded';
  ex c being Element of L st for a being Element of L holds c "\/" a = a &
  a "\/" c = a
  proof
    set b = the Element of L;
    take c = (b | (b | b)) | (b | (b | b));
    let a be Element of L;
    c "\/" a = ((b | (b | b))")" | (a | a) by Def12
      .= ((a | (a | a))")" | (a | a) by Th32
      .= (a | (a | a)) | (a | a) by Def13
      .= (a | a) | (a | (a | a)) by Th31
      .= (a | a) | (a | a) by Def14
      .= a by Def13;
    hence c "\/" a = a;
    a "\/" c = (a | a) | ((b | (b | b))")" by Def12
      .= (a | a) | ((a | (a | a))")" by Th32
      .= (a | a) | (a | (a | a)) by Def13
      .= (a | a) | (a | a) by Def14
      .= a by Def13;
    hence a "\/" c = a;
  end;
  then
A3: L is lower-bounded';
  for b being Element of L ex a being Element of L st a is_a_complement'_of b
  proof
    let b be Element of L;
    set a = b | b;
    take a;
A4: Top' L = (b | b) | ((b | b)|(b | b))
    proof
      set X = (b | b) | ((b | b)|(b | b));
      for a being Element of L holds X "/\" a = a & a "/\" X = a
      proof
        let a be Element of L;
        set Y = b";
        thus X "/\" a = (((b | b) | (Y | Y)) | a)" by Def12
        .= (a | (Y | (Y | Y)))" by Th31
        .= (a | a)" by Def14
        .= a by Def13;
        thus a "/\" X = (a | ((b | b) | (Y | Y)))" by Def12
        .= (a | a)" by Def14
        .= a by Def13;
      end;
      hence thesis by A2,Def2;
    end;
    then
A5: b "\/" a = Top' L by Def12;
A6: Bot' L = (b | (b | b)) | (b | (b | b))
    proof
      set X = (b | (b | b)) | (b | (b | b));
      for a being Element of L holds X "\/" a = a & a "\/" X = a
      proof
        let a be Element of L;
        thus X "\/" a = ((b | (b | b))")" | (a | a) by Def12
          .= ((a | (a | a))")" | (a | a) by Th32
          .= (a | (a | a)) | (a | a) by Def13
          .= (a | a) | (a | (a | a)) by Th31
          .= (a | a) | (a | a) by Def14
          .= a by Def13;
        thus a "\/" X = (a | a) | ((b | (b | b))")" by Def12
          .= (a | a) | ((a | (a | a))")" by Th32
          .= (a | a) | (a | (a | a)) by Def13
          .= (a | a) | (a | a) by Def14
          .= a by Def13;
      end;
      hence thesis by A3,Def4;
    end;
    then
A7: b "/\" a = Bot' L by Def12;
A8: a "\/" b = ((b | b) | (b | b)) | (b | b) by Def12
      .= Top' L by A4,Th31;
    a "/\" b = ((b | b) | b) | ((b | b) | b) by Def12
      .= (b | (b | b)) | ((b | b) | b) by Th31
      .= Bot' L by A6,Th31;
    hence thesis by A8,A5,A7;
  end;
  then
A9: L is complemented';
  L is join-commutative & L is meet-commutative by Th33,Th34;
  hence thesis by A3,A2,A9,A1;
end;
