reserve T for TopSpace;
reserve A,B for Subset of T;
reserve T for non empty TopSpace;
reserve P,Q for Element of Topology_of T;
reserve p,q for Element of Open_setLatt(T);
reserve L for D_Lattice;
reserve F for Filter of L;
reserve a,b for Element of L;
reserve x,X,X1,X2,Y,Z for set;

theorem Th23:
  LattStr(#StoneS(L),Set_Union L,Set_Meet L#) is Lattice
proof
  set SL = LattStr(#StoneS(L),Set_Union L,Set_Meet L#);
A1: now
    let p,q be Element of SL;
    thus p "\/" q = q \/ p by Def9
      .= q"\/"p by Def9;
  end;
A2: now
    let p,q be Element of SL;
    thus (p"/\"q)"\/"q = (p"/\"q) \/ q by Def9
      .= (p /\ q) \/ q by Def10
      .= q by XBOOLE_1:22;
  end;
A3: now
    let p,q be Element of SL;
    thus p"/\"(p"\/"q) = p /\ (p"\/"q) by Def10
      .= p /\ (p \/ q) by Def9
      .= p by XBOOLE_1:21;
  end;
A4: now
    let p,q,r be Element of SL;
    thus p"/\"(q"/\"r) = p /\ (q "/\" r) by Def10
      .= p /\ (q /\ r) by Def10
      .= (p /\ q) /\ r by XBOOLE_1:16
      .= (p "/\" q) /\ r by Def10
      .= (p"/\"q)"/\"r by Def10;
  end;
A5: now
    let p,q be Element of SL;
    thus p "/\" q =q /\ p by Def10
      .= q"/\"p by Def10;
  end;
  now
    let p,q,r be Element of SL;
    thus p"\/"(q"\/"r) = p \/ (q "\/" r) by Def9
      .= p \/ (q \/ r) by Def9
      .= (p \/ q) \/ r by XBOOLE_1:4
      .= (p "\/" q) \/ r by Def9
      .= (p"\/"q)"\/"r by Def9;
  end;
  then
  SL is join-commutative join-associative meet-absorbing meet-commutative
  meet-associative join-absorbing by A1,A2,A5,A4,A3;
  hence thesis;
end;
