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;

theorem Th3:
  for T being non empty TopSpace holds LattStr(#Topology_of T,
    Top_Union T,Top_Meet T#) is Lattice
proof
  let T;
  set L = LattStr(#Topology_of T,Top_Union T,Top_Meet T#);
A1: now
    let p,q be Element of L;
    thus p "\/" q = q \/ p by Def2
      .= q"\/"p by Def2;
  end;
A2: now
    let p,q be Element of L;
    thus (p"/\"q)"\/"q = p"/\"q \/ q by Def2
      .= (p /\ q) \/ q by Def3
      .= q by XBOOLE_1:22;
  end;
A3: now
    let p,q be Element of L;
    thus p"/\"(p"\/"q) = p /\ (p"\/"q) by Def3
      .= p /\ (p \/ q) by Def2
      .= p by XBOOLE_1:21;
  end;
A4: now
    let p,q,r be Element of L;
    thus p"/\"(q"/\"r) = p /\ (q "/\" r) by Def3
      .= p /\ (q /\ r) by Def3
      .= (p /\ q) /\ r by XBOOLE_1:16
      .= p "/\" q /\ r by Def3
      .= (p"/\"q)"/\"r by Def3;
  end;
A5: now
    let p,q be Element of L;
    thus p "/\" q =q /\ p by Def3
      .= q"/\"p by Def3;
  end;
  now
    let p,q,r be Element of L;
    thus p"\/"(q"\/"r) = p \/ q "\/" r by Def2
      .= p \/ (q \/ r) by Def2
      .= (p \/ q) \/ r by XBOOLE_1:4
      .= p "\/" q \/ r by Def2
      .= (p"\/"q)"\/"r by Def2;
  end;
  then L is join-commutative join-associative meet-absorbing meet-commutative
  meet-associative join-absorbing by A1,A2,A5,A4,A3;
  hence thesis;
end;
