reserve T for non empty TopSpace,
  X,Z for Subset of T;
reserve x,y for Element of OpenClosedSet(T);

theorem Th9:
  LattStr(#OpenClosedSet(T),T_join T,T_meet T#) is Lattice
proof
  set L = LattStr(#OpenClosedSet(T),T_join T,T_meet T#);
A1: for p,q be Element of L holds p"\/"q = q"\/"p
  proof
    let p,q be Element of L;
    consider p9,q9 being Element of OpenClosedSet(T) such that
A2: p=p9 and
A3: q=q9;
    thus p "\/" q = q9 \/ p9 by A2,A3,Def2
      .= q"\/"p by A2,A3,Def2;
  end;
A4: for p,q,r be Element of L holds p"\/"(q"\/"r) = (p"\/"q)"\/"r
  proof
    let p,q,r be Element of L;
    consider p9,q9,r9,k9,l9 being Element of OpenClosedSet(T) such that
A5: p=p9 and
A6: q=q9 and
A7: r=r9 and
A8: q "\/" r=k9 and
A9: p "\/" q=l9;
    thus p"\/"(q"\/"r) = p9 \/ k9 by A5,A8,Def2
      .= p9 \/ (q9 \/ r9) by A6,A7,A8,Def2
      .= (p9 \/ q9) \/ r9 by XBOOLE_1:4
      .= l9 \/ r9 by A5,A6,A9,Def2
      .= (p"\/"q)"\/"r by A7,A9,Def2;
  end;
A10: for p,q be Element of L holds (p"/\"q)"\/"q = q
  proof
    let p,q be Element of L;
    consider p9,q9,k9 being Element of OpenClosedSet(T) such that
A11: p=p9 and
A12: q=q9 and
A13: p"/\"q=k9;
    thus (p"/\"q)"\/"q = k9 \/ q9 by A12,A13,Def2
      .= (p9 /\ q9) \/ q9 by A11,A12,A13,Def3
      .= q by A12,XBOOLE_1:22;
  end;
A14: for p,q be Element of L holds p"/\"q = q"/\"p
  proof
    let p,q be Element of L;
    consider p9,q9 being Element of OpenClosedSet(T) such that
A15: p=p9 and
A16: q=q9;
    thus p "/\" q =q9 /\ p9 by A15,A16,Def3
      .= q"/\"p by A15,A16,Def3;
  end;
A17: for p,q,r be Element of L holds p"/\"(q"/\"r) = (p"/\"q)"/\"r
  proof
    let p,q,r be Element of L;
    consider p9,q9,r9,k9,l9 being Element of OpenClosedSet(T) such that
A18: p=p9 and
A19: q=q9 and
A20: r=r9 and
A21: q "/\" r=k9 and
A22: p "/\" q=l9;
    thus p"/\"(q"/\"r) = p9 /\ k9 by A18,A21,Def3
      .= p9 /\ (q9 /\ r9) by A19,A20,A21,Def3
      .= (p9 /\ q9) /\ r9 by XBOOLE_1:16
      .= l9 /\ r9 by A18,A19,A22,Def3
      .= (p"/\"q)"/\"r by A20,A22,Def3;
  end;
  for p,q be Element of L holds p"/\"(p"\/"q)=p
  proof
    let p,q be Element of L;
    consider p9,q9,k9 being Element of OpenClosedSet(T) such that
A23: p=p9 and
A24: q=q9 and
A25: p"\/"q=k9;
    thus p"/\"(p"\/"q) = p9 /\ k9 by A23,A25,Def3
      .= p9 /\ (p9 \/ q9) by A23,A24,A25,Def2
      .= p by A23,XBOOLE_1:21;
  end;
  then L is join-commutative join-associative meet-absorbing
  meet-commutative meet-associative join-absorbing
  by A1,A4,A10,A14,A17;
  hence thesis;
end;
