reserve T for TopSpace;
reserve T for non empty TopSpace;
reserve F for Subset-Family of T;
reserve T for non empty TopSpace;
reserve T for non empty TopSpace;
reserve T for non empty TopSpace;

theorem Th105:
  for a, b being Element of Open_Domains_Lattice T for A, B being
  Element of Open_Domains_of T st a = A & b = B holds a [= b iff A c= B
proof
  let a, b be Element of Open_Domains_Lattice T;
  let A, B be Element of Open_Domains_of T;
  reconsider A1 = A as Subset of T;
  assume that
A1: a = A and
A2: b = B;
  A in Open_Domains_of T;
  then A in {C where C is Subset of T : C is open_condensed} by TDLAT_1:def 9;
  then ex Q being Subset of T st Q = A & Q is open_condensed;
  then
A3: A1 is open by TOPS_1:67;
  thus a [= b implies A c= B
  proof
    assume a [= b;
    then a "\/" b = b by LATTICES:def 3;
    then Int(Cl(A \/ B)) = B by A1,A2,Th103;
    hence thesis by A3,Th4;
  end;
  B in Open_Domains_of T;
  then B in {C where C is Subset of T : C is open_condensed} by TDLAT_1:def 9;
  then
A4: ex P being Subset of T st P = B & P is open_condensed;
  thus A c= B implies a [= b
  proof
    assume A c= B;
    then Int(Cl(A \/ B)) = B by A4,Th63;
    then a "\/" b = b by A1,A2,Th103;
    hence thesis by LATTICES:def 3;
  end;
end;
