reserve X for TopSpace;
reserve X for non empty TopSpace;
reserve X1, X2, X3 for non empty SubSpace of X;
reserve X1, X2, X3 for non empty SubSpace of X;
reserve X for TopSpace;
reserve A1, A2 for Subset of X;
reserve A1,A2 for Subset of X;
reserve X for TopSpace,
  A1, A2 for Subset of X;
reserve X for non empty TopSpace,
  A1, A2 for Subset of X;

theorem Th61:
  A1 \/ A2 = the carrier of X & A1,A2 are_weakly_separated & not
A1 c= A2 & not A2 c= A1 implies ex C1, C2 being non empty Subset of X st C1 is
  open & C2 is open & C1 c= A1 & C2 c= A2 & (A1 \/ A2 = C1 \/ C2 or ex C being
  non empty Subset of X st A1 \/ A2 = (C1 \/ C2) \/ C & C c= A1 /\ A2 & C is
  closed)
proof
  assume
A1: A1 \/ A2 = the carrier of X;
  assume A1,A2 are_weakly_separated & not A1 c= A2 & not A2 c= A1;
  then consider C1, C2 being non empty Subset of X such that
A2: C1 is open & C2 is open & C1 /\ (A1 \/ A2) c= A1 & C2 /\ (A1 \/ A2)
  c= A2 and
A3: A1 \/ A2 c= C1 \/ C2 or ex C being non empty Subset of X st C is
  closed & C /\ (A1 \/ A2) c= A1 /\ A2 & the carrier of X = (C1 \/ C2) \/ C by
Th59;
  take C1,C2;
  now
    assume not A1 \/ A2 = C1 \/ C2;
    then consider C being non empty Subset of X such that
A4: C is closed & C /\ (A1 \/ A2) c= A1 /\ A2 & the carrier of X = (C1
    \/ C2 ) \/ C by A1,A3,XBOOLE_0:def 10;
    thus ex C being non empty Subset of X st A1 \/ A2 = (C1 \/ C2) \/ C & C c=
    A1 /\ A2 & C is closed
    proof
      take C;
      thus thesis by A1,A4,XBOOLE_1:28;
    end;
  end;
  hence thesis by A1,A2,XBOOLE_1:28;
end;
