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 Th60:
  A1 \/ A2 = the carrier of X implies (A1,A2 are_weakly_separated
  iff ex C1, C2, C being Subset of X st A1 \/ A2 = (C1 \/ C2) \/ C & C1 c= A1 &
  C2 c= A2 & C c= A1 /\ A2 & C1 is open & C2 is open & C is closed)
proof
  assume
A1: A1 \/ A2 = the carrier of X;
  thus A1,A2 are_weakly_separated implies ex C1, C2, C being Subset of X st A1
\/ A2 = (C1 \/ C2) \/ C & C1 c= A1 & C2 c= A2 & C c= A1 /\ A2 & C1 is open & C2
  is open & C is closed
  proof
    assume A1,A2 are_weakly_separated;
    then consider C1, C2, C being Subset of X such that
A2: C1 /\ (A1 \/ A2) c= A1 & C2 /\ (A1 \/ A2) c= A2 & C /\ (A1 \/ A2)
c= A1 /\ A2 & the carrier of X = (C1 \/ C2) \/ C & C1 is open & C2 is open & C
    is closed by Th58;
    take C1,C2,C;
    thus thesis by A1,A2,XBOOLE_1:28;
  end;
  given C1, C2, C being Subset of X such that
A3: A1 \/ A2 = (C1 \/ C2) \/ C & C1 c= A1 & C2 c= A2 & C c= A1 /\ A2 &
  C1 is open & C2 is open & C is closed;
  ex C1, C2, C being Subset of X st C1 /\ (A1 \/ A2) c= A1 & C2 /\ (A1 \/
A2) c= A2 & C /\ (A1 \/ A2) c= A1 /\ A2 & the carrier of X = (C1 \/ C2) \/ C &
  C1 is open & C2 is open & C is closed
  proof
    take C1,C2,C;
    thus thesis by A1,A3,XBOOLE_1:28;
  end;
  hence thesis by Th58;
end;
