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;

theorem Th43:
  A1,A2 are_separated iff ex C1, C2 being Subset of X st A1 c= C1
  & A2 c= C2 & C1 /\ C2 misses A1 \/ A2 & C1 is closed & C2 is closed
proof
  thus A1,A2 are_separated implies ex C1, C2 being Subset of X st A1 c= C1 &
  A2 c= C2 & C1 /\ C2 misses A1 \/ A2 & C1 is closed & C2 is closed
  proof
    assume A1,A2 are_separated;
    then consider C1, C2 being Subset of X such that
A1: A1 c= C1 & A2 c= C2 and
A2: C1 misses A2 & C2 misses A1 and
A3: C1 is closed & C2 is closed by Th42;
    take C1,C2;
    C1 /\ C2 misses A1 & C1 /\ C2 misses A2 by A2,XBOOLE_1:17,63;
    hence thesis by A1,A3,XBOOLE_1:70;
  end;
  given C1, C2 being Subset of X such that
A4: A1 c= C1 and
A5: A2 c= C2 and
A6: C1 /\ C2 misses A1 \/ A2 and
A7: C1 is closed & C2 is closed;
  ex C1, C2 being Subset of X st A1 c= C1 & A2 c= C2 & C1 misses A2 & C2
  misses A1 & C1 is closed & C2 is closed
  proof
    take C1,C2;
A8: now
      A1 /\ C2 c= C1 /\ C2 & A1 /\ C2 c= A1 by A4,XBOOLE_1:17,26;
      then
A9:   A1 /\ C2 c= (C1 /\ C2) /\ A1 by XBOOLE_1:19;
      assume C2 meets A1;
      then
A10:  A1 /\ C2 <> {} by XBOOLE_0:def 7;
      (C1 /\ C2) /\ A1 c= (C1 /\ C2) /\ (A1 \/ A2) by XBOOLE_1:7,26;
      then (C1 /\ C2) /\ (A1 \/ A2) <> {} by A10,A9,XBOOLE_1:1,3;
      hence contradiction by A6,XBOOLE_0:def 7;
    end;
    now
      C1 /\ A2 c= C1 /\ C2 & C1 /\ A2 c= A2 by A5,XBOOLE_1:17,26;
      then
A11:  C1 /\ A2 c= (C1 /\ C2) /\ A2 by XBOOLE_1:19;
      assume C1 meets A2;
      then
A12:  C1 /\ A2 <> {} by XBOOLE_0:def 7;
      (C1 /\ C2) /\ A2 c= (C1 /\ C2) /\ (A1 \/ A2) by XBOOLE_1:7,26;
      then (C1 /\ C2) /\ (A1 \/ A2) <> {} by A12,A11,XBOOLE_1:1,3;
      hence contradiction by A6,XBOOLE_0:def 7;
    end;
    hence thesis by A4,A5,A7,A8;
  end;
  hence thesis by Th42;
end;
