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 Th62:
  A1,A2 are_separated iff ex B1, B2 being Subset of X st B1,B2
  are_weakly_separated & A1 c= B1 & A2 c= B2 & B1 /\ B2 misses A1 \/ A2
proof
  thus A1,A2 are_separated implies ex B1, B2 being Subset of X st B1,B2
  are_weakly_separated & A1 c= B1 & A2 c= B2 & B1 /\ B2 misses A1 \/ A2
  proof
    assume A1,A2 are_separated;
    then consider B1, B2 being Subset of X such that
A1: A1 c= B1 & A2 c= B2 & B1 /\ B2 misses A1 \/ A2 & B1 is open & B2
    is open by Th45;
    take B1,B2;
    thus thesis by A1,Th49;
  end;
  given B1, B2 being Subset of X such that
A2: B1,B2 are_weakly_separated and
A3: A1 c= B1 and
A4: A2 c= B2 and
A5: B1 /\ B2 misses A1 \/ A2;
  B1 /\ B2 misses A1 by A5,XBOOLE_1:7,63;
  then
A6: (B1 /\ B2) /\ A1 = {} by XBOOLE_0:def 7;
  B1 /\ B2 misses A2 by A5,XBOOLE_1:7,63;
  then
A7: (B1 /\ B2) /\ A2 = {} by XBOOLE_0:def 7;
  B1 /\ A2 c= A2 & B1 /\ A2 c= B1 /\ B2 by A4,XBOOLE_1:17,26;
  then
A8: B1 /\ A2 = {} by A7,XBOOLE_1:3,19;
  A2 \ B1 c= B2 \ B1 by A4,XBOOLE_1:33;
  then
A9: A2 \ B1 /\ A2 c= B2 \ B1 by XBOOLE_1:47;
  A1 /\ B2 c= A1 & A1 /\ B2 c= B1 /\ B2 by A3,XBOOLE_1:17,26;
  then
A10: A1 /\ B2 = {} by A6,XBOOLE_1:3,19;
  A1 \ B2 c= B1 \ B2 by A3,XBOOLE_1:33;
  then
A11: A1 \ A1 /\ B2 c= B1 \ B2 by XBOOLE_1:47;
  B1 \ B2,B2 \ B1 are_separated by A2;
  hence thesis by A10,A8,A11,A9,CONNSP_1:7;
end;
