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 Th42:
  A1,A2 are_separated iff 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
  thus A1,A2 are_separated implies 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
    set C1 = Cl A1, C2 = Cl A2;
    assume
A1: A1,A2 are_separated;
    take C1,C2;
    thus thesis by A1,CONNSP_1:def 1,PRE_TOPC:18;
  end;
  given C1, C2 being Subset of X such that
A2: A1 c= C1 & A2 c= C2 & C1 misses A2 & C2 misses A1 & C1 is closed &
  C2 is closed;
  Cl A1 misses A2 & A1 misses Cl A2 by A2,TOPS_1:5,XBOOLE_1:63;
  hence thesis by CONNSP_1:def 1;
end;
