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 Th44:
  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 open & C2 is open
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 open & C2 is open
  proof
    assume A1,A2 are_separated;
    then consider C1, C2 being Subset of X such that
A1: A1 c= C1 & A2 c= C2 & C1 misses A2 & C2 misses A1 & C1 is closed &
    C2 is closed by Th42;
    take C2`,C1`;
    thus thesis by A1,SUBSET_1:23,24;
  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 open & C2
  is open;
  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 C2`,C1`;
    thus thesis by A2,SUBSET_1:23,24;
  end;
  hence thesis by Th42;
end;
