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;

theorem Th49:
  for A1,A2 being Subset of X holds A1 is open & A2 is open
  implies A1,A2 are_weakly_separated
proof
  let A1,A2 be Subset of X;
  assume that
A1: A1 is open and
A2: A2 is open;
  (A2 \ A1) misses A1 by XBOOLE_1:79;
  then Cl(A2 \ A1) misses A1 by A1,Th36;
  then
A3: (A1 \ A2) misses Cl(A2 \ A1) by XBOOLE_1:36,63;
  A2 misses (A1 \ A2) by XBOOLE_1:79;
  then A2 misses Cl(A1 \ A2) by A2,Th36;
  then Cl(A1 \ A2) misses (A2 \ A1) by XBOOLE_1:36,63;
  then A1 \ A2,A2 \ A1 are_separated by A3,CONNSP_1:def 1;
  hence thesis;
end;
