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;

theorem Th38:
  A1 \/ A2 is open & A1,A2 are_separated implies A1 is open & A2 is open
proof
  assume
A1: A1 \/ A2 is open;
A2: A1 c= Cl A1 by PRE_TOPC:18;
  assume
A3: A1,A2 are_separated;
  then A2 misses Cl A1 by CONNSP_1:def 1;
  then
A4: A2 c= (Cl A1)` by SUBSET_1:23;
  A1 misses Cl A2 by A3,CONNSP_1:def 1;
  then
A5: A1 c= (Cl A2)` by SUBSET_1:23;
A6: A2 c= Cl A2 by PRE_TOPC:18;
A7: (A1 \/ A2) /\ (Cl A2)` = (A1 \/ A2) \ Cl A2 by SUBSET_1:13
    .= (A1 \ Cl A2) \/ (A2 \ Cl A2) by XBOOLE_1:42
    .= (A1 \ Cl A2) \/ {} by A6,XBOOLE_1:37
    .= A1 /\ (Cl A2)` by SUBSET_1:13
    .= A1 by A5,XBOOLE_1:28;
  (A1 \/ A2) /\ (Cl A1)` = (A1 \/ A2) \ Cl A1 by SUBSET_1:13
    .= (A1 \ Cl A1) \/ (A2 \ Cl A1) by XBOOLE_1:42
    .= {} \/ (A2 \ Cl A1) by A2,XBOOLE_1:37
    .= A2 /\ (Cl A1)` by SUBSET_1:13
    .= A2 by A4,XBOOLE_1:28;
  hence thesis by A1,A7;
end;
