reserve TS for 1-sorted,
  K, Q for Subset of TS;
reserve TS for TopSpace,
  GX for TopStruct,
  x for set,
  P, Q for Subset of TS,
  K , L for Subset of TS,
  R, S for Subset of GX,
  T, W for Subset of GX;

theorem Th43:
  P is closed iff Fr P = P \ Int P
proof
A1: (P`) misses P by XBOOLE_1:79;
  (P`) /\ Int P c= (P`) /\ P by Th16,XBOOLE_1:26;
  then
A2: (P`) /\ Int P c= {} TS by A1;
  thus P is closed implies Fr P = P \ Int P
  proof
    assume P is closed;
    then P = Cl P by PRE_TOPC:22;
    hence thesis by Lm2;
  end;
A3: (P``) \/ (Int P)` = ((P`) /\ Int P)` by XBOOLE_1:54;
  assume Fr P = P \ Int P;
  then Cl P = P \/ (P \ Int P) by Th31
    .= P \/ ((Int P)` /\ P) by SUBSET_1:13
    .= (P \/ (Int P)`) /\ (P \/ P) by XBOOLE_1:24
    .= ({} TS)` /\ P by A2,A3
    .= P by XBOOLE_1:28;
  hence thesis;
end;
