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 Th42:
  P is open iff Fr P = Cl P \ P
proof
A1: Fr P misses (Fr P)` by XBOOLE_1:79;
A2: Int P c= P by Th16;
  hereby
    assume P is open;
    then P = Int P by Th23;
    hence Fr P = Cl P \ P by Lm2;
  end;
  assume
A3: Fr P = Cl P \ P;
  Cl P \ Fr P = (P \/ Fr P) \ Fr P by Th31
    .= (Fr P)` /\ (P \/ Fr P) by SUBSET_1:13
    .= (P /\ (Fr P)`) \/ ((Fr P)` /\ (Fr P)) by XBOOLE_1:23
    .= (P \ Fr P) \/ (Fr P /\ (Fr P)`) by SUBSET_1:13
    .= Int P \/ (Fr P /\ (Fr P)`) by Th40
    .= Int P \/ {} TS by A1
    .= Int P;
  then P c= Int P by A3,Lm5,PRE_TOPC:18;
  hence thesis by A2,XBOOLE_0:def 10;
end;
