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
  R is boundary iff R c= Fr R
proof
A1: Cl R /\ Cl(R`) c= Cl(R`) by XBOOLE_1:17;
  hereby
    assume R is boundary;
    then R` is dense;
    then Cl R /\ Cl R` = Cl R /\ ([#] GX);
    then Cl R = Cl R /\ Cl R` by XBOOLE_1:28;
    hence R c= Fr R by PRE_TOPC:18;
  end;
A2: R` c= Cl(R`) by PRE_TOPC:18;
  assume R c= Fr R;
  then R c= Cl(R`) by A1;
  then R \/ (R`) c= Cl(R`) by A2,XBOOLE_1:8;
  then [#] GX c= Cl(R`) by PRE_TOPC:2;
  then [#] GX = Cl(R`);
  then R` is dense;
  hence thesis;
end;
