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 Th58:
  P is open implies Fr P is nowhere_dense
proof
A1: Int(Cl P) c= Cl P by Th16;
  assume P is open;
  then
A2: Int P = P by Th23;
  then P misses Fr P by Th39;
  then
A3: P /\ Fr P = {} TS;
  Int (P /\ Fr P) = P /\ Int(Fr P) by A2,Th17;
  then P /\ Int(Fr P) = {} by A3;
  then
A4: P misses Int(Fr P);
  Int(Fr P) c= Int(Cl P) by Th19,XBOOLE_1:17;
  then
A5: Int(Fr P) c= Cl P by A1;
  Fr P is boundary
  proof
    set x = the Element of Int(Fr P);
    assume
A6: not Fr P is boundary;
    then
A7: TS is non empty;
A8: Int (Fr P) <> {} by A6,Th48;
    then x in Cl P by A5;
    hence contradiction by A4,A8,A7,Th12;
  end;
  hence thesis;
end;
