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 Th31:
  Cl T = T \/ Fr T
proof
A1: (T \/ Cl T) /\ (T \/ Cl(T`)) = Cl T /\ (T \/ Cl(T`)) by PRE_TOPC:18
,XBOOLE_1:12;
  T \/ (Cl T \ T) c= T \/ (Cl T /\ Cl(T`))
  proof
    let x be object;
    assume
A2: x in T \/ (Cl T \ T);
    then reconsider x99=x as Point of GX;
    x99 in T or x99 in Cl T \ T by A2,XBOOLE_0:def 3;
    then
A3: x99 in T or x99 in Cl T & x99 in T` by XBOOLE_0:def 5;
    T` c= Cl(T`) by PRE_TOPC:18;
    then x99 in T or x99 in (Cl T /\ Cl (T`)) by A3,XBOOLE_0:def 4;
    hence thesis by XBOOLE_0:def 3;
  end;
  then
A4: Cl T c= T \/ Fr T by PRE_TOPC:18,XBOOLE_1:45;
  T \/ Fr T = (T \/ Cl T) /\ (T \/ Cl (T`)) by XBOOLE_1:24;
  then T \/ Fr T c= Cl T by A1,XBOOLE_1:17;
  hence thesis by A4;
end;
