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
  Fr T = Cl(T`) /\ T \/ (Cl T \ T)
proof
  for x being object holds x in Fr T iff x in ((Cl(T`) /\ T) \/ (Cl T\T))
  proof
    let x be object;
A1: T` c= Cl(T`) by PRE_TOPC:18;
    thus x in Fr T implies x in ((Cl(T`) /\ T) \/ (Cl T \ T))
    proof
      assume
A2:   x in Fr T;
      then reconsider x99= x as Point of GX;
      x99 in Cl T & x99 in Cl(T`) & x99 in T or x99 in Cl T & x99 in Cl(T`
      ) & x99 in T` by A2,SUBSET_1:29,XBOOLE_0:def 4;
      then x99 in Cl(T`) /\ T or not x99 in T & x99 in Cl T by XBOOLE_0:def 4;
      then x99 in (Cl(T`) /\ T) or x99 in (Cl T \ T) by XBOOLE_0:def 5;
      hence thesis by XBOOLE_0:def 3;
    end;
A3: T c= Cl T by PRE_TOPC:18;
    thus x in ((Cl(T`) /\ T) \/ (Cl T \ T)) implies x in Fr T
    proof
      assume
A4:   x in (Cl(T`) /\ T) \/ (Cl T \ T);
      then reconsider x99= x as Point of GX;
      x99 in (Cl(T`) /\ T) or x99 in (Cl T \ T) by A4,XBOOLE_0:def 3;
      then x99 in Cl(T`) & x99 in T or x99 in Cl T & not x99 in T by
XBOOLE_0:def 4,def 5;
      then x99 in Cl(T`) & x99 in T or x99 in Cl T & x99 in T` by SUBSET_1:29;
      hence thesis by A3,A1,XBOOLE_0:def 4;
    end;
  end;
  hence thesis by TARSKI:2;
end;
