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 K \/ Fr L = Fr(K \/ L) \/ Fr(K /\ L) \/ (Fr K /\ Fr L)
proof
A1: Fr L c= Fr(K \/ L) \/ Fr(K /\ L) \/ (Fr K /\ Fr L) by Lm1;
A2: Fr(K \/ L) \/ Fr(K /\ L) \/ (Fr K /\ Fr L) c= Fr K \/ Fr L
  proof
    let x be object;
A3: now
      assume x in (Fr K /\ Fr L);
      then x in Fr K by XBOOLE_0:def 4;
      hence thesis by XBOOLE_0:def 3;
    end;
A4: now
      assume
A5:   x in Fr(K \/ L);
      Fr(K \/ L) c= (Fr K \/ Fr L) by Th33;
      hence thesis by A5;
    end;
A6: now
      assume
A7:   x in Fr(K /\ L);
      Fr(K /\ L) c= (Fr K \/ Fr L) by Th32;
      hence thesis by A7;
    end;
    assume x in (Fr(K \/ L) \/ Fr(K /\ L)) \/ (Fr K /\ Fr L);
    then x in (Fr(K \/ L) \/ Fr(K /\ L)) or x in (Fr K /\ Fr L) by
XBOOLE_0:def 3;
    hence thesis by A4,A6,A3,XBOOLE_0:def 3;
  end;
  Fr K c= Fr(K \/ L) \/ Fr(K /\ L) \/ (Fr K /\ Fr L) by Lm1;
  then Fr K \/ Fr L c= Fr(K \/ L) \/ Fr(K /\ L) \/ (Fr K /\ Fr L) by A1,
XBOOLE_1:8;
  hence thesis by A2;
end;
