reserve FT for non empty RelStr;
reserve A for Subset of FT;
reserve T for non empty TopStruct;
reserve FMT for non empty FMT_Space_Str;
reserve x, y for Element of FMT;
reserve A, B, W, V for Subset of FMT;

theorem
  for FMT being non empty FMT_Space_Str st FMT is Fo_filled holds (for A
  ,B being Subset of FMT holds (A \/ B)^Fodelta = ((A^Fodelta) \/ (B^Fodelta)))
implies for x being Element of FMT, V1,V2 being Subset of FMT st V1 in U_FMT x
& V2 in U_FMT x holds ex W being Subset of FMT st W in U_FMT x & W c= (V1 /\ V2
  )
proof
  let FMT be non empty FMT_Space_Str;
  assume
A1: FMT is Fo_filled;
  (ex x being Element of FMT, V1,V2 being Subset of FMT st (V1 in U_FMT x)
& (V2 in U_FMT x) & (for W being Subset of FMT st W in U_FMT x holds (not(W c=
V1 /\ V2)) ) ) implies ex A,B being Subset of FMT st ((A \/ B)^Fodelta) <> ((A
  ^Fodelta) \/ (B^Fodelta))
  proof
    given x0 being Element of FMT, V1,V2 being Subset of FMT such that
A2: V1 in U_FMT x0 and
A3: V2 in U_FMT x0 and
A4: for W being Subset of FMT st W in U_FMT x0 holds not W c= V1 /\ V2;
    take (V1)`,(V2)`;
A5: not x0 in ((V2)`)^Fodelta
    proof
      assume x0 in ((V2)`)^Fodelta;
      then V2 meets (V2)` by A3,Th19;
      hence contradiction by XBOOLE_1:79;
    end;
    for W being Subset of FMT st W in U_FMT x0 holds W meets ((V1)` \/ (V2
    )`) & W meets ((V1)` \/ (V2)`)`
    proof
      let W being Subset of FMT;
      assume
A6:   W in U_FMT x0;
      then
A7:   not W c= V1 /\ V2 by A4;
A8:   W meets (V1 /\ V2)`
      proof
        assume W /\ (V1 /\ V2)` = {};
        then W \ (V1 /\ V2) = {} by SUBSET_1:13;
        hence contradiction by A7,XBOOLE_1:37;
      end;
      x0 in V1 & x0 in V2 by A1,A2,A3;
      then
A9:   x0 in V1 /\ V2 by XBOOLE_0:def 4;
      x0 in W by A1,A6;
      then W /\ ((V1 /\ V2)`)` <> {} by A9,XBOOLE_0:def 4;
      then W meets ((V1 /\ V2)`)`;
      hence thesis by A8,XBOOLE_1:54;
    end;
    then
A10: x0 in ((V1)` \/ (V2)`)^Fodelta;
    not x0 in ((V1)`)^Fodelta
    proof
      assume x0 in ((V1)`)^Fodelta;
      then V1 meets (V1)` by A2,Th19;
      hence contradiction by XBOOLE_1:79;
    end;
    hence thesis by A10,A5,XBOOLE_0:def 3;
  end;
  hence thesis;
end;
