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, A,B being Subset of FMT holds (
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
  )))) implies (A \/ B)^Fodelta c= ((A^Fodelta) \/ (B^Fodelta))
proof
  let FMT be non empty FMT_Space_Str;
  let A,B be Subset of FMT;
  assume
A1: 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;
  for x be Element of FMT holds x in (A \/ B)^Fodelta implies x in (A
  ^Fodelta) \/ (B^Fodelta)
  proof
    let x be Element of FMT;
    assume
A2: x in (A \/ B)^Fodelta;
A3: for W1 being Subset of FMT st W1 in U_FMT x holds ( W1 meets A & W1
    meets A` or W1 meets B & W1 meets B` )
    proof
      let W1 being Subset of FMT;
      assume
A4:   W1 in U_FMT x;
      then W1 meets (A \/ B)` by A2,Th19;
      then W1 /\ (A \/ B)` <> {};
      then
A5:   W1 /\ (A` /\ B`) <> {} by XBOOLE_1:53;
      then (W1 /\ A`) /\ B` <> {} by XBOOLE_1:16;
      then (W1 /\ A`) meets B`;
      then
A6:   ex z1 being object st z1 in (W1 /\ A`) /\ B` by XBOOLE_0:4;
      (W1 /\ B`) /\ A` <> {} by A5,XBOOLE_1:16;
      then (W1 /\ B`) meets A`;
      then
A7:   ex z2 being object st z2 in (W1 /\ B`) /\ A` by XBOOLE_0:4;
      W1 meets (A \/ B) by A2,A4,Th19;
      then W1 /\ (A \/ B) <> {};
      then (W1 /\ A \/ W1 /\ B) <> {} by XBOOLE_1:23;
      then
      W1 /\ A <> {} & W1 /\ A` <> {} or W1 /\ B <> {} & W1 /\ B` <> {} by A6,A7
;
      hence thesis;
    end;
    for V1,V2 being Subset of FMT st V1 in U_FMT x & V2 in U_FMT x holds
    V1 meets A & V1 meets A` or V2 meets B & V2 meets B`
    proof
      let V1,V2 be Subset of FMT;
      assume V1 in U_FMT x & V2 in U_FMT x;
      then consider W being Subset of FMT such that
A8:   W in U_FMT x and
A9:   W c= V1 /\ V2 by A1;
      V1 /\ V2 c= V2 by XBOOLE_1:17;
      then W c= V2 by A9;
      then
A10:  W /\ B c= V2 /\ B & W /\ B` c= V2 /\ B` by XBOOLE_1:26;
      V1 /\ V2 c= V1 by XBOOLE_1:17;
      then W c= V1 by A9;
      then
A11:  W /\ A c= V1 /\ A & W /\ A` c= V1 /\ A` by XBOOLE_1:26;
      V1 meets A & V1 meets A` or V2 meets B & V2 meets B`
      proof
        now
          per cases by A3,A8;
          case
            W meets A & W meets A`;
            then (ex z1 being object st z1 in W /\ A )&
ex z2 being object st z2 in
            W /\ A` by XBOOLE_0:4;
            hence thesis by A11;
          end;
          case
            W meets B & W meets B`;
            then (ex z3 being object st z3 in W /\ B )&
ex z4 being object st z4 in
            W /\ B` by XBOOLE_0:4;
            hence thesis by A10;
          end;
        end;
        hence thesis;
      end;
      hence thesis;
    end;
    then (for V1 being Subset of FMT st V1 in U_FMT x holds (V1 meets A & V1
meets A`)) or for V2 being Subset of FMT st V2 in U_FMT x holds V2 meets B & V2
    meets B`;
    then x in (A^Fodelta) or x in (B^Fodelta);
    hence thesis by XBOOLE_0:def 3;
  end;
  hence thesis;
end;
