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 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)) iff for A,B being Subset
  of FMT holds (A^Foi) /\ (B^Foi) = (A /\ B)^Foi
proof
  let FMT be non empty FMT_Space_Str;
  thus (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 for A,B being Subset of FMT holds (A^Foi) /\ (B^Foi) = (A /\ B)^Foi
  proof
    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;
    let A,B be Subset of FMT;
    for x be Element of FMT holds x in (A^Foi) /\ (B^Foi) iff x in (A /\ B
    )^Foi
    proof
      let x be Element of FMT;
A2:   x in (A^Foi) /\ (B^Foi) implies x in (A /\ B)^Foi
      proof
        assume
A3:     x in (A^Foi) /\ (B^Foi);
        then x in B^Foi by XBOOLE_0:def 4;
        then
A4:     ex W2 being Subset of FMT st W2 in U_FMT x & W2 c= B by Th21;
        x in A^Foi by A3,XBOOLE_0:def 4;
        then ex W1 being Subset of FMT st W1 in U_FMT x & W1 c= A by Th21;
        then consider W1,W2 being Subset of FMT such that
A5:     W1 in U_FMT x & W2 in U_FMT x and
A6:     W1 c= A and
A7:     W2 c= B by A4;
        consider W being Subset of FMT such that
A8:     W in U_FMT x and
A9:     W c= W1 /\ W2 by A1,A5;
        W1 /\ W2 c= W2 by XBOOLE_1:17;
        then W c= W2 by A9;
        then
A10:    W c= B by A7;
        W1 /\ W2 c= W1 by XBOOLE_1:17;
        then W c= W1 by A9;
        then W c= A by A6;
        then W c= A /\ B by A10,XBOOLE_1:19;
        hence thesis by A8;
      end;
      x in (A /\ B)^Foi implies x in (A^Foi) /\ (B^Foi)
      proof
        assume
A11:    x in (A /\ B)^Foi;
        (A /\ B)^Foi c= (A^Foi) /\ (B^Foi) by Th29;
        hence thesis by A11;
      end;
      hence thesis by A2;
    end;
    hence (A^Foi) /\ (B^Foi) = (A /\ B)^Foi by SUBSET_1:3;
  end;
  (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^Foi) /\ (B^Foi)) <> (A
  /\ B)^Foi
  proof
    given x0 being Element of FMT, V1,V2 being Subset of FMT such that
A12: V1 in U_FMT x0 & V2 in U_FMT x0 and
A13: for W being Subset of FMT st W in U_FMT x0 holds not W c= V1 /\ V2;
    take V1,V2;
    x0 in (V1)^Foi & x0 in (V2)^Foi by A12;
    then x0 in ( ((V1)^Foi) /\ (V2^Foi) ) by XBOOLE_0:def 4;
    hence thesis by A13,Th21;
  end;
  hence
  (for A,B being Subset of FMT holds ((A^Foi) /\ (B^Foi)) = (A /\ B) ^Foi
) 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;
end;
