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 x be Element of FMT, A being Subset of FMT holds x in A^Fos iff x
  in A & not x in (A\{x})^Fob
proof
  let x be Element of FMT;
  let A be Subset of FMT;
A1: x in A & not x in (A\{x})^Fob implies x in A^Fos
  proof
    assume that
A2: x in A and
A3: not x in (A\{x})^Fob;
    consider V9 being Subset of FMT such that
A4: V9 in U_FMT x and
A5: V9 misses (A\{x}) by A3;
    V9 misses (A /\ {x}`) by A5,SUBSET_1:13;
    then (V9 /\ (A /\ {x}`)) = {};
    then (V9 /\ {x}`)/\ A = {} by XBOOLE_1:16;
    then (V9 \ {x}) /\ A = {} by SUBSET_1:13;
    then (V9 \ {x}) misses A;
    hence thesis by A2,A4;
  end;
  x in A^Fos implies x in A & not x in (A\{x})^Fob
  proof
    assume
A6: x in A^Fos;
    then consider V9 being Subset of FMT such that
A7: V9 in U_FMT x and
A8: V9 \ {x} misses A by Th22;
    V9 /\ {x}` misses A by A8,SUBSET_1:13;
    then V9 /\ {x}` /\ A = {};
    then V9 /\ ({x}`/\ A) = {} by XBOOLE_1:16;
    then V9 misses {x}`/\ A;
    then V9 misses A\{x} by SUBSET_1:13;
    hence thesis by A6,A7,Th20,Th22;
  end;
  hence thesis by A1;
end;
