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
  x in A^Fon iff x in A & for V st V in U_FMT x holds (V \ {x}) meets A
proof
  thus x in A^Fon implies x in A & for V st V in U_FMT x holds (V \ {x}) meets
  A
  proof
    assume x in A^Fon;
    then x in A & not x in A^Fos by XBOOLE_0:def 5;
    hence thesis;
  end;
  assume that
A1: x in A and
A2: for V st V in U_FMT x holds (V \ {x}) meets A;
  not x in A^Fos by A2,Th22;
  hence thesis by A1,XBOOLE_0:def 5;
end;
