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 st FMT
is Fo_filled holds (for x being Element of FMT holds {x} in U_FMT x ) implies (
  A^Foi) \/ (B^Foi) = (A \/ B)^Foi
proof
  let FMT be non empty FMT_Space_Str;
  let A,B be Subset of FMT;
  assume
A1: FMT is Fo_filled;
  assume
A2: for x being Element of FMT holds {x} in U_FMT x;
A3: for C being Subset of FMT holds C c= C^Foi
  proof
    let C be Subset of FMT;
    for y being Element of FMT holds y in C implies y in C^Foi
    proof
      let y be Element of FMT;
      assume y in C;
      then
A4:   {y} is Subset of C by SUBSET_1:41;
      {y} in U_FMT y by A2;
      hence thesis by A4;
    end;
    hence thesis;
  end;
A5: for C being Subset of FMT holds C = C^Foi
  by A1,A3,Th36;
  then (A \/ B)^Foi = (A \/ B) & (A^Foi) = A;
  hence thesis by A5;
end;
