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 Th36:
  for FMT being non empty FMT_Space_Str holds FMT is Fo_filled iff
  for A being Subset of FMT holds A^Foi c= A
proof
  let FMT be non empty FMT_Space_Str;
A1: FMT is Fo_filled implies for A being Subset of FMT holds A^Foi c= A
  proof
    assume
A2: FMT is Fo_filled;
    let A be Subset of FMT;
    let x be object;
    assume
A3: x in A^Foi;
    then reconsider y=x as Element of FMT;
    consider V be Subset of FMT such that
A4: V in U_FMT y and
A5: V c= A by A3,Th21;
    y in V by A2,A4;
    hence thesis by A5;
  end;
  (for A being Subset of FMT holds A^Foi c= A) implies FMT is Fo_filled
  proof
    assume
A6: for A being Subset of FMT holds A^Foi c= A;
    assume not FMT is Fo_filled;
    then consider y being Element of FMT, V being Subset of FMT such that
A7: V in U_FMT y and
A8: not y in V;
    y in V^Foi by A7;
    then not V^Foi c= V by A8;
    hence contradiction by A6;
  end;
  hence thesis by A1;
end;
