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 Th25:
  for FMT being non empty FMT_Space_Str, A,B being Subset of FMT
  holds A c= B implies A^Foi c= B^Foi
proof
  let FMT be non empty FMT_Space_Str;
  let A,B be Subset of FMT;
  assume
A1: A c= B;
  let x be object;
  assume
A2: x in A^Foi;
  then reconsider y=x as Element of FMT;
  consider V9 be Subset of FMT such that
A3: V9 in U_FMT y and
A4: V9 c= A by A2,Th21;
  V9 c= B by A1,A4;
  hence thesis by A3;
end;
