reserve A for Tolerance_Space,
  X, Y for Subset of A;
reserve A for Approximation_Space,
  X for Subset of A;
reserve A for finite Tolerance_Space,
  X for Subset of A,
  x for Element of A;
reserve A for finite Approximation_Space,
  X, Y for Subset of A,
  x for Element of A;

theorem Th47:
  X c= Y implies MemberFunc (X, A).x <= MemberFunc (Y, A).x
proof
  set CI = Class (the InternalRel of A, x);
  assume X c= Y;
  then card (Y /\ CI) >= card (X /\ CI) by NAT_1:43,XBOOLE_1:26;
  then
A1: card (Y /\ CI) / (card CI) >= card (X /\ CI) / (card CI) by XREAL_1:72;
  MemberFunc (X, A).x = card (X /\ CI) / (card CI) by Def9;
  hence thesis by A1,Def9;
end;
