reserve X,Y,Z,Z1,Z2,D for set,x,y for object;
reserve SFX,SFY,SFZ for set;
reserve F,G for Subset-Family of D;
reserve P for Subset of D;
reserve E for set;

theorem
  for TS being set, F, G being Subset-Family of TS st F is
  with_proper_subsets & G c= F holds G is with_proper_subsets;
