reserve X for set;
reserve G for Group;
reserve H for Subgroup of G;
reserve h,x,y for object;

theorem Th11:
  H is proper iff (the carrier of G) \ (the carrier of H) is non empty set
proof
  set UG = the carrier of G;
  set UH = the carrier of H;
  thus H is proper implies UG \ UH is non empty set
  proof
    assume A1: H is proper;
    UH c= UG & UH <> UG by A1,Th10, GROUP_2:def 5;
    then (for x holds x in UH implies x in UG) &
    not (for x holds x in UH iff x in UG) by TARSKI:2;
    hence (the carrier of G) \ (the carrier of H) is non empty set
    by XBOOLE_0:def 5;
  end;
  thus UG \ UH is non empty set implies H is proper;
end;
