reserve G for Group;
reserve A,B for non empty Subset of G;
reserve N,H,H1,H2 for Subgroup of G;
reserve x,a,b for Element of G;
reserve N1,N2 for Subgroup of G;

theorem Th54:
  carr(H) c= N ~ H
proof
  let x be object;
  assume x in carr(H);
  then reconsider x as Element of H;
  reconsider x as Element of G by GROUP_2:42;
  x in x * N by GROUP_2:108;
  then x * N meets carr(H) by XBOOLE_0:3;
  hence thesis;
end;
