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 Th56:
   H1 is Subgroup of H2 implies N ~ H1 c= N ~ H2
proof
  assume H1 is Subgroup of H2; then
A1: carr(H1) c= carr(H2) by GROUP_2:def 5;
  let x be object;
  assume
A2: x in N ~ H1;
  then reconsider x as Element of G;
  x * N meets carr(H1) by A2,Th51;
  then x * N meets carr(H2) by A1,XBOOLE_1:63;
  hence thesis;
end;
