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 Th57:
  N1 is Subgroup of N2 implies N1 ~ H c= N2 ~ H
proof
  assume
A1:N1 is Subgroup of N2;
  let x be object;
  assume
A2: x in N1 ~ H;
  then reconsider x as Element of G;
  x * N1 meets carr(H) by A2,Th51;
  then x * N2 meets carr(H) by A1,GROUP_3:6,XBOOLE_1:63;
  hence thesis;
end;
