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 Th60:
  N1 is Subgroup of N2 implies N2 ` H c= N1 ` H
proof
  assume
A1: N1 is Subgroup of N2;
  let x be object;
  assume
A2: x in N2 ` H;
  then reconsider x as Element of G;
A3: x * N1 c= x * N2 by A1,GROUP_3:6;
  x * N2 c= carr(H) by A2,Th49;
  then x * N1 c= carr(H) by A3;
  hence thesis;
end;
