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;

theorem Th25:
  N is Subgroup of H implies H ` A c= N ` A
proof
  assume
A1: N is Subgroup of H;
  let x be object;
  assume
A2: x in H ` A;
  then reconsider x as Element of G;
A3: x * N c= x * H by A1,GROUP_3:6;
  x * H c= A by A2,Th12;
  then x * N c= A by A3;
  hence thesis;
end;
