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 Th31:
  for x being Element of G st x in N ` (N ~ A) holds x * N c= N ~ A
proof
  let x be Element of G;
  assume x in N ` (N ~ A);
  then ex x1 being Element of G st x1 = x & x1 * N c= N ~ A;
  hence thesis;
end;
