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 Th1:
  for N be normal Subgroup of G,x1,x2 be Element of G holds
  (x1 * N) * (x2 * N) = (x1 * x2) * N
proof
  let N be normal Subgroup of G,x1,x2 be Element of G;
  (x1 * N) * (x2 * N) = x1 * N * x2 * N by GROUP_2:10
                     .= x1 * (N * x2) * N by GROUP_2:29
                     .= x1 * (x2 * N) * N by GROUP_3:117
                     .= (x1 * x2) * N * N by GROUP_2:105
                     .= (x1 * x2) * (N * N) by GROUP_2:29;
  hence thesis by GROUP_2:76;
end;
