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 Th4:
  for x,y being Element of G, N be normal Subgroup of G
  st y in N holds x * y * x" in N
proof
  let x,y be Element of G, N be normal Subgroup of G;
  assume y in N;
then x * y in x * N by GROUP_2:103;
  then x * y in N * x by GROUP_3:117;
  then consider y1 be Element of G such that
A1: x * y = y1 * x & y1 in N by GROUP_2:104;
   x * y * x" = y1 * (x * x") by A1,GROUP_1:def 3
             .= y1 * 1_G  by GROUP_1:def 5
             .= y1 by GROUP_1:def 4;
  hence thesis by A1;
end;
