theorem
  for N being normal Subgroup of G for x holds x * N * x" c= carr(N)
proof
  let N be normal Subgroup of G;
  let x;
  x * N c= N * x by GROUP_3:118; then
A1: x * N * x" c= N * x * x" by GROUP_3:5;
  N * x * x" = N * (x * x") by GROUP_2:107
            .= N * 1_G by GROUP_1:def 5;
  hence thesis by A1,GROUP_2:109;
end;
