C# で外部プロセスをパイプしたい時

コマンドプロンプトで実行する場合

netstat -an | find /c /v ""

C# で実行する場合

ProcessStartInfo psInfo1 = new ProcessStartInfo();
psInfo1.FileName = "netstat";
psInfo1.Arguments = "-an";
psInfo1.CreateNoWindow = true;
psInfo1.UseShellExecute = false;

psInfo1.RedirectStandardOutput = true;
Process p1 = Process.Start(psInfo1);

ProcessStartInfo psInfo2 = new ProcessStartInfo();
psInfo2.FileName = "find ";
psInfo2.Arguments = "/c /v \"\"";
psInfo2.CreateNoWindow = true;
psInfo2.UseShellExecute = false;

psInfo2.RedirectStandardOutput = true;
psInfo2.RedirectStandardInput = true;
Process p2 = Process.Start(psInfo2);

string output1 = p1.StandardOutput.ReadToEnd();
StreamWriter p2Writer = p2.StandardInput;

p2Writer.AutoFlush = true;
p2Writer.Write(output1);
p2Writer.Close();
string output2 = p2.StandardOutput.ReadToEnd();