Sorry but you've got it wrong.
1, The script MUST NOT be pasted in the form page, but in the "Thankyou" or "confirmation" or whatever you call it page.
2. The action of your form is STILL set to "mailto:". You must change that. Please follow Andy's tutorial to the letter. The form action must be set to "thankyou.php" if you called your thankyou (and processing, at the same time) page this way.

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