in the update you are using a form value which wont exist unless the form is submitted, instead of using the form values in the update, use the recordset values that you are using to populate the form.
for example, the code you are using for the CurrInvNum form element:
<?php echo $row_rsArt['Art_InvNum']+1; ?>
use that in the update for the value of the artist_invnumcurrent instead of binding to the CurrInvNum form element