let i be set ; :: according to PBOOLE:def 16 :: thesis: ( not i in the carrier of S or not (ParsedTerms X) . i is empty )
assume i in the carrier of S ; :: thesis: not (ParsedTerms X) . i is empty
then reconsider s = i as Element of S ;
(ParsedTerms X) . s = ParsedTerms X,s by Def8;
hence not (ParsedTerms X) . i is empty ; :: thesis: verum