let i be object ; :: according to PBOOLE:def 13 :: thesis: ( not i in the carrier of S or not (FreeSort X) . i is empty )
assume i in the carrier of S ; :: thesis: not (FreeSort X) . i is empty
then reconsider s = i as SortSymbol of S ;
(FreeSort X) . s = FreeSort (X,s) by Def11;
hence not (FreeSort X) . i is empty ; :: thesis: verum