let S be non empty non void ManySortedSign ; :: thesis: for X being non-empty ManySortedSet of the carrier of S holds FreeGen X is non-empty
let X be non-empty ManySortedSet of the carrier of S; :: thesis: FreeGen X is non-empty
let x be object ; :: according to PBOOLE:def 13 :: thesis: ( not x in the carrier of S or not (FreeGen X) . x is empty )
assume x in the carrier of S ; :: thesis: not (FreeGen X) . x is empty
then reconsider s = x as SortSymbol of S ;
(FreeGen X) . s = FreeGen (s,X) by Def16;
hence not (FreeGen X) . x is empty ; :: thesis: verum