let S be non empty non void ManySortedSign ; :: thesis: for X being V5() ManySortedSet of the carrier of S holds FreeGen X is non-empty
let X be V5() ManySortedSet of the carrier of S; :: thesis: FreeGen X is non-empty
let x be set ; :: according to PBOOLE:def 16 :: 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 Def18;
hence not (FreeGen X) . x is empty ; :: thesis: verum