let S be non void Signature; :: thesis: for X being V9() ManySortedSet of the carrier of S
for Y being ManySortedSet of the carrier of S
for t being Element of (Free (S,X)) holds S variables_in t c= X

let X be V9() ManySortedSet of the carrier of S; :: thesis: for Y being ManySortedSet of the carrier of S
for t being Element of (Free (S,X)) holds S variables_in t c= X

let Y be ManySortedSet of the carrier of S; :: thesis: for t being Element of (Free (S,X)) holds S variables_in t c= X
let t be Element of (Free (S,X)); :: thesis:
set Z = X (\/) ( the carrier of S --> );
reconsider t = t as Term of S,(X (\/) ( the carrier of S --> )) by Th8;
t in Union the Sorts of (Free (S,X)) ;
then A1: t in Union (S -Terms (X,(X (\/) ( the carrier of S --> )))) by Th24;
dom (S -Terms (X,(X (\/) ( the carrier of S --> )))) = the carrier of S by PARTFUN1:def 2;
then ex s being object st
( s in the carrier of S & t in (S -Terms (X,(X (\/) ( the carrier of S --> )))) . s ) by ;
then variables_in t c= X by Th17;
hence S variables_in t c= X ; :: thesis: verum