let S be non void Signature; :: thesis: for X being V9() ManySortedSet of the carrier of S
for x being Element of (Free (S,X)) holds x is Term of S,(X (\/) ( the carrier of S --> ))

let X be V9() ManySortedSet of the carrier of S; :: thesis: for x being Element of (Free (S,X)) holds x is Term of S,(X (\/) ( the carrier of S --> ))
set Y = X (\/) ( the carrier of S --> );
let x be Element of (Free (S,X)); :: thesis: x is Term of S,(X (\/) ( the carrier of S --> ))
A1: S -Terms (X (\/) ( the carrier of S --> )) = TS (DTConMSA (X (\/) ( the carrier of S --> ))) by MSATERM:def 1
.= union (rng (FreeSort (X (\/) ( the carrier of S --> )))) by MSAFREE:11
.= Union (FreeSort (X (\/) ( the carrier of S --> ))) by CARD_3:def 4 ;
A2: ( FreeMSA (X (\/) ( the carrier of S --> )) = MSAlgebra(# (FreeSort (X (\/) ( the carrier of S --> ))),(FreeOper (X (\/) ( the carrier of S --> ))) #) & dom the Sorts of (FreeMSA (X (\/) ( the carrier of S --> ))) = the carrier of S ) by ;
consider y being object such that
A3: y in dom the Sorts of (Free (S,X)) and
A4: x in the Sorts of (Free (S,X)) . y by CARD_5:2;
ex A being MSSubset of (FreeMSA (X (\/) ( the carrier of S --> ))) st
( Free (S,X) = GenMSAlg A & A = (Reverse (X (\/) ( the carrier of S --> ))) "" X ) by Def1;
then the Sorts of (Free (S,X)) is MSSubset of (FreeMSA (X (\/) ( the carrier of S --> ))) by MSUALG_2:def 9;
then the Sorts of (Free (S,X)) c= the Sorts of (FreeMSA (X (\/) ( the carrier of S --> ))) by PBOOLE:def 18;
then the Sorts of (Free (S,X)) . y c= the Sorts of (FreeMSA (X (\/) ( the carrier of S --> ))) . y by A3;
hence x is Term of S,(X (\/) ( the carrier of S --> )) by A1, A3, A4, A2, CARD_5:2; :: thesis: verum