reserve x,y,z for set;

theorem Th16:
  for S being non void Signature for Y being non-empty
ManySortedSet of the carrier of S for X being ManySortedSet of the carrier of S
  for s being SortSymbol of S st x in S-Terms(X,Y).s holds x is Term of S,Y
proof
  let S be non void Signature;
  let Y be non-empty ManySortedSet of the carrier of S;
  let X be ManySortedSet of the carrier of S;
  let s be SortSymbol of S;
  assume x in S-Terms(X,Y).s;
  then
  x in {t where t is Term of S,Y: the_sort_of t = s & variables_in t c= X}
  by Def5;
  then
  ex t being Term of S,Y st x = t & the_sort_of t = s & variables_in t c= X;
  hence thesis;
end;
