reserve S for non void non empty ManySortedSign,
  V for non-empty ManySortedSet of the carrier of S;
reserve A for MSAlgebra over S,
  t for Term of S,V;

theorem Th15:
  for t being c-Term of A,V for s being SortSymbol of S, x be set
  st x in (the Sorts of A).s & t = root-tree [x,s] holds the_sort_of t = s
proof
  let t be c-Term of A,V;
  let s be SortSymbol of S, x be set;
  assume x in (the Sorts of A).s;
  then x in (the Sorts of A).s \/ V.s by XBOOLE_0:def 3;
  then x in ((the Sorts of A) (\/) V).s by PBOOLE:def 4;
  hence thesis by Th14;
end;
