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 Th11:
  for t being Term of S,V ex s being SortSymbol of S st t in FreeSort (V, s)
proof
  let t be Term of S,V;
  set X = V;
  set G = DTConMSA X;
  reconsider e = {} as Node of t by TREES_1:22;
  per cases;
  suppose
    t.{} is Terminal of G;
    then reconsider ts = t.{} as Terminal of G;
    consider s being SortSymbol of S, x being set such that
A1: x in X.s and
A2: ts = [x,s] by MSAFREE:7;
    take s;
    t = root-tree [x,s] by A2,DTCONSTR:9;
    then t in {a where a is Element of TS G: (ex x be set st x in X.s & a =
root-tree [x,s]) or ex o be OperSymbol of S st [o,the carrier of S] = a.{} &
    the_result_sort_of o = s} by A1;
    hence thesis by MSAFREE:def 10;
  end;
  suppose
    t.{} is not Terminal of G;
    then reconsider nt = t.e as NonTerminal of G by Lm6;
    nt in [:the carrier' of S,{the carrier of S}:] by Lm5;
    then consider
    o being OperSymbol of S, x2 being Element of {the carrier of S}
    such that
A3: nt = [o,x2] by DOMAIN_1:1;
    take s = the_result_sort_of o;
    x2 = the carrier of S by TARSKI:def 1;
    then t in {a where a is Element of TS G: (ex x be set st x in X.s & a =
root-tree [x,s]) or ex o be OperSymbol of S st [o,the carrier of S] = a.{} &
    the_result_sort_of o = s} by A3;
    hence thesis by MSAFREE:def 10;
  end;
end;
