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 Th29:
  for p being Node of t holds t|p is Term of S,V
proof
  defpred P[set] means for q being Node of t st q = $1 holds t|q is Term of S,
  V;
A1: for p be Node of t,n be Nat st P[p] & p^<*n*> in dom t holds
  P[p^<*n*>]
  proof
    let p be Node of t, n be Nat;
    assume that
A2: for q being Node of t st q = p holds t|q is Term of S,V and
A3: p^<*n*> in dom t;
    reconsider u = t|p as Term of S,V by A2;
A4: dom u = (dom t)|p by TREES_2:def 10;
    reconsider nn=n as Element of NAT by ORDINAL1:def 12;
A5: <*nn*> in (dom t)|p by A3,TREES_1:def 6;
A6: now
      given s being SortSymbol of S, x being Element of V.s such that
A7:   u.{} = [x,s];
      u = root-tree [x,s] by A7,Th5;
      then <*n*> in {{}} by A5,A4,TREES_1:29,TREES_4:3;
      hence contradiction by TARSKI:def 1;
    end;
    (ex s being SortSymbol of S, v being Element of V.s st u.{} = [v,s])
    or u.{} in [:the carrier' of S,{the carrier of S}:] by Th2;
    then consider
    o being OperSymbol of S, x2 being Element of {the carrier of S}
    such that
A8: u.{} = [o,x2] by A6,DOMAIN_1:1;
    x2 = the carrier of S by TARSKI:def 1;
    then consider a being ArgumentSeq of Sym(o,V) such that
A9: u = [o,the carrier of S]-tree a by A8,Th10;
    consider i being Nat, T being DecoratedTree, r being Node of T
    such that
A10: i < len a and
    T = a.(i+1) and
A11: <*n*> = <*i*>^r by A5,A4,A9,TREES_4:11;
A12: n = <*n*>.1
      .= i by A11,FINSEQ_1:41;
    then
A13: u|<*nn*> = a.(nn+1) by A9,A10,TREES_4:def 4;
    let q be Node of t;
    nn+1 in dom a by A10,A12,Lm9;
    then ex t being Term of S,V
    st t = u|<*nn*> & t = (a qua FinSequence of S
-Terms V qua non empty set)/.(n+1) & the_sort_of t = (the_arity_of o).(n+1) &
    the_sort_of t = (the_arity_of o)/.(n+1) by A13,Lm8;
    hence thesis by TREES_9:3;
  end;
A14: P[{}] by TREES_9:1;
  for p being Node of t holds P[p] from TREES_2:sch 1(A14,A1);
  hence thesis;
end;
