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 Th26:
  for S being non void non empty ManySortedSign, V1,V2 being
non-empty ManySortedSet of the carrier of S st V1 c= V2 for t being Term of S,
  V1 holds t is Term of S, V2
proof
  let S be non void non empty ManySortedSign;
  let V1,V2 be non-empty ManySortedSet of the carrier of S such that
A1: for s being object st s in the carrier of S holds V1.s c= V2.s;
  defpred P[set] means $1 is Term of S,V2;
A2: for o being OperSymbol of S, p being ArgumentSeq of Sym(o,V1) st for t
being Term of S,V1 st t in rng p holds P[t] holds P[[o,the carrier of S]-tree p
  ]
  proof
    let o be OperSymbol of S, p be ArgumentSeq of Sym(o,V1);
    assume
A3: for t being Term of S,V1 st t in rng p holds t is Term of S,V2;
    rng p c= S-Terms V2
    proof
      let x be object;
      assume
A4:   x in rng p;
      rng p c= S-Terms V1 by FINSEQ_1:def 4;
      then reconsider x as Term of S,V1 by A4;
      x is Term of S,V2 by A3,A4;
      hence thesis;
    end;
    then reconsider q = p as FinSequence of S-Terms V2 by FINSEQ_1:def 4;
A5: now
      let i be Nat;
      assume
A6:   i in dom q;
      then consider t being Term of S,V1 such that
A7:   t = q.i and
      t = (p qua FinSequence of S-Terms V1 qua non empty set)/.i and
A8:   the_sort_of t = (the_arity_of o).i and
      the_sort_of t = (the_arity_of o)/.i by Lm8;
      t in rng p by A6,A7,FUNCT_1:def 3;
      then reconsider T = t as Term of S,V2 by A3;
      take T;
      thus T = q.i by A7;
      thus the_sort_of T = (the_arity_of o).i
      proof
        per cases by Th2;
        suppose
          ex s being SortSymbol of S, v being Element of V1.s st t.{} = [v,s];
          then consider
          s being SortSymbol of S, v being Element of V1.s such that
A9:       t.{} = [v,s];
A10:      t = root-tree [v,s] by A9,Th5;
          V1.s c= V2.s by A1;
          then v in V2.s;
          hence the_sort_of T = s by A10,Th14
            .= (the_arity_of o).i by A8,A10,Th14;
        end;
        suppose
          t.{} in [:the carrier' of S,{the carrier of S}:];
          then consider
          o9 being OperSymbol of S, x2 being Element of {the carrier
          of S} such that
A11:      t.{} = [o9,x2] by DOMAIN_1:1;
A12:      x2 = the carrier of S by TARSKI:def 1;
          hence the_sort_of T = the_result_sort_of o9 by A11,Th17
            .= (the_arity_of o).i by A8,A11,A12,Th17;
        end;
      end;
    end;
    len p = len the_arity_of o by Lm8;
    then q is ArgumentSeq of Sym(o,V2) by A5,Th24;
    hence thesis by Th1;
  end;
A13: for s being SortSymbol of S, v being Element of V1.s holds P[root-tree [
  v,s]]
  proof
    let s be SortSymbol of S, v be Element of V1.s;
    V1.s c= V2.s by A1;
    then v in V2.s;
    hence thesis by Th4;
  end;
  for t being Term of S,V1 holds P[t] from TermInd(A13,A2);
  hence thesis;
end;
