reserve i for Nat,
  j for Element of NAT,
  X,Y,x,y,z for set;
reserve C for initialized ConstructorSignature,
  s for SortSymbol of C,
  o for OperSymbol of C,
  c for constructor OperSymbol of C;
reserve a,b for expression of C, an_Adj C;
reserve t, t1,t2 for expression of C, a_Type C;
reserve p for FinSequence of QuasiTerms C;
reserve e for expression of C;
reserve a,a9 for expression of C, an_Adj C;
reserve q for pure expression of C, a_Type C,
  A for finite Subset of QuasiAdjs C;
reserve T for quasi-type of C;

theorem Th116:
  for D1,D2 being non empty DTConstrStr st Terminals D1 c= Terminals D2 &
  the Rules of D1 c= the Rules of D2
  holds TS D1 c= TS D2
proof
  let G,G9 be non empty DTConstrStr such that
A1: Terminals G c= Terminals G9 and
A2: the Rules of G c= the Rules of G9;
A3: the carrier of G9 = (Terminals G9) \/ NonTerminals G9 by LANG1:1;
A4: the carrier of G c= the carrier of G9 by A1,A2,Th115;
  defpred P[set] means $1 in TS G9;
A5: for s being Symbol of G st s in Terminals G holds P[root-tree s]
  proof
    let s be Symbol of G;
    assume
A6: s in Terminals G;
    then reconsider s9 = s as Symbol of G9 by A1,A3,XBOOLE_0:def 3;
    root-tree s = root-tree s9;
    hence thesis by A1,A6,DTCONSTR:def 1;
  end;
A7: for nt being Symbol of G,
  ts being FinSequence of TS(G) st nt ==> roots ts &
  for t being DecoratedTree of the carrier of G st t in rng ts
  holds P[t]
  holds P[nt-tree ts]
  proof
    let n be Symbol of G;
    let s be FinSequence of TS(G) such that
A8: [n, roots s] in the Rules of G and
    A9: for t being DecoratedTree of the carrier of G st t in rng s holds P[t];
    rng s c= TS G9
    by A9;
    then reconsider s9 = s as FinSequence of TS G9 by FINSEQ_1:def 4;
    reconsider n9 = n as Symbol of G9 by A4;
    n9 ==> roots s9 by A2,A8;
    hence thesis by DTCONSTR:def 1;
  end;
A10: for t being DecoratedTree of the carrier of G st t in TS(G) holds P[t]
  from DTCONSTR:sch 7(A5,A7);
  let x be object;
  assume
A11: x in TS G;
  then reconsider t = x as Element of FinTrees(the carrier of G);
  P[t] by A10,A11;
  hence thesis;
end;
