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;
reserve S for non void non empty ManySortedSign,
  A for non-empty MSAlgebra over S,
  V for Variables of A,
  t for c-Term of A,V,
  f for ManySortedFunction of V, the Sorts of A;

theorem Th34:
  for t being c-Term of A,V, e being finite DecoratedTree st e
is_an_evaluation_of t,f for p being Node of t, n being Node of e st n = p holds
  e|n is_an_evaluation_of t|p, f
proof
  let t be c-Term of A,V, e be finite DecoratedTree such that
A1: dom e = dom t and
A2: for p being Node of e holds (for s being SortSymbol of S, v being
Element of V.s st t.p = [v,s] holds e.p = f.s.v) & (for s being SortSymbol of S
  , x being Element of (the Sorts of A).s st t.p = [x,s] holds e.p = x) & for o
being OperSymbol of S st t.p = [o,the carrier of S] holds e.p = Den(o, A).succ(
  e,p);
  let p be Node of t, n be Node of e;
  set vt = e|n, tp = t|p;
A3: dom vt = (dom e)|n by TREES_2:def 10;
  assume
A4: n = p;
  hence dom vt = dom tp by A1,A3,TREES_2:def 10;
  let q be Node of vt;
  reconsider nq = n^q as Node of e by A3,TREES_1:def 6;
  reconsider pq = p^q as Node of t by A1,A4,A3,TREES_1:def 6;
  hereby
    let s be SortSymbol of S, v be Element of V.s;
    assume tp.q = [v,s];
    then t.pq = [v,s] by A1,A4,A3,TREES_2:def 10;
    then e.nq = f.s.v by A2,A4;
    hence vt.q = f.s.v by A3,TREES_2:def 10;
  end;
  hereby
    let s be SortSymbol of S, x be Element of (the Sorts of A).s;
    assume tp.q = [x,s];
    then t.pq = [x,s] by A1,A4,A3,TREES_2:def 10;
    then e.nq = x by A2,A4;
    hence vt.q = x by A3,TREES_2:def 10;
  end;
  let o be OperSymbol of S;
  assume tp.q = [o,the carrier of S];
  then t.pq = [o,the carrier of S] by A1,A4,A3,TREES_2:def 10;
  then e.nq = Den(o, A).succ(e,nq) by A2,A4;
  then vt.q = Den(o, A).succ(e,n^q) by A3,TREES_2:def 10;
  hence thesis by TREES_9:31;
end;
