reserve
  a,b for object, I,J for set, f for Function, R for Relation,
  i,j,n for Nat, m for (Element of NAT),
  S for non empty non void ManySortedSign,
  s,s1,s2 for SortSymbol of S,
  o for OperSymbol of S,
  X for non-empty ManySortedSet of the carrier of S,
  x,x1,x2 for (Element of X.s), x11 for (Element of X.s1),
  T for all_vars_including inheriting_operations free_in_itself
  (X,S)-terms MSAlgebra over S,
  g for Translation of Free(S,X),s1,s2,
  h for Endomorphism of Free(S,X);
reserve
  r,r1,r2 for (Element of T),
  t,t1,t2 for (Element of Free(S,X));
reserve
  Y for infinite-yielding ManySortedSet of the carrier of S,
  y,y1 for (Element of Y.s), y11 for (Element of Y.s1),
  Q for all_vars_including inheriting_operations free_in_itself
  (Y,S)-terms MSAlgebra over S,
  q,q1 for (Element of Args(o,Free(S,Y))),
  u,u1,u2 for (Element of Q),
  v,v1,v2 for (Element of Free(S,Y)),
  Z for non-trivial ManySortedSet of the carrier of S,
  z,z1 for (Element of Z.s),
  l,l1 for (Element of Free(S,Z)),
  R for all_vars_including inheriting_operations free_in_itself
  (Z,S)-terms MSAlgebra over S,
  k,k1 for Element of Args(o,Free(S,Z));
reserve c,c1,c2 for set, d,d1 for DecoratedTree;
reserve
  w for (Element of Args(o,T)),
  p,p1 for Element of Args(o,Free(S,X));
reserve C for (context of x), C1 for (context of y), C9 for (context of z),
  C11 for (context of x11), C12 for (context of y11), D for context of s,X;
reserve
  S9 for sufficiently_rich non empty non void ManySortedSign,
  s9 for SortSymbol of S9,
  o9 for s9-dependent OperSymbol of S9,
  X9 for non-trivial ManySortedSet of the carrier of S9,
  x9 for (Element of X9.s9);
reserve h1 for x-constant Homomorphism of Free(S,X), T,
  h2 for y-constant Homomorphism of Free(S,Y), Q;
reserve
  s2 for s1-reachable SortSymbol of S,
  g1 for Translation of Free(S,Y),s1,s2,
  g for Translation of Free(S,X),s1,s2;

theorem Th132:
  for xi being Node of t st t.xi = [x,s] holds t|xi = x-term
  proof
    let xi be Node of t;
    assume Z0: t.xi = [x,s];
    reconsider tx = t|xi as Element of Free(S,X) by MSAFREE4:44;
    per cases by Th16;
    suppose ex s1,x11 st tx = x11-term;
      then consider s1,x11 such that
A1:   tx = x11-term;
      <*>NAT in dom tx = (dom t)|xi by TREES_1:22,TREES_2:def 10;
      then tx.{} = t.(xi^{}) by TREES_2:def 10;
      hence t|xi = x-term by Z0,A1,TREES_4:3;
    end;
    suppose ex o,p st tx = o-term p;
      then consider o,p such that
A2:   tx = o-term p;
      <*>NAT in dom tx = (dom t)|xi by TREES_1:22,TREES_2:def 10;
      then tx.{} = t.(xi^{}) by TREES_2:def 10;
      then [o,the carrier of S] = [x,s] by Z0,A2,TREES_4:def 4;
      then s in the carrier of S = s by XTUPLE_0:1;
      hence thesis;
    end;
  end;
