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 Th133:
  for t,t1 being DecoratedTree, xi,nu being Node of t st not xi c= nu &
  not nu c= xi holds (t with-replacement(xi,t1))|nu = t|nu
  proof
    let t,t1 be DecoratedTree;
    let xi,nu be Node of t;
    assume Z0: not xi c= nu;
    assume Z1: not nu c= xi;
    not xi c< nu by Z0,XBOOLE_0:def 8;
    then
A1: nu in dom t with-replacement(xi,dom t1)
    = dom(t with-replacement(xi,t1)) by TREES_1:def 9,TREES_2:def 11;
A2: dom((t with-replacement(xi,t1))|nu) = (dom(t with-replacement(xi,t1)))|nu
    by TREES_2:def 10;
    hence
A4: dom((t with-replacement(xi,t1))|nu) = (dom t)|nu by Z0,Z1,A1,Th134
    .= dom (t|nu) by TREES_2:def 10;
    let p be Node of (t with-replacement(xi,t1))|nu;
A8: (dom t)|nu = dom (t|nu) by TREES_2:def 10;
    then
A5: nu^p in dom t & (t|nu).p = t.(nu^p) by A4,TREES_1:def 6,TREES_2:def 10;
    not xi c< nu^p by Z0,Z1,Lem8B,XBOOLE_0:def 8;
    then
A7: nu^p in dom t with-replacement(xi,dom t1) by A5,TREES_1:def 9;
A6: not ex r being FinSequence of NAT st r in dom t1 & nu^p = xi^r &
    (t with-replacement(xi,t1)).(nu^p) = t1.r by Z0,Z1,Lem8B,TREES_1:1;
    thus ((t with-replacement(xi,t1))|nu).p
    = (t with-replacement(xi,t1)).(nu^p) by A2,TREES_2:def 10
    .= t.(nu^p) by A6,A7,TREES_2:def 11
    .= (t|nu).p by A8,TREES_2:def 10,A4;
  end;
