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;

theorem Th52:
  for h being Endomorphism of T st
  (for s,x holds h.s.(x-term) = x-term) holds h = id the Sorts of T
  proof
    let h be Endomorphism of T;
A1: id the Sorts of T is_homomorphism T,T by MSUALG_3:9;
A2: h is_homomorphism T,T by MSUALG_6:def 2;
    assume Z0: for s,x holds h.s.(x-term) = x-term;
    h||FreeGen T = (id the Sorts of T)||FreeGen T
    proof
      let s be SortSymbol of S;
      thus (h||FreeGen T).s = ((id the Sorts of T)||FreeGen T).s
      proof
        let t be Element of (FreeGen T).s;
A6:     (FreeGen T).s c= (the Sorts of T).s by PBOOLE:def 2,def 18;
        (FreeGen T).s = FreeGen(s,X) by MSAFREE:def 16;
        then consider x being set such that
A4:     x in X.s & t = root-tree [x,s] by MSAFREE:def 15;
        reconsider x as Element of X.s by A4;
        thus (h||FreeGen T).s.t = ((h.s)|((FreeGen T).s)).t by MSAFREE:def 1
        .= h.s.(x-term) by A4,FUNCT_1:49
        .= x-term by Z0
        .= (id ((the Sorts of T).s)).t by A4,A6,FUNCT_1:18
        .= ((id the Sorts of T).s).t by MSUALG_3:def 1
        .= (((id the Sorts of T).s)|((FreeGen T).s)).t by FUNCT_1:49
        .= ((id the Sorts of T)||FreeGen T).s.t by MSAFREE:def 1;
      end;
    end;
    hence h = id the Sorts of T by A1,A2,EXTENS_1:19;
  end;
