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);

theorem Th23:
  t is x-omitting implies (t,[x,s])<-t1 = t
  proof assume
A1: Coim(t,[x,s]) = {};
    reconsider dt = dom t as set;
AA: dom ((t,[x,s])<-t1) = dt
    proof
      thus dom ((t,[x,s])<-t1) c= dt
      proof let a; assume a in dom ((t,[x,s])<-t1);
        then reconsider r = a as Node of (t,[x,s])<-t1;
        per cases by TREES_4:def 7;
        suppose r in dom t;
          hence thesis;
        end;
        suppose ex q being Node of t, p being Node of t1 st
          q in Leaves dom t & t.q = [x,s] & r = q^p;
          then consider q being Node of t, p being Node of t1 such that
A2:       q in Leaves dom t & t.q = [x,s] & r = q^p;
          [x,s] in {[x,s]} by TARSKI:def 1;
          hence thesis by A1,A2,FUNCT_1:def 7;
        end;
      end;
      thus dt c= dom ((t,[x,s])<-t1) by TREES_4:def 7;
    end;
    now let a; assume a in dom t;
      then reconsider r = a as Node of t;
      [x,s] in {[x,s]} by TARSKI:def 1;
      then t.r <> [x,s] by A1,FUNCT_1:def 7;
      hence ((t,[x,s])<-t1).a = t.a by TREES_4:def 7;
    end;
    hence thesis by AA,FUNCT_1:2;
  end;
