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));

theorem Th18:
  for I,J being FinSequence-membered set
  for f being FinSequence holds I c= J iff f^^I c= f^^J
  proof
    let I,J be FinSequence-membered set;
    let f be FinSequence;
    thus I c= J implies f^^I c= f^^J
    proof assume
A1:   I c= J;
      let a; assume a in f^^I;
      then ex b being Element of I st a = f^b & b in I;
      hence a in f^^J by A1;
    end;
    assume
A2: f^^I c= f^^J;
    let a; assume
A3: a in I;
    then reconsider a as FinSequence;
    f^a in f^^I by A3;
    then f^a in f^^J by A2;
    then consider b being Element of J such that
A4: f^a = f^b & b in J;
    thus thesis by A4,FINSEQ_1:33;
  end;
