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));

theorem Lem8A:
  for p,q,r being FinSequence st p^q is_a_prefix_of p^r holds
  q is_a_prefix_of r
  proof
    let p,q,r be FinSequence;
    assume p^q is_a_prefix_of p^r;
    then consider n being FinSequence such that
A1: p^r = p^q^n by TREES_1:1;
    p^r = p^(q^n) by A1,FINSEQ_1:32;
    then r = q^n by FINSEQ_1:33;
    hence q is_a_prefix_of r by TREES_1:1;
  end;
