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;
