reserve p,q,r for FinSequence,
  x,y for object;

theorem Th8:
  for R being Relation, p,q being RedSequence of R st p.len p = q.1
  holds p$^q is RedSequence of R
proof
  let R be Relation, p,q be RedSequence of R;
  defpred P[set,set] means [$1,$2] in R;
  set r = p$^q;
  consider p1 being FinSequence, x being object such that
A1: p = p1^<*x*> by FINSEQ_1:46;
  assume p.len p = q.1;
  then
A2: len p > 0 & len q > 0 & p.len p = q.1;
  r = p1^q by A1,Th2;
  hence len r > 0;
A3: for i being Nat st i in dom q & i+1 in dom q holds P[q.i, q.(
  i+1)] by Def2;
A4: for i being Nat st i in dom p & i+1 in dom p holds P[p.i, p.(
  i+1)] by Def2;
  for i being Nat st i in dom (p$^q) & i+1 in dom (p$^q) for x,
y being set st x = (p$^q).i & y = (p$^q).(i+1) holds P[x,y] from PathCatenation
  (A4,A3,A2);
  hence thesis;
end;
