reserve i, n for Nat,
  x, y, a for Real,
  v for Element of n-tuples_on REAL,
  p, p1, p2, p3, q, q1, q2 for Point of TOP-REAL n;

theorem Th8:
  for x being real-valued FinSequence holds |(x, 0*(len x))| = 0
proof
  let x be real-valued FinSequence;
  set n=len x;
  x is FinSequence of REAL by Lm1;
  then reconsider p1=x as Element of n-tuples_on REAL by FINSEQ_2:92;
A1: 0*n = n |-> 0 by EUCLID:def 4;
  |(x, 0*n)| = Sum mlt(p1,0*n) .= Sum 0*n by Th2
    .= 0 by A1,RVSUM_1:81;
  hence thesis;
end;
