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 Th6:
  for x being real-valued FinSequence holds |(x,x)|=0 iff x=0*(len x)
proof
  let x be real-valued FinSequence;
  thus |(x,x)|=0 implies x=0*(len x)
  proof
    x is FinSequence of REAL by Lm1;
    then reconsider y=x as Element of REAL len x by EUCLID:76;
    assume |(x,x)|=0;
    then |.x.|^2=0 by Th4;
    then |.x.|=0 by XCMPLX_1:6;
    then y=0*(len x) by EUCLID:8;
    hence thesis;
  end;
  assume x=0*(len x);
  then |.x.|=0 by EUCLID:7;
  then |(x,x)|=0^2 by Th4;
  hence thesis;
end;
