theorem
  for x being real-valued FinSequence holds |(x,x)| = 0 iff |.x.| = 0
proof
  let x be real-valued FinSequence;
A1: |(x,x)| = 0^2 implies |.x.| = 0
  proof
    assume |(x,x)| = 0^2;
    then x=0*(len x) by Th6;
    hence thesis by EUCLID:7;
  end;
  |.x.| = 0 implies |(x,x)| = 0^2 by Th4;
  hence thesis by A1;
end;
