reserve s for set,
  i,j for natural Number,
  k for Nat,
  x,x1,x2,x3 for Real,
  r,r1,r2,r3,r4 for Real,
  F,F1,F2,F3 for real-valued FinSequence,
  R,R1,R2 for Element of i-tuples_on REAL;
reserve z,z1,z2 for Element of COMPLEX;
reserve n for Nat,
  x, y, a for Real,
  p, p1, p2, p3, q, q1, q2 for Element of n-tuples_on REAL;

theorem Th119:
  for x being real-valued FinSequence holds |(x,x)| >= 0
proof
  let x be real-valued FinSequence;
  set n=len x;
  x is FinSequence of REAL by Lm2;
  then reconsider w = x as Element of n-tuples_on REAL by FINSEQ_2:92;
  |(x, x)| = Sum sqr(w);
  hence thesis by Th86;
end;
