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
  for x,y being real-valued FinSequence st len x=len y holds |(x-y, x-
  y)| = |(x, x)| - 2*|(x, y)| + |(y, y)|
proof
  let x,y be real-valued FinSequence;
  assume len x=len y;
  then |(x-y, x-y)| = |(x,x)| - |(x,y)| - |(y,x)| + |(y, y)| by Th127
    .= |(x,x)| - 2*|(x,y)| + |(y, y)|;
  hence thesis;
end;
