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 Th11:
  for x,y being real-valued FinSequence st len x=len y holds |.x-y.|^2
  = |.x.|^2 - 2*|(y, x)| + |.y.|^2
proof
  let x,y be real-valued FinSequence;
  assume
A1: len x=len y;
  |.x-y.|^2 = |(x-y, x-y)| by Th4
    .= |(x, x)| - 2*|(x, y)| + |(y, y)| by A1,RVSUM_1:129
    .= |.x.|^2 - 2*|(y, x)| + |(y, y)| by Th4
    .= |.x.|^2 - 2*|(y, x)| + |.y.|^2 by Th4;
  hence thesis;
end;
