theorem Th10:
  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_1:128
    .= |.x.|^2 + 2*|(y, x)| + |(y, y)| by Th4
    .= |.x.|^2 + 2*|(y, x)| + |.y.|^2 by Th4;
  hence thesis;
end;
