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
  |((x*p1+y*p2), p3)| = x*|(p1,p3)| + y*|(p2,p3)|
proof
  |(x*p1 + y*p2, p3)| = |(x*p1, p3)| + |(y*p2, p3)| by Th16
    .= x*|(p1,p3)| + |(y*p2,p3)| by Th17
    .= x*|(p1,p3)| + y*|(p2,p3)| by Th17;
  hence thesis;
end;
