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 Th54:
  p, q are_orthogonal implies a*p,q are_orthogonal
proof
  assume p, q are_orthogonal;
  then |(p,q)|=0;
  then a*(|(p,q)|)=0;
  then |(a*p,q)|=0 by Th17;
  hence thesis;
end;
