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
  |((x*p1+y*p2), p3)| = x*|(p1,p3)| + y*|(p2,p3)|
proof
  |(x*p1 + y*p2, p3)| = |(x*p1, p3)| + |(y*p2, p3)| by Th130
    .= x*|(p1,p3)| + |(y*p2,p3)| by Th131
    .= x*|(p1,p3)| + y*|(p2,p3)| by Th131;
  hence thesis;
end;
