reserve r, r1, r2, x, y, z,
        x1, x2, x3, y1, y2, y3 for Real;
reserve R, R1, R2, R3 for Element of 3-tuples_on REAL;
reserve p, q, p1, p2, p3, q1, q2 for Element of REAL 3;
reserve f1, f2, f3, g1, g2, g3, h1, h2, h3 for PartFunc of REAL,REAL;
reserve t, t0, t1, t2 for Real;

theorem
  -|[ x,y,z ]| = -x*<e1> - y*<e2> - z*<e3>
proof
  -|[ x,y,z ]| = (-1)*x*<e1> + (-1)*y*<e2> + (-1)*z*<e3> by Th32
  .=-x*<e1> + ((-1)*y)*<e2> + (-z)*<e3> by RVSUM_1:49
  .=-x*<e1> + -y*<e2> + ((-1)*z)*<e3> by RVSUM_1:49
  .=-x*<e1> - y*<e2> - z*<e3> by RVSUM_1:49;
  hence thesis;
end;
