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
  |{ p2,p1,p2 }| = 0
proof
A1:(p1<X>p2).1 = (p1.2*p2.3)-(p1.3*p2.2);
A2:(p1<X>p2).2 = (p1.3*p2.1)-(p1.1*p2.3);
(p1<X>p2).3 = (p1.1*p2.2)-(p1.2*p2.1);
   then |{ p2,p1,p2 }| = p2.1*((p1.2*p2.3)-(p1.3*p2.2))
  +p2.2*((p1.3*p2.1)-(p1.1*p2.3))
  +p2.3*((p1.1*p2.2)-(p1.2*p2.1)) by A2,A1,Lm5
.= 0;
  hence thesis;
end;
