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