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,p3 }| = |(p1<X>p2,p3)|
proof
    |{ p1,p2,p3 }| = |(|[ p1.1, p1.2, p1.3 ]|,
    |[ (p2.2*p3.3)-(p2.3*p3.2),(p2.3*p3.1)-(p2.1*p3.3),
    (p2.1*p3.2)-(p2.2*p3.1) ]|)| by Th1
 .= p1.1*((p2.2*p3.3)-(p2.3*p3.2))+p1.2*((p2.3*p3.1)-(p2.1*p3.3))+
    p1.3*((p2.1*p3.2)-(p2.2*p3.1)) by EUCLID_5:30
 .= p2.2*(p1.1*p3.3)-p2.3*(p1.1*p3.2)+(p2.3*(p1.2*p3.1)-
    p2.1*(p1.2*p3.3))+(p2.1*(p1.3*p3.2)-p2.2*(p1.3*p3.1));
    then |{ p1,p2,p3 }|
  = (p2.3*p1.2-p2.2*p1.3)*p3.1+(p2.1*p1.3-p2.3*p1.1)*p3.2+
    (p2.2*p1.1-p2.1*p1.2)*p3.3
 .= |( p1<X>p2, |[ p3.1, p3.2, p3.3 ]| )| by EUCLID_5:30
 .= |( p1<X>p2, p3 )| by Th1;
    hence thesis;
end;
