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 Th84:
  |{ p1,p2,p3 }| = |{ p2,p3,p1 }|
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
 .= p2.1*(p3.2*p1.3-p3.3*p1.2)+p2.2*(p3.3*p1.1-p3.1*p1.3)+
    p2.3*(p3.1*p1.2-p3.2*p1.1)
 .= |(|[ p2.1, p2.2, p2.3 ]|,
    |[ p3.2*p1.3-p3.3*p1.2, p3.3*p1.1-p3.1*p1.3, p3.1*p1.2-p3.2*p1.1 ]| )|
    by EUCLID_5:30
 .= |(p2, p3<X>p1)| by Th1;
  hence thesis;
end;
