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
  <e1><X><e3> = -<e2>
proof
     <e1><X><e3>
    = |[ (0 * 1)-(0 * 0),(0 * 0)-(1 * 1),(1 * 0)-(0 * 0 )]|
   .= |[ (-1)*<e2>.1,(-1)*<e2>.2,(-1)*<e2>.3 ]|
   .= -<e2> by Lm1;
    hence thesis;
end;
