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 Th51:
  -p = |[ -p.1,-p.2,-p.3 ]|
proof
    reconsider r = -1 as Element of REAL by XREAL_0:def 1;
    r*p = |[ (-1)*p.1,(-1)*p.2,(-1)*p.3 ]| by Th49
    .= |[ -p.1,-p.2,-p.3 ]|;
    hence thesis;
end;
