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 Th77:
  p1<X>(p2+p3) = (p1<X>p2)+(p1<X>p3)
proof
A1: p2+p3 = |[ p2.1+p3.1, p2.2+p3.2, p2.3+p3.3 ]| by Lm2; then
A2: (p2+p3).1 = p2.1+p3.1;
A3: (p2+p3).2 = p2.2+p3.2 by A1;
A4: (p2+p3).3 = p2.3+p3.3 by A1;
    (p1<X>p2)+(p1<X>p3) = |[ (p1.2*p2.3-p1.3*p2.2)+(p1.2*p3.3-p1.3*p3.2),
      (p1.3*p2.1-p1.1*p2.3)+(p1.3*p3.1-p1.1*p3.3),
      (p1.1*p2.2-p1.2*p2.1)+(p1.1*p3.2-p1.2*p3.1) ]| by Lm8
   .= |[ p1.2*p2.3-p1.3*p2.2+p1.2*p3.3-p1.3*p3.2,
      p1.3*p2.1-p1.1*p2.3+p1.3*p3.1-p1.1*p3.3,
      p1.1*p2.2-p1.2*p2.1+p1.1*p3.2-p1.2*p3.1 ]|;
    hence thesis by A2,A3,A4;
end;
