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)<X>(q1+q2) = p1<X>q1+p1<X>q2+p2<X>q1+p2<X>q2
proof
 (p1+p2)<X>(q1+q2) = (p1<X>(q1+q2))+(p2<X>(q1+q2)) by Th78;
then  (p1+p2)<X>(q1+q2) = (p1<X>q1+p1<X>q2)+(p2<X>(q1+q2)) by Th77;
    then (p1+p2)<X>(q1+q2) = p1<X>q1+p1<X>q2+(p2<X>q1+p2<X>q2) by Th77;
    hence thesis by RVSUM_1:15;
end;
