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 Th70:
  |(p,p)| = 0 iff p = 0.REAL 3
proof
  thus |(p,p)|=0 implies p =0.REAL 3
  proof
    assume |(p,p)|= 0; then
    Sum sqr(p) = 0;
    hence thesis by RVSUM_1:91;
  end;
  assume p= 0.REAL 3; then
A1:p = |[ 0,0,0 ]| by FINSEQ_2:62;
then A2:p.1 = 0;
A3:p.2 = 0 by A1;
   p.3 = 0 by A1; then
   |(p,p)| = 0^2 + 0^2 + 0^2 by A2,A3,Th55;
  hence |(p,p)| = 0;
end;
