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
  |.p.| = 0 iff p = 0.REAL 3
proof
  thus |.p.| = 0 implies p = 0.REAL 3
  proof
    assume
A1: |.p.| = 0;
 |(p, p)| >= 0 by RVSUM_1:119;
    then (sqrt|(p,p)|)^2 = |(p,p)| by SQUARE_1:def 2;
    hence thesis by A1,Th70;
  end;
    assume p = 0.REAL 3; then
A2: p = |[ 0,0,0 ]| by FINSEQ_2:62;
then A3:p.1 = 0;
A4:p.2 = 0 by A2;
p.3 = 0 by A2;
   then |(p,p)| = 0^2 + 0^2 + 0^2 by A3,A4,Th55;
   hence thesis;
end;
