reserve x,y,z for Real,
  x3,y3 for Real,
  p for Point of TOP-REAL 3;
reserve p1,p2,p3,p4 for Point of TOP-REAL 3,
  x1,x2,y1,y2,z1,z2 for Real;

theorem Th29:
  |(p1,p2)| = p1`1*p2`1 + p1`2*p2`2 + p1`3*p2`3
proof
  reconsider f1=p1, f2=p2 as FinSequence of REAL by EUCLID:24;
A1: len f1 = len <* p1`1, p1`2, p1`3 *> by Th27
    .= 3 by FINSEQ_1:45;
A2: len f2 = len <* p2`1, p2`2, p2`3 *> by Th27
    .= 3 by FINSEQ_1:45;
  |(p1,p2)| = Sum mlt(f1, f2) by RVSUM_1:def 16
    .= Sum <* f1.1*f2.1, f1.2*f2.2, f1.3*f2.3 *> by A1,A2,Th28
    .= p1`1*p2`1 + p1`2*p2`2 + p1`3*f2.3 by RVSUM_1:78;
  hence thesis;
end;
