reserve z,z1,z2 for Complex;
reserve r,x1,x2 for Real;
reserve p0,p,p1,p2,p3,q for Point of TOP-REAL 2;

theorem
  for p1,p2,p3 st p1<>0.TOP-REAL 2 & p2<>0.TOP-REAL 2 holds ( |(p1,p2)|=
  0 iff angle(p1,0.TOP-REAL 2,p2)=PI/2 or angle(p1,0.TOP-REAL 2,p2)=3/2*PI)
proof
  let p1,p2,p3;
  assume p1<>0.TOP-REAL 2 & p2<>0.TOP-REAL 2;
  then
A1: euc2cpx(p1)<> 0c & euc2cpx(p2)<> 0c by Th2,Th16;
  |(p1,p2)| = Re ((euc2cpx(p1)) .|. (euc2cpx(p2))) by Th42;
  hence thesis by A1,Th17,COMPLEX2:75;
end;
