reserve p1,p2,p3,p4,p5,p6,p,pc for Point of TOP-REAL 2;
reserve a,b,c,r,s for Real;

theorem Th37:
  p1 in circle(a,b,r) & p2 in circle(a,b,r) & p3 in circle(a,b,r)
  & angle(p1,p2,p3) = 0 & p1<>p2 & p2<>p3 implies p1=p3
proof
  assume
A1: p1 in circle(a,b,r) & p2 in circle(a,b,r) & p3 in circle(a,b,r);
  assume
A2: angle(p1,p2,p3) = 0;
  assume
A3: p1<>p2 & p2<>p3;
  then
A4: euc2cpx(p1)<> euc2cpx(p2) & euc2cpx(p2)<> euc2cpx(p3) by EUCLID_3:4;
  assume
A5: p1<>p3;
  then euc2cpx(p1)<> euc2cpx(p3) by EUCLID_3:4;
  then angle(p2,p3,p1) = 0 & angle(p3,p1,p2) = PI or angle(p2,p3,p1) = PI &
  angle(p3,p1,p2) = 0 by A2,A4,COMPLEX2:87;
  hence contradiction by A1,A3,A5,Th35;
end;
