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

theorem Th13:
  p in LSeg(p1,p3) & p<>p1 & p<>p3 implies angle(p1,p,p2)+angle(p2
  ,p,p3)=PI or angle(p1,p,p2)+angle(p2,p,p3)=3*PI
proof
  assume p in LSeg(p1,p3) & p<>p1 & p<>p3;
  then
A1: angle(p1,p,p3) = PI by Th8;
  angle(p1,p,p2)+angle(p2,p,p3) = angle(p1,p,p3) or angle(p1,p,p2)+angle(
  p2,p,p3) = angle(p1,p,p3) + 2*PI by Th4;
  hence thesis by A1;
end;
