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

theorem Th35:
  p1 in circle(a,b,r) & p2 in circle(a,b,r) & p3 in circle(a,b,r)
  & p1<>p2 & p2<>p3 implies angle(p1,p2,p3)<>PI
proof
  assume
A1: p1 in circle(a,b,r);
  assume
A2: p2 in circle(a,b,r);
  assume p3 in circle(a,b,r);
  then
A3: LSeg(p1,p3) \ {p1,p3} c= inside_of_circle(a,b,r) by A1,TOPREAL9:60;
  assume p1<>p2 & p2<>p3;
  then
A4: not p2 in {p1,p3} by TARSKI:def 2;
  inside_of_circle(a,b,r) misses circle(a,b,r) by TOPREAL9:54;
  then
A5: inside_of_circle(a,b,r) /\ circle(a,b,r) = {} by XBOOLE_0:def 7;
  assume angle(p1,p2,p3)=PI;
  then p2 in LSeg(p1,p3) by Th11;
  then p2 in LSeg(p1,p3) \ {p1,p3} by A4,XBOOLE_0:def 5;
  hence contradiction by A2,A3,A5,XBOOLE_0:def 4;
end;
