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

theorem Th34:
  p1 in circle(a,b,r) & p2 in circle(a,b,r) & p3 in circle(a,b,r)
& p4 in circle(a,b,r) & p1<>p3 & p1<>p4 & p2<>p3 & p2<>p4 implies angle(p1,p3,
p2) = angle(p1,p4,p2) or angle(p1,p3,p2) = angle(p1,p4,p2) - PI or angle(p1,p3,
  p2) = angle(p1,p4,p2) + PI
proof
  assume
A1: p1 in circle(a,b,r);
  set pc=|[a,b]|;
  assume
A2: p2 in circle(a,b,r);
  assume
A3: p3 in circle(a,b,r);
  assume
A4: p4 in circle(a,b,r);
  assume that
A5: p1<>p3 and
A6: p1<>p4 and
A7: p2<>p3 and
A8: p2<>p4;
  per cases by A1,A2,A3,A5,A7,Th33;
  suppose
    2*angle(p1,p3,p2) = angle(p1,pc,p2);
    then
    2*angle(p1,p4,p2) = 2*angle(p1,p3,p2) or 2*(angle(p1,p4,p2) - PI) = 2
    *angle(p1,p3,p2) by A1,A2,A4,A6,A8,Th33;
    hence thesis;
  end;
  suppose
    2*(angle(p1,p3,p2) - PI) = angle(p1,pc,p2);
    then 2*angle(p1,p4,p2) = 2*(angle(p1,p3,p2) - PI) or 2*(angle(p1,p4,p2) -
    PI) = 2*(angle(p1,p3,p2) - PI) by A1,A2,A4,A6,A8,Th33;
    hence thesis;
  end;
end;
