theorem
  s in circle(a,b,r) & t in inside_of_circle(a,b,r) implies LSeg(s,t) /\
  circle(a,b,r) = {s}
proof
  assume
A1: s in circle(a,b,r) & t in inside_of_circle(a,b,r);
  reconsider e = |[a,b]| as Point of Euclid 2 by TOPREAL3:8;
A2: inside_of_circle(a,b,r) = Ball(e,r) by Th46
    .= Ball(|[a,b]|,r) by Th11;
  circle(a,b,r) = Sphere(e,r) by Th47
    .= Sphere(|[a,b]|,r) by Th13;
  hence thesis by A1,A2,Th31;
end;
