reserve n for Nat,
  a, b, r, w for Real,
  x, y, z for Point of TOP-REAL n,
  e for Point of Euclid n;
reserve V for RealLinearSpace,
        p,q,x for Element of V;
reserve p, q, x for Point of TOP-REAL n;
reserve s, t for Point of TOP-REAL 2;

theorem
  s in circle(a,b,r) & t in circle(a,b,r) implies LSeg(s,t) \ {s,t} c=
  inside_of_circle(a,b,r)
proof
  assume
A1: s in circle(a,b,r) & t in circle(a,b,r);
  reconsider G = |[a,b]| as Point of Euclid 2 by TOPREAL3:8;
  Sphere(G,r) = circle(a,b,r) by Th47;
  then
A2: Sphere(|[a,b]|,r) = circle(a,b,r) by Th13;
  Ball(|[a,b]|,r) = inside_of_circle(a,b,r) by Th48;
  hence thesis by A1,A2,Th32;
end;
