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;

theorem Th33:
  y in Sphere(x,r) & z in Sphere(x,r) implies LSeg(y,z) /\ Sphere( x,r) = {y,z}
proof
A1: y in LSeg(y,z) & z in LSeg(y,z) by RLTOPSP1:68;
  assume
A2: y in Sphere(x,r) & z in Sphere(x,r);
  then
A3: LSeg(y,z) \ {y,z} c= Ball(x,r) by Th32;
  hereby
    let a be object;
    assume
A4: a in LSeg(y,z) /\ Sphere(x,r);
    assume
A5: not a in {y,z};
    a in LSeg(y,z) by A4,XBOOLE_0:def 4;
    then
A6: a in LSeg(y,z) \ {y,z} by A5,XBOOLE_0:def 5;
A7: Ball(x,r) misses Sphere(x,r) by Th17;
    a in Sphere(x,r) by A4,XBOOLE_0:def 4;
    hence contradiction by A3,A6,A7,XBOOLE_0:3;
  end;
  let a be object;
  assume a in {y,z};
  then a = y or a = z by TARSKI:def 2;
  hence thesis by A2,A1,XBOOLE_0:def 4;
end;
