Original sentence: The only such third-degree polynomial is zero.Two distinct third-degree polynomials satisfying the given boundary conditions would differ only by a third-degree polynomial which is zero with zero derivative at two distinct points.Using the procedure outlined above, two control points and two tangents specify a third-degree polynomial
Edited sentence: Here's an outline of the proof: Using the procedure outlined above, two control points and two tangents specify a third-degree polynomial; Two distinct third-degree polynomials satisfying the given boundary conditions would differ only by a third-degree polynomial which is zero with zero derivative at two distinct points; The only such third-degree polynomial is zero: QED.The formulas specified above are guaranteed to produce a unique path between the two points