  We provide a computer-assisted proof of the holomorphy of the quartic and the octic meromorphic differentials arising in the main Theorem 4.11 of our paper 'The Classification of Branched Willmore spheres in the $3$-Sphere and the $4$-Sphere' (arXiv:1706.01405), using the free mathematical software Sage. 
Computer-Assisted Proof of the Main Theorem of 'The Classification of   Branched Willmore Spheres in the $3$-Sphere and the $4$-Sphere'