Proof.
The proof is given by an explicit calculation. As an example (the remaining cases are checked analogously) we show that \(\mathrm{slide}^{-1}_{\mathbf{1}_1,B_1}\circ\; \mathrm{slide}_{\mathbf{1}_1,B_1}\) is homotopic to the identity by computing their difference and exhibiting an explicit null-homotopy: 
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