Let $S₁$ and $S₂$ witness the two flips and put $S = S₁ ∪ S₂$. The partition into atomic types over $S$ in $G₀$ refines the one over $S₁$ in $G₀$, and also the one over $S₂$ in $G₁$: two vertices outside $S$ with the same neighbors in $S$ lie in the same part over $S₁$, so their adjacency to a vertex of $S₂$ is preserved or complemented alike when passing to $G₁$. A flip of a flip is a flip for a common refinement.