I think you could make the first theorem in the post (simplified fundamental theorem on two variables) easier to understand to a novice if you explicitly clarified that the conclusion diagram Λ′ → Λ → X is the same as Λ′ ← Λ → X by the chain rerooting rule, and perhaps use the latter diagram in the picture as it more directly makes clear the idea of mediation/inducing independence.
I think you could make the first theorem in the post (simplified fundamental theorem on two variables) easier to understand to a novice if you explicitly clarified that the conclusion diagram Λ′ → Λ → X is the same as Λ′ ← Λ → X by the chain rerooting rule, and perhaps use the latter diagram in the picture as it more directly makes clear the idea of mediation/inducing independence.