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Proof of Proposition 2.8

Proof.
Suppose that ω ∈ L*_N(E). By definition, there is a basis proposition A* such that ω ∈ A*. It suffices to show that for each m ≥ 1 and for all agents i₁, i₂, … , i_m ∈ N,

ω ∈ Ki1Ki2 … Kim(E)

We prove the result by induction on m. The m = 1 case follows at once from (L1) and (L3). Now if we assume that for m = k, ω ∈ L*_N(E) implies ω ∈ K_i1K_i2 … K_ik(E), then L*_N(E) ⊆ K_i1K_i2 … K_ik(E) because ω is an arbitrary possible world, so K_i1(A*) ⊆ K_i1K_i2 … K_ik(E) by (L3). Since (L2) is the case and the agents of N are A*-symmetric reasoners,

Ki1(A*) ⊆ Ki1Ki2 … Kik(E)

for any i_k+1 ∈ N, so ω ∈ K_i1K_i2 … K_ik(E) by (L1), which completes the induction since i₁, i_k+1, i₂, … , i_k are k + 1 arbitrary agents of N.

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