these cases, u = true and v = false, and in all the other cases, p is true. Thus we have: lemma not_uv_or_p : u ̸= v ∨ p := or.elim u_def (assume hut : u = true, or.elim v_def (assume hvf : v = false are equal. By the definition of u and v, this implies that they are equal as well. lemma p_implies_uv : p → u = v := assume hp : p, have hpred : U = V, from funext (assume x : Prop, (continues on next the desired conclusion: theorem em : p ∨ ¬p := have h : ¬(u = v) → ¬p, from mt p_implies_uv, or.elim not_uv_or_p (assume hne : ¬(u = v), or.inr (h hne)) (assume hp : p, or.inl hp) Consequences of excluded