lemmaLemma 3.2State equality preserves every futureLemma 3.2 (State equality preserves every future). If δ∗(s0,u)=δ∗(s0,u′)\delta^*(s_0,u)=\delta^*(s_0,u')δ∗(s0,u)=δ∗(s0,u′), then for all v∈Σ∗v\in\Sigma^*v∈Σ∗, uv∈L(A)⟺u′v∈L(A).uv\in L(\mathcal A)\quad\Longleftrightarrow\quad u'v\in L(\mathcal A).uv∈L(A)⟺u′v∈L(A).
Lemma 3.2 (State equality preserves every future). If δ∗(s0,u)=δ∗(s0,u′)\delta^*(s_0,u)=\delta^*(s_0,u')δ∗(s0,u)=δ∗(s0,u′), then for all v∈Σ∗v\in\Sigma^*v∈Σ∗, uv∈L(A)⟺u′v∈L(A).uv\in L(\mathcal A)\quad\Longleftrightarrow\quad u'v\in L(\mathcal A).uv∈L(A)⟺u′v∈L(A).