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If <span id="MathJax-Element-32-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="A∈Δ" role="presentation" style="display: inline-block; line-height: 0; text-indent: 0px; font-size: 16.16px; overflow-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding-top: 1px; padding-bottom: 1px; position: relative;">A∈ΔA∈Δ, then clearly <span id="MathJax-Element-33-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="Δ⊢A" role="presentation" style="display: inline-block; line-height: 0; text-indent: 0px; font-size: 16.16px; overflow-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding-top: 1px; padding-bottom: 1px; position: relative;">Δ⊢AΔ⊢A. Conversely, suppose <span id="MathJax-Element-34-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="Δ⊢A" role="presentation" style="display: inline-block; line-height: 0; text-indent: 0px; font-size: 16.16px; overflow-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding-top: 1px; padding-bottom: 1px; position: relative;">Δ⊢AΔ⊢A. Let <span id="MathJax-Element-35-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="ℰ" role="presentation" style="display: inline-block; line-height: 0; text-indent: 0px; font-size: 16.16px; overflow-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding-top: 1px; padding-bottom: 1px; position: relative;">Eℰ be a [url=]deduction[/url] of <span id="MathJax-Element-36-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="A" role="presentation" style="display: inline-block; line-height: 0; text-indent: 0px; font-size: 16.16px; overflow-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding-top: 1px; padding-bottom: 1px; position: relative;">AA from <span id="MathJax-Element-37-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="Δ" role="presentation" style="display: inline-block; line-height: 0; text-indent: 0px; font-size: 16.16px; overflow-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding-top: 1px; padding-bottom: 1px; position: relative;">ΔΔ, and <span id="MathJax-Element-38-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="Γ:=Δ∪{A}" role="presentation" style="display: inline-block; line-height: 0; text-indent: 0px; font-size: 16.16px; overflow-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding-top: 1px; padding-bottom: 1px; position: relative;">Γ:=Δ∪{A}Γ:=Δ∪{A}. Suppose <span id="MathJax-Element-39-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="Γ⊢B" role="presentation" style="display: inline-block; line-height: 0; text-indent: 0px; font-size: 16.16px; overflow-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding-top: 1px; padding-bottom: 1px; position: relative;">Γ⊢BΓ⊢B. Let <span id="MathJax-Element-40-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="ℰ1" role="presentation" style="display: inline-block; line-height: 0; text-indent: 0px; font-size: 16.16px; overflow-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding-top: 1px; padding-bottom: 1px; position: relative;">E1ℰ1 be a deduction of <span id="MathJax-Element-41-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="B" role="presentation" style="display: inline-block; line-height: 0; text-indent: 0px; font-size: 16.16px; overflow-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding-top: 1px; padding-bottom: 1px; position: relative;">BB from <span id="MathJax-Element-42-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="Γ" role="presentation" style="display: inline-block; line-height: 0; text-indent: 0px; font-size: 16.16px; overflow-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding-top: 1px; padding-bottom: 1px; position: relative;">ΓΓ, then <span id="MathJax-Element-43-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="ℰ,ℰ1" role="presentation" style="display: inline-block; line-height: 0; text-indent: 0px; font-size: 16.16px; overflow-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding-top: 1px; padding-bottom: 1px; position: relative;">E,E1ℰ,ℰ1 is a deduction of <span id="MathJax-Element-44-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="B" role="presentation" style="display: inline-block; line-height: 0; text-indent: 0px; font-size: 16.16px; overflow-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding-top: 1px; padding-bottom: 1px; position: relative;">BB from <span id="MathJax-Element-45-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="Δ" role="presentation" style="display: inline-block; line-height: 0; text-indent: 0px; font-size: 16.16px; overflow-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding-top: 1px; padding-bottom: 1px; position: relative;">ΔΔ, so <span id="MathJax-Element-46-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="Δ⊢B" role="presentation" style="display: inline-block; line-height: 0; text-indent: 0px; font-size: 16.16px; overflow-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding-top: 1px; padding-bottom: 1px; position: relative;">Δ⊢BΔ⊢B. Since <span id="MathJax-Element-47-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="Δ⊬⟂" role="presentation" style="display: inline-block; line-height: 0; text-indent: 0px; font-size: 16.16px; overflow-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding-top: 1px; padding-bottom: 1px; position: relative;">Δ⊬⟂Δ⊬⟂, <span id="MathJax-Element-48-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="Γ⊬⟂" role="presentation" style="display: inline-block; line-height: 0; text-indent: 0px; font-size: 16.16px; overflow-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding-top: 1px; padding-bottom: 1px; position: relative;">Γ⊬⟂Γ⊬⟂, so <span id="MathJax-Element-49-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="Γ" role="presentation" style="display: inline-block; line-height: 0; text-indent: 0px; font-size: 16.16px; overflow-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding-top: 1px; padding-bottom: 1px; position: relative;">ΓΓ is consistent. Since <span id="MathJax-Element-50-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="Δ" role="presentation" style="display: inline-block; line-height: 0; text-indent: 0px; font-size: 16.16px; overflow-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding-top: 1px; padding-bottom: 1px; position: relative;">ΔΔ is maximal, <span id="MathJax-Element-51-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="Γ=Δ" role="presentation" style="display: inline-block; line-height: 0; text-indent: 0px; font-size: 16.16px; overflow-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding-top: 1px; padding-bottom: 1px; position: relative;">Γ=ΔΓ=Δ, or <span id="MathJax-Element-52-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="A∈Δ" role="presentation" style="display: inline-block; line-height: 0; text-indent: 0px; font-size: 16.16px; overflow-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding-top: 1px; padding-bottom: 1px; position: relative;">A∈ΔA∈Δ.
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