让$\mathcal{<trans\; data-src="𝒜">𝒜</trans>}$成为首相$<trans\; data-src="\ast ">\ast </trans>$-代数和let$\mathrm{<trans\; data-src="\xi ">\xi </trans>}<trans\; data-src="\in ">\in </trans>\mathrm{<trans\; data-src="ℂ">ℂ</trans>}<trans\; data-src="\setminus ">\setminus </trans><trans\; data-src="\{">\{</trans>\mathrm{<trans\; data-src="0">0</trans>}<trans\; data-src=",">,</trans><trans\; data-src="\pm ">\pm </trans>\mathrm{<trans\; data-src="1">1</trans>}<trans\; data-src="\}">\}</trans>$.我们证明了映射$\mathrm{<trans\; data-src="\varphi ">\varphi </trans>}<trans\; data-src=":">:</trans>\mathcal{<trans\; data-src="𝒜">𝒜</trans>}<trans\; data-src="\to ">\to </trans>\mathcal{<trans\; data-src="𝒜">𝒜</trans>}$满足

$$\mathrm{<trans\; data-src="\varphi ">\varphi </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="A">A类</trans>}{<trans\; data-src="◇">◇</trans>}_{\mathrm{<trans\; data-src="\xi ">\xi </trans>}}\mathrm{<trans\; data-src="B">B类</trans>}{<trans\; data-src="◇">◇</trans>}_{\mathrm{<trans\; data-src="\xi ">\xi </trans>}}\mathrm{<trans\; data-src="C">C类</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="\varphi ">\varphi </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="A">A类</trans>}<trans\; data-src=")">)</trans>{<trans\; data-src="◇">◇</trans>}_{\mathrm{<trans\; data-src="\xi ">\xi </trans>}}\mathrm{<trans\; data-src="B">B类</trans>}{<trans\; data-src="◇">◇</trans>}_{\mathrm{<trans\; data-src="\xi ">\xi </trans>}}\mathrm{<trans\; data-src="C">C类</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="A">A类</trans>}{<trans\; data-src="◇">◇</trans>}_{\mathrm{<trans\; data-src="\xi ">\xi </trans>}}\mathrm{<trans\; data-src="\varphi ">\varphi </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="B">B类</trans>}<trans\; data-src=")">)</trans>{<trans\; data-src="◇">◇</trans>}_{\mathrm{<trans\; data-src="\xi ">\xi </trans>}}\mathrm{<trans\; data-src="C">C类</trans>}<trans\; data-src="+">+</trans>\mathrm{<trans\; data-src="A">A类</trans>}{<trans\; data-src="◇">◇</trans>}_{\mathrm{<trans\; data-src="\xi ">\xi </trans>}}\mathrm{<trans\; data-src="B">B类</trans>}{<trans\; data-src="◇">◇</trans>}_{\mathrm{<trans\; data-src="\xi ">\xi </trans>}}\mathrm{<trans\; data-src="\varphi ">\varphi </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="C">C类</trans>}<trans\; data-src=")">)</trans>$$

为所有人$\mathrm{<trans\; data-src="A">A类</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="B">B类</trans>}<trans\; data-src=",">,</trans>\mathrm{<trans\; data-src="C">C类</trans>}<trans\; data-src="\in ">\in </trans>\mathcal{<trans\; data-src="𝒜">𝒜</trans>}$当且仅当$\mathrm{<trans\; data-src="\varphi ">\varphi </trans>}$是添加剂$<trans\; data-src="\ast ">\ast </trans>$-推导和$\mathrm{<trans\; data-src="\varphi ">\varphi </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="\xi ">\xi </trans>}\mathrm{<trans\; data-src="A">A类</trans>}<trans\; data-src=")">)</trans><trans\; data-src="=">=</trans>\mathrm{<trans\; data-src="\xi ">\xi </trans>}\mathrm{<trans\; data-src="\varphi ">\varphi </trans>}<trans\; data-src="(">(</trans>\mathrm{<trans\; data-src="A">A类</trans>}<trans\; data-src=")">)</trans>$为所有人$\mathrm{<trans\; data-src="A">A类</trans>}<trans\; data-src="\in ">\in </trans>\mathcal{<trans\; data-src="𝒜">𝒜</trans>}$.