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<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="142_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:142_1_1_1"/>
( <font color="Maroon">X</font> is <a href="xboole_0.html#V1">empty</a> or  not <font color="Maroon">X</font> is <a href="xboole_0.html#V1">empty</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="142_1_1_1_1"><b>suppose </b></a><a NAME="E1:142_1_1_1_1"/>
<font color="Maroon">X</font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/><div class="add"><a NAME="E2:142_1_1_1_1"/><b>then </b>
 <a href="polynom1.html#K18" target="_self">NatMinor</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a>  <a href="fraenkel.html#K1">Funcs</a> <a href="xboole_0.html#K1">{}</a> ,<a href="numbers.html#K5">NAT</a> 
 ;<br/><a NAME="E3:142_1_1_1_1"/><b>then </b>
 <a href="polynom1.html#K18" target="_self">NatMinor</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="funct_5.html#T64">FUNCT_5:64</a></i>;<br/><b>hence </b><a NAME="E4:142_1_1_1_1"/>
 <a href="polynom1.html#K18" target="_self">NatMinor</a> <font color="Maroon">f</font> is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i><a class="ref" href="finset_1.html#T13">FINSET_1:13</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="142_1_1_1_2"><b>suppose </b></a><a NAME="E1:142_1_1_1_2"/>
not <font color="Maroon">X</font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">X</font> = <font color="Maroon">X</font> as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  ;<br/><b>reconsider </b><font color="Maroon">f</font> = <font color="Maroon">f</font> as  <a href="polynom1.html#V1" target="_self">finite-support</a> <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">X</font>, <a href="numbers.html#K5">NAT</a>  ;<br/><a NAME="E4:142_1_1_1_2"/>
 <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="polynom1.html#K18" target="_self">NatMinor</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a> <a href="binop_2.html#K48">multnat</a>  <a href="setwiseo.html#K7">$$</a> <span class="p1">(<span class="default"><a href="polynom1.html#K19" target="_self">support</a> <font color="Maroon">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="binop_2.html#K47">addnat</a>  <a href="funcop_1.html#K7">[:]</a> <font color="Maroon">f</font>,1</span>)</span>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E5:142_1_1_1_2"/>
 <a href="polynom1.html#K18" target="_self">NatMinor</a> <font color="Maroon">f</font> is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i><a class="ref" href="card_4.html#T1">CARD_4:1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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