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<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="141_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:141_1_1_1"/>
( <font color="Maroon" title="c1">X</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> or  not <font color="Maroon" title="c1">X</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="141_1_1_1_1"><b>suppose </b></a><a NAME="E1:141_1_1_1_1"/>
<font color="Maroon" title="c1">X</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ;<br/><div class="add"><a NAME="E2:141_1_1_1_1"/><b>then </b>
 <a href="polynom1.html#K18" title="POLYNOM1:func.18">NatMinor</a> <font color="Maroon" title="c2">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,<a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 ;<br/><a NAME="E3:141_1_1_1_1"/><b>then </b>
 <a href="polynom1.html#K18" title="POLYNOM1:func.18">NatMinor</a> <font color="Maroon" title="c2">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="funct_5.html#T64" title="FUNCT_5:th.64">FUNCT_5:64</a></i>;<br/><b>hence </b><a NAME="E4:141_1_1_1_1"/>
 <a href="polynom1.html#K18" title="POLYNOM1:func.18">NatMinor</a> <font color="Maroon" title="c2">f</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 <b>by </b><i><a class="ref" href="finset_1.html#T13" title="FINSET_1:th.13">FINSET_1:13</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="141_1_1_1_2"><b>suppose </b></a><a NAME="E1:141_1_1_1_2"/>
not <font color="Maroon" title="c1">X</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c3">X</font> = <font color="Maroon" title="c1">X</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ;<br/><b>reconsider </b><font color="Maroon" title="c4">f</font> = <font color="Maroon" title="c2">f</font> as  <a href="polynom1.html#V1" title="POLYNOM1:attr.1">finite-support</a> <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c3">X</font>, <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/><a NAME="E4:141_1_1_1_2"/>
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><a href="polynom1.html#K18" title="POLYNOM1:func.18">NatMinor</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="binop_2.html#K48" title="BINOP_2:func.48">multnat</a>  <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">support</a> <font color="Maroon" title="c4">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="binop_2.html#K47" title="BINOP_2:func.47">addnat</a>  <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">[:]</a> <font color="Maroon" title="c4">f</font>,1</span>)</span>
 <b>by </b><i><a class="ref" href="polynom1.html#T67" target="_self" title="POLYNOM1:th.67">Th67</a></i>;<br/><b>hence </b><a NAME="E5:141_1_1_1_2"/>
 <a href="polynom1.html#K18" title="POLYNOM1:func.18">NatMinor</a> <font color="Maroon" title="c2">f</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 <b>by </b><i><a class="ref" href="card_4.html#T1" title="CARD_4:th.1">CARD_4:1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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