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<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>





<b>thus </b><a NAME="E1:57"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="card_fin.html#K1" target="_self">Choose</a> <font color="Maroon">n</font>,<font color="Maroon">k</font>,1,0 implies  ex <font color="Olive">F</font> being  <a href="afinsq_1.html#NM2">XFinSequence</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">F</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp;  <a href="afinsq_1.html#K2">dom</a> <font color="Olive">F</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Olive">F</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default">0,1</span><a href="domain_1.html#K7">}</a></span> &amp;  <a href="stirl2_1.html#K7">Sum</a> <font color="Olive">F</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:57_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="card_fin.html#K1" target="_self">Choose</a> <font color="Maroon">n</font>,<font color="Maroon">k</font>,1,0
 ;<br/>

<b>then consider </b><font color="Maroon">F</font> being   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">n</font>,<span class="p1"><a href="domain_1.html#K7">{</a><span class="default">0,1</span><a href="domain_1.html#K7">}</a></span><b> such that </b><br/><a NAME="E3:57_1"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> &amp;  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="stirl2_1.html#K1">"</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> )
 <b>by </b><i/>;<br/>
<a NAME="E4:57_1"/><i><font color="Green">E50</font></i>: 
(  <a href="relset_1.html#K5">rng</a> <font color="Maroon">F</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default">0,1</span><a href="domain_1.html#K7">}</a></span> &amp;  <a href="relset_1.html#K4">dom</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">F</font> = <font color="Maroon">F</font> as   <a href="afinsq_1.html#NM1">XFinSequence</a> <b>by </b><i><a class="ref" href="afinsq_1.html#T7">AFINSQ_1:7</a></i>;<br/>
<a NAME="E6:57_1"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font> is   <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="ref" href="card_fin.html#T1" target="_self">Th1</a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">F</font> = <font color="Maroon">F</font> as   <a href="afinsq_1.html#NM2">XFinSequence</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D8">ORDINAL1:def 8</a></i>;<br/>
<b>take </b>
<font color="Maroon">F</font>
;<br/>

<a NAME="E8:57_1"/>
 <a href="stirl2_1.html#K7">Sum</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> 1 <a href="binop_2.html#K24">*</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="relat_1.html#K10">"</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="card_fin.html#T1" target="_self">Th1</a>, <a class="txt" href="card_fin.html#E1"><i><font color="Green">Lemma44</font></i></a></i>;<br/>
<b>hence </b><a NAME="E9:57_1"/>
( <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp;  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default">0,1</span><a href="domain_1.html#K7">}</a></span> &amp;  <a href="stirl2_1.html#K7">Sum</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> )
 <b>by </b><i><a class="txt" href="card_fin.html#E2"><i><font color="Green">Lemma45</font></i></a>, <a class="ref" href="card_fin.html#T1" target="_self">Th1</a></i>;<br/>


</div><b>end;</b></div>

<b>given </b><font color="Maroon">F</font> being   <a href="afinsq_1.html#NM2">XFinSequence</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><a NAME="E2:57"/><i><font color="Green">E51</font></i>: 
( <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp;  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default">0,1</span><a href="domain_1.html#K7">}</a></span> &amp;  <a href="stirl2_1.html#K7">Sum</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> )
 ;<br/>

<a NAME="E3:57"/>
1 <a href="binop_2.html#K24">*</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="relat_1.html#K10">"</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font>
 
<b>by </b><i><a class="txt" href="card_fin.html#E2"><i><font color="Green">Lemma45</font></i></a>, <a class="txt" href="card_fin.html#E1"><i><font color="Green">Lemma44</font></i></a></i>;<br/>
<a NAME="E4:57"/><b>then </b>
(  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="relat_1.html#K10">"</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">F</font> is   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">n</font>,<span class="p1"><a href="domain_1.html#K7">{</a><span class="default">0,1</span><a href="domain_1.html#K7">}</a></span> )
 
<b>by </b><i><a class="txt" href="card_fin.html#E2"><i><font color="Green">Lemma45</font></i></a>, <a class="ref" href="funct_2.html#T4">FUNCT_2:4</a></i>;<br/>
<b>hence </b><a NAME="E5:57"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="card_fin.html#K1" target="_self">Choose</a> <font color="Maroon">n</font>,<font color="Maroon">k</font>,1,0
 <b>by </b><i><a class="txt" href="card_fin.html#E2"><i><font color="Green">Lemma45</font></i></a>, </i>;<br/>


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