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<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>let </b><font color="Maroon">Fy</font> be  <a href="finset_1.html#V2">finite-yielding</a> <a href="funct_1.html#NM1">Function</a>;<br/>



<b>assume </b><a NAME="E1:63"/><i><font color="Green">E47</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">Fy</font> is <a href="finset_1.html#V1">finite</a> &amp; <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 0 )
 ;<br/>

<b>then reconsider </b><font color="Maroon">X</font> =  <a href="relat_1.html#K1">dom</a> <font color="Maroon">Fy</font> as  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>  ;<br/>
<b>set </b><font color="Maroon">Ch</font> =  <a href="card_fin.html#K1" target="_self">Choose</a> <font color="Maroon">X</font>,<font color="Maroon">k</font>,0,1;<br/>
<b>consider </b><font color="Maroon">P</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="card_fin.html#K1" target="_self">Choose</a> <font color="Maroon">X</font>,<font color="Maroon">k</font>,0,1</span>)</span>, <a href="card_fin.html#K1" target="_self">Choose</a> <font color="Maroon">X</font>,<font color="Maroon">k</font>,0,1<b> such that </b><br/><a NAME="E4:63"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">P</font> is <a href="funct_1.html#V2">one-to-one</a>
 <b>by </b><i><a class="txt" href="card_fin.html#E1:63"><i><font color="Green">E47</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">XFS</font> being   <a href="afinsq_1.html#NM2">XFinSequence</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E6:63"/><i><font color="Green">E51</font></i>: 
 <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">XFS</font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">P</font>
 <b>and </b><br/><a NAME="E7:63"/><i><font color="Green">E53</font></i>: 
for <font color="Olive">z</font> being   <a href="hidden.html#M1">set</a> <br/> for <font color="Olive">f</font> being  <a href="funct_1.html#NM1">Function</a>  st <font color="Olive">z</font> <a href="hidden.html#R2">in</a>  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">XFS</font> &amp; <font color="Olive">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Olive">z</font> holds <br/><font color="Maroon">XFS</font> <a href="stirl2_1.html#K8">.</a> <font color="Olive">z</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="card_fin.html#K2" target="_self">Intersection</a> <font color="Maroon">Fy</font>,<font color="Olive">f</font>,0</span>)</span>
 <b>and </b><br/><a NAME="E8:63"/><i><font color="Green">E57</font></i>: 
 <a href="card_fin.html#K5" target="_self">Card_Intersection</a> <font color="Maroon">Fy</font>,<font color="Maroon">k</font> <a href="hidden.html#R1">=</a>  <a href="stirl2_1.html#K7">Sum</a> <font color="Maroon">XFS</font>
 <b>by </b><i><a class="txt" href="card_fin.html#E4:63"><i><font color="Green">E50</font></i></a>, <a class="ref" href="card_fin.html#T2" target="_self">Th2</a></i>;<br/>
<a NAME="E9:63"/>
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="card_fin.html#K1" target="_self">Choose</a> <font color="Maroon">X</font>,<font color="Maroon">k</font>,0,1</span>)</span> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i>, </i>;<br/>
<a NAME="E10:63"/><b>then </b><i><font color="Green">E58</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <font color="Maroon">P</font> <a href="hidden.html#R1">=</a> 1 &amp; 0 <a href="hidden.html#R2">in</a> 1 )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a>, <a class="ref" href="card_5.html#T1">CARD_5:1</a>, <a class="ref" href="card_1.html#T47">CARD_1:47</a></i>;<br/>
<a NAME="E11:63"/><b>then </b>
<font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">P</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>then consider </b><font color="Maroon">P0</font> being   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">X</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="E13:63"/><i><font color="Green">E59</font></i>: 
( <font color="Maroon">P0</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> 0 &amp;  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><font color="Maroon">P0</font> <a href="relset_1.html#K11">"</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i>, </i>;<br/>
<a NAME="E14:63"/>
<font color="Maroon">P0</font> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i>, <a class="ref" href="card_2.html#T59">CARD_2:59</a></i>;<br/>
<a NAME="E15:63"/><b>then </b><i><font color="Green">E60</font></i>: 
(  <a href="card_fin.html#K2" target="_self">Intersection</a> <font color="Maroon">Fy</font>,<font color="Maroon">P0</font>,0 <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">Fy</font></span>)</span> &amp; <font color="Maroon">XFS</font> <a href="stirl2_1.html#K8">.</a> 0 <a href="hidden.html#R1">=</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="card_fin.html#K2" target="_self">Intersection</a> <font color="Maroon">Fy</font>,<font color="Maroon">P0</font>,0</span>)</span> )
 
<b>by </b><i><a class="txt" href="card_fin.html#E6:63"><i><font color="Green">E51</font></i></a>, , , , </i>;<br/>
<a NAME="E16:63"/>
 <a href="afinsq_1.html#K1">len</a> <font color="Maroon">XFS</font> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="txt" href="card_fin.html#E6:63"><i><font color="Green">E51</font></i></a>, , <a class="ref" href="afinsq_1.html#D1">AFINSQ_1:def 1</a></i>;<br/>
<a NAME="E17:63"/><b>then </b>
( <font color="Maroon">XFS</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="afinsq_1.html#K9">&lt;%</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">XFS</font> <a href="stirl2_1.html#K8">.</a> 0</span>)</span></span><a href="afinsq_1.html#K9">%&gt;</a></span> &amp; <font color="Maroon">XFS</font> <a href="stirl2_1.html#K8">.</a> 0 <a href="hidden.html#R1">=</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="card_fin.html#K2" target="_self">Intersection</a> <font color="Maroon">Fy</font>,<font color="Maroon">P0</font>,0</span>)</span> )
 
<b>by </b><i><a class="txt" href="card_fin.html#E6:63"><i><font color="Green">E51</font></i></a>, , , , <a class="ref" href="afinsq_1.html#T38">AFINSQ_1:38</a></i>;<br/>
<a NAME="E18:63"/><b>then </b>
<a href="binop_2.html#K47">addnat</a>  <a href="stirl2_1.html#K4">"**"</a> <font color="Maroon">XFS</font> <a href="hidden.html#R1">=</a> <font color="Maroon">XFS</font> <a href="stirl2_1.html#K8">.</a> 0
 
<b>by </b><i><a class="ref" href="stirl2_1.html#T44">STIRL2_1:44</a></i>;<br/>
<b>hence </b><a NAME="E19:63"/>
 <a href="card_fin.html#K5" target="_self">Card_Intersection</a> <font color="Maroon">Fy</font>,<font color="Maroon">k</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">Fy</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="txt" href="card_fin.html#E7:63"><i><font color="Green">E53</font></i></a>, <a class="ref" href="stirl2_1.html#D4">STIRL2_1:def 4</a></i>;<br/>


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