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<b>let </b><font color="Maroon" title="c1">k</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>let </b><font color="Maroon" title="c2">Fy</font> be  <a href="finset_1.html#V2" title="FINSET_1:attr.2">finite-yielding</a> <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>;<br/>



<b>assume </b><a NAME="E1:63"/><i><font color="Green" title="E47">A1</font></i>: 
(  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c2">Fy</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> &amp; <font color="Maroon" title="c1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 )
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c3">X</font> =  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c2">Fy</font> as  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ;<br/>
<b>set </b><font color="Maroon" title="c4">Ch</font> =  <a href="card_fin.html#K1" title="CARD_FIN:func.1">Choose</a> <font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c1">k</font>,0,1;<br/>
<b>consider </b><font color="Maroon" title="c5">P</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1">(<span class="default"><a href="card_fin.html#K1" title="CARD_FIN:func.1">Choose</a> <font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c1">k</font>,0,1</span>)</span>, <a href="card_fin.html#K1" title="CARD_FIN:func.1">Choose</a> <font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c1">k</font>,0,1<b> such that </b><br/><a NAME="E4:63"/><i><font color="Green" title="E48">A2</font></i>: 
<font color="Maroon" title="c5">P</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 <b>by </b><i><a class="txt" href="card_fin.html#E54"><i><font color="Green" title="E45">Lm3</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c6">XFS</font> being   <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E6:63"/><i><font color="Green" title="E49">A3</font></i>: 
 <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c6">XFS</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c5">P</font>
 <b>and </b><br/><a NAME="E7:63"/><i><font color="Green" title="E50">A4</font></i>: 
for <font color="Olive" title="b1">z</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <br/> for <font color="Olive" title="b2">f</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>  st <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c6">XFS</font> &amp; <font color="Olive" title="b2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">z</font> holds <br/><font color="Maroon" title="c6">XFS</font> <a href="stirl2_1.html#K8" title="STIRL2_1:func.8">.</a> <font color="Olive" title="b1">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><a href="card_fin.html#K2" title="CARD_FIN:func.2">Intersection</a> <font color="Maroon" title="c2">Fy</font>,<font color="Olive" title="b2">f</font>,0</span>)</span>
 <b>and </b><br/><a NAME="E8:63"/><i><font color="Green" title="E51">A5</font></i>: 
 <a href="card_fin.html#K5" title="CARD_FIN:func.5">Card_Intersection</a> <font color="Maroon" title="c2">Fy</font>,<font color="Maroon" title="c1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="stirl2_1.html#K7" title="STIRL2_1:func.7">Sum</a> <font color="Maroon" title="c6">XFS</font>
 <b>by </b><i><a class="txt" href="card_fin.html#E4:63"><i><font color="Green" title="E48">A2</font></i></a>, <a class="ref" href="card_fin.html#D4" target="_self" title="CARD_FIN:def.4">Def4</a></i>;<br/>
<a NAME="E9:63"/>
 <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1">(<span class="default"><a href="card_fin.html#K1" title="CARD_FIN:func.1">Choose</a> <font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c1">k</font>,0,1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="txt" href="card_fin.html#E1:63"><i><font color="Green" title="E47">A1</font></i></a>, <a class="ref" href="card_fin.html#T11" target="_self" title="CARD_FIN:th.11">Th11</a></i>;<br/>
<a NAME="E10:63"/><b>then </b><i><font color="Green" title="E52">A6</font></i>: 
(  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c5">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; 0 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> 1 )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a>, <a class="ref" href="card_5.html#T1" title="CARD_5:th.1">CARD_5:1</a>, <a class="ref" href="card_1.html#T47" title="CARD_1:th.47">CARD_1:47</a></i>;<br/>
<a NAME="E11:63"/><b>then </b>
<font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 0 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c5">P</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c7">P0</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c3">X</font>,<span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default">0,1</span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span><b> such that </b><br/><a NAME="E13:63"/><i><font color="Green" title="E53">A7</font></i>: 
( <font color="Maroon" title="c7">P0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 0 &amp;  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">P0</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">0</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 )
 <b>by </b><i><a class="txt" href="card_fin.html#E1:63"><i><font color="Green" title="E47">A1</font></i></a>, <a class="ref" href="card_fin.html#D1" target="_self" title="CARD_FIN:def.1">Def1</a></i>;<br/>
<a NAME="E14:63"/>
<font color="Maroon" title="c7">P0</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">0</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="txt" href="card_fin.html#E13:63"><i><font color="Green" title="E53">A7</font></i></a>, <a class="ref" href="card_2.html#T59" title="CARD_2:th.59">CARD_2:59</a></i>;<br/>
<a NAME="E15:63"/><b>then </b><i><font color="Green" title="E54">A8</font></i>: 
(  <a href="card_fin.html#K2" title="CARD_FIN:func.2">Intersection</a> <font color="Maroon" title="c2">Fy</font>,<font color="Maroon" title="c7">P0</font>,0 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">Fy</font></span>)</span> &amp; <font color="Maroon" title="c6">XFS</font> <a href="stirl2_1.html#K8" title="STIRL2_1:func.8">.</a> 0 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><a href="card_fin.html#K2" title="CARD_FIN:func.2">Intersection</a> <font color="Maroon" title="c2">Fy</font>,<font color="Maroon" title="c7">P0</font>,0</span>)</span> )
 
<b>by </b><i><a class="txt" href="card_fin.html#E6:63"><i><font color="Green" title="E49">A3</font></i></a>, <a class="txt" href="card_fin.html#E7:63"><i><font color="Green" title="E50">A4</font></i></a>, <a class="txt" href="card_fin.html#E10:63"><i><font color="Green" title="E52">A6</font></i></a>, <a class="txt" href="card_fin.html#E13:63"><i><font color="Green" title="E53">A7</font></i></a>, <a class="ref" href="card_fin.html#T36" target="_self" title="CARD_FIN:th.36">Th36</a></i>;<br/>
<a NAME="E16:63"/>
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c6">XFS</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="txt" href="card_fin.html#E6:63"><i><font color="Green" title="E49">A3</font></i></a>, <a class="txt" href="card_fin.html#E10:63"><i><font color="Green" title="E52">A6</font></i></a>, <a class="ref" href="afinsq_1.html#D1" title="AFINSQ_1:def.1">AFINSQ_1:def 1</a></i>;<br/>
<a NAME="E17:63"/><b>then </b>
( <font color="Maroon" title="c6">XFS</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">&lt;%</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">XFS</font> <a href="stirl2_1.html#K8" title="STIRL2_1:func.8">.</a> 0</span>)</span></span><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">%&gt;</a></span> &amp; <font color="Maroon" title="c6">XFS</font> <a href="stirl2_1.html#K8" title="STIRL2_1:func.8">.</a> 0 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><a href="card_fin.html#K2" title="CARD_FIN:func.2">Intersection</a> <font color="Maroon" title="c2">Fy</font>,<font color="Maroon" title="c7">P0</font>,0</span>)</span> )
 
<b>by </b><i><a class="txt" href="card_fin.html#E6:63"><i><font color="Green" title="E49">A3</font></i></a>, <a class="txt" href="card_fin.html#E10:63"><i><font color="Green" title="E52">A6</font></i></a>, <a class="txt" href="card_fin.html#E13:63"><i><font color="Green" title="E53">A7</font></i></a>, <a class="txt" href="card_fin.html#E7:63"><i><font color="Green" title="E50">A4</font></i></a>, <a class="ref" href="afinsq_1.html#T38" title="AFINSQ_1:th.38">AFINSQ_1:38</a></i>;<br/>
<a NAME="E18:63"/><b>then </b>
<a href="binop_2.html#K47" title="BINOP_2:func.47">addnat</a>  <a href="stirl2_1.html#K4" title="STIRL2_1:func.4">"**"</a> <font color="Maroon" title="c6">XFS</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">XFS</font> <a href="stirl2_1.html#K8" title="STIRL2_1:func.8">.</a> 0
 
<b>by </b><i><a class="ref" href="stirl2_1.html#T44" title="STIRL2_1:th.44">STIRL2_1:44</a></i>;<br/>
<b>hence </b><a NAME="E19:63"/>
 <a href="card_fin.html#K5" title="CARD_FIN:func.5">Card_Intersection</a> <font color="Maroon" title="c2">Fy</font>,<font color="Maroon" title="c1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">Fy</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="card_fin.html#E15:63"><i><font color="Green" title="E54">A8</font></i></a>, <a class="txt" href="card_fin.html#E8:63"><i><font color="Green" title="E51">A5</font></i></a>, <a class="ref" href="stirl2_1.html#D4" title="STIRL2_1:def.4">STIRL2_1:def 4</a></i>;<br/>


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