<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c<sub>5</sub></font> be  <a href="card_fin.html#V1" target="_self">finite-yielding</a> <a href="funct_1.html#NM1">Function</a>;<br/>









<b>assume </b><a NAME="E1:68"/><i><font color="Green">E53</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="funct_4.html#K4">--&gt;</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> )
 ;<br/>

<b>set </b><font color="Maroon">c<sub>6</sub></font> = 0,1 <a href="funct_4.html#K4">--&gt;</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>;<br/>
<a NAME="E2:68"/>
(  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default">0,1 <a href="funct_4.html#K4">--&gt;</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</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="relat_1.html#K2">rng</a> <span class="p1">(<span class="default">0,1 <a href="funct_4.html#K4">--&gt;</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="tarski.html#K2">}</a></span> &amp;  <a href="card_1.html#K4">card</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="tarski.html#K2">}</a></span> <a href="hidden.html#R1">=</a> 2 &amp; 2 <a href="hidden.html#R1">=</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> )
 
<b>by </b><i><a class="txt" href="card_fin.html#E1:68"><i><font color="Green">E53</font></i></a>, <a class="ref" href="funct_4.html#T65">FUNCT_4:65</a>, <a class="ref" href="funct_4.html#T67">FUNCT_4:67</a>, <a class="ref" href="card_5.html#T1">CARD_5:1</a>, <a class="ref" href="card_2.html#T76">CARD_2:76</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>7</sub></font> = 0,1 <a href="funct_4.html#K4">--&gt;</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> as   <a href="funct_2.html#NM1">Function</a> of  <a href="card_1.html#K4">card</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="tarski.html#K2">}</a></span>,<span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="tarski.html#K2">}</a></span> <b>by </b><i><a class="ref" href="funct_2.html#T3">FUNCT_2:3</a></i>;<br/>
<a NAME="E4:68"/>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="tarski.html#K2">}</a></span>
 
<b>by </b><i><a class="ref" href="funct_4.html#T67">FUNCT_4:67</a></i>;<br/>
<a NAME="E5:68"/><b>then </b>
( <font color="Maroon">c<sub>7</sub></font> is <a href="funct_2.html#V2">onto</a> &amp;  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="tarski.html#K2">}</a></span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="tarski.html#K2">}</a></span> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D3">FUNCT_2:def 3</a>, <a class="ref" href="card_1.html#T66">CARD_1:66</a></i>;<br/>
<a NAME="E6:68"/><b>then </b>
( <font color="Maroon">c<sub>7</sub></font> is <a href="funct_1.html#V2">one-to-one</a> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="tarski.html#K2">}</a></span> )
 
<b>by </b><i><a class="txt" href="card_fin.html#E1:68"><i><font color="Green">E53</font></i></a>, <a class="ref" href="stirl2_1.html#T70">STIRL2_1:70</a>, <a class="ref" href="funct_4.html#T65">FUNCT_4:65</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>8</sub></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="E8:68"/><i><font color="Green">E54</font></i>: 
 <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="tarski.html#K2">}</a></span>
 <b>and </b><br/><a NAME="E9:68"/><i><font color="Green">E55</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">c<sub>8</sub></font> holds <br/><font color="Maroon">c<sub>8</sub></font> <a href="stirl2_1.html#K8">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E10:68"/><i><font color="Green">E56</font></i>: 
 <a href="card_fin.html#K5" target="_self">Card_Intersection</a> <font color="Maroon">c<sub>5</sub></font>,1 <a href="hidden.html#R1">=</a>  <a href="stirl2_1.html#K7">Sum</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="ref" href="card_fin.html#T52" target="_self">Th52</a></i>;<br/>
<a NAME="E11:68"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</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>
 
<b>by </b><i><a class="ref" href="funct_4.html#T65">FUNCT_4:65</a></i>;<br/>
<a NAME="E12:68"/><b>then </b>
(  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> 2 &amp; 2 <a href="hidden.html#R1">=</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; 0 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>7</sub></font> &amp; 1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>7</sub></font> )
 
<b>by </b><i><a class="txt" href="card_fin.html#E1:68"><i><font color="Green">E53</font></i></a>, <a class="txt" href="card_fin.html#E8:68"><i><font color="Green">E54</font></i></a>, <a class="ref" href="card_2.html#T76">CARD_2:76</a>, <a class="ref" href="card_5.html#T1">CARD_5:1</a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>
<a NAME="E13:68"/><b>then </b>
(  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> 2 &amp; 0 <a href="hidden.html#R2">in</a>  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">c<sub>8</sub></font> &amp; 1 <a href="hidden.html#R2">in</a>  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> 0</span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> 1</span>)</span> )
 
<b>by </b><i><a class="txt" href="card_fin.html#E1:68"><i><font color="Green">E53</font></i></a>, <a class="ref" href="afinsq_1.html#D1">AFINSQ_1:def 1</a>, <a class="ref" href="funct_4.html#T66">FUNCT_4:66</a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>;<br/>
<a NAME="E14:68"/><b>then </b>
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="stirl2_1.html#K5">&lt;%</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="stirl2_1.html#K8">.</a> 0</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="stirl2_1.html#K8">.</a> 1</span>)</span></span><a href="stirl2_1.html#K5">%&gt;</a></span> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="stirl2_1.html#K8">.</a> 0 <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="stirl2_1.html#K8">.</a> 1 <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>4</sub></font> )
 
<b>by </b><i><a class="txt" href="card_fin.html#E9:68"><i><font color="Green">E55</font></i></a>, <a class="ref" href="afinsq_1.html#T42">AFINSQ_1:42</a></i>;<br/>
<a NAME="E15:68"/><b>then </b>
<a href="binop_2.html#K47">addnat</a>  <a href="stirl2_1.html#K4">"**"</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <a href="binop_2.html#K47">addnat</a>  <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="stirl2_1.html#T48">STIRL2_1:48</a></i>;<br/>
<a NAME="E16:68"/><b>then </b><i><font color="Green">E57</font></i>: 
<a href="binop_2.html#K47">addnat</a>  <a href="stirl2_1.html#K4">"**"</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="binop_2.html#D23">BINOP_2:def 23</a></i>;<br/>
<a NAME="E17:68"/>
0 <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E18:68"/><b>then </b>
( <span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="tarski.html#K2">}</a></span> <a href="finsop_1.html#K3">--&gt;</a> 0</span>)</span> <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> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="tarski.html#K2">}</a></span> &amp;  <a href="card_1.html#K4">card</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="tarski.html#K2">}</a></span> <a href="hidden.html#R1">=</a> 2 &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="tarski.html#K2">}</a></span> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> )
 
<b>by </b><i><a class="txt" href="card_fin.html#E1:68"><i><font color="Green">E53</font></i></a>, <a class="ref" href="card_2.html#T76">CARD_2:76</a>, <a class="ref" href="funct_4.html#T65">FUNCT_4:65</a>, <a class="ref" href="funct_4.html#T66">FUNCT_4:66</a>, <a class="ref" href="funcop_1.html#T20">FUNCOP_1:20</a></i>;<br/>
<a NAME="E19:68"/><b>then </b>
(  <a href="card_fin.html#K2" target="_self">Intersection</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="tarski.html#K2">}</a></span> <a href="finsop_1.html#K3">--&gt;</a> 0</span>)</span>,0 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">c<sub>4</sub></font> &amp;  <a href="card_fin.html#K5" target="_self">Card_Intersection</a> <font color="Maroon">c<sub>5</sub></font>,2 <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">c<sub>5</sub></font>,<span class="p2">(<span class="default"><span class="p3"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="tarski.html#K2">}</a></span> <a href="finsop_1.html#K3">--&gt;</a> 0</span>)</span>,0</span>)</span> )
 
<b>by </b><i><a class="ref" href="card_fin.html#T38" target="_self">Th38</a>, <a class="ref" href="card_fin.html#T53" target="_self">Th53</a></i>;<br/>
<b>hence </b><a NAME="E20:68"/>
(  <a href="card_fin.html#K5" target="_self">Card_Intersection</a> <font color="Maroon">c<sub>5</sub></font>,1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> &amp;  <a href="card_fin.html#K5" target="_self">Card_Intersection</a> <font color="Maroon">c<sub>5</sub></font>,2 <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="card_fin.html#E16:68"><i><font color="Green">E57</font></i></a>, <a class="txt" href="card_fin.html#E10:68"><i><font color="Green">E56</font></i></a>, <a class="ref" href="stirl2_1.html#D4">STIRL2_1:def 4</a></i>;<br/>


</div>
