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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  <a href="card_fin.html#V1" target="_self">finite-yielding</a> <a href="funct_1.html#NM1">Function</a>;<br/><b>let </b><font color="Maroon">c<sub>2</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>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>







<b>assume </b><a NAME="E1:83"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<b>assume </b><a NAME="E2:83"/>
ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b<sub>2</sub></font> &amp; ( for <font color="Olive">b<sub>3</sub></font> being  <a href="funct_1.html#NM1">Function</a> st <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="card_fin.html#K1" target="_self">Choose</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> holds <br/> <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>1</sub></font>,<font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> ) )
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:83"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>6</sub></font>
 <b>and </b><br/><a NAME="E5:83"/><i><font color="Green">E68</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="funct_1.html#NM1">Function</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="card_fin.html#K1" target="_self">Choose</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font> holds <br/> <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>1</sub></font>,<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>
<b>set </b><font color="Maroon">c<sub>7</sub></font> =  <a href="card_fin.html#K1" target="_self">Choose</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>;<br/>
<b>consider </b><font color="Maroon">c<sub>8</sub></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">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span>, <a href="card_fin.html#K1" target="_self">Choose</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font><b> such that </b><br/><a NAME="E7:83"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">c<sub>8</sub></font> is <a href="funct_1.html#V2">one-to-one</a>
 <b>by </b><i><a class="txt" href="card_fin.html#E56"><i><font color="Green">Lemma47</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>9</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="E9:83"/><i><font color="Green">E70</font></i>: 
 <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>8</sub></font>
 <b>and </b><br/><a NAME="E10:83"/><i><font color="Green">E71</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a> <br/>for <font color="Olive">b<sub>2</sub></font> being  <a href="funct_1.html#NM1">Function</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>9</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Maroon">c<sub>9</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#K1">Card</a> <span class="p1">(<span class="default"><a href="card_fin.html#K2" target="_self">Intersection</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E11:83"/><i><font color="Green">E72</font></i>: 
 <a href="card_fin.html#K5" target="_self">Card_Intersection</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="stirl2_1.html#K7">Sum</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="card_fin.html#E1:83"><i><font color="Green">E66</font></i></a>, <a class="txt" href="card_fin.html#E4:83"><i><font color="Green">E67</font></i></a>, <a class="txt" href="card_fin.html#E7:83"><i><font color="Green">E69</font></i></a>, <a class="ref" href="card_fin.html#D4" target="_self">Def4</a></i>;<br/>
<a NAME="E12:83"/>
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>9</sub></font> holds <br/><font color="Maroon">c<sub>9</sub></font> <a href="stirl2_1.html#K8">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="83_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>10</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:83_1"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">c<sub>9</sub></font>
 ;<br/>

<a NAME="E2:83_1"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="txt" href="card_fin.html#E9:83"><i><font color="Green">E70</font></i></a>, <a class="txt" href="card_fin.html#E1:83_1"><i><font color="Green">E73</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>11</sub></font> being   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c<sub>2</sub></font>,<span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="tarski.html#K2">}</a></span><b> such that </b><br/><a NAME="E4:83_1"/><i><font color="Green">E75</font></i>: 
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font> &amp;  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="relset_1.html#K11">"</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="ref" href="card_fin.html#D1" target="_self">Def1</a></i>;<br/>
<a NAME="E5:83_1"/>
( <font color="Maroon">c<sub>9</sub></font> <a href="stirl2_1.html#K8">.</a> <font color="Maroon">c<sub>10</sub></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">c<sub>1</sub></font>,<font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> &amp;  <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>1</sub></font>,<font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> )
 
<b>by </b><i><a class="txt" href="card_fin.html#E5:83"><i><font color="Green">E68</font></i></a>, <a class="txt" href="card_fin.html#E10:83"><i><font color="Green">E71</font></i></a>, <a class="txt" href="card_fin.html#E1:83_1"><i><font color="Green">E73</font></i></a>, <a class="txt" href="card_fin.html#E2:83_1"><i><font color="Green">E74</font></i></a>, <a class="txt" href="card_fin.html#E4:83_1"><i><font color="Green">E75</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:83_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="stirl2_1.html#K8">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E13:83"/><b>then </b>
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K2">dom</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="domain_1.html#K6">}</a></span> )
 
<b>by </b><i><a class="ref" href="funcop_1.html#T17">FUNCOP_1:17</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E14:83"/><b>then </b>
( <font color="Maroon">c<sub>9</sub></font> <a href="relat_1.html#K10">"</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="domain_1.html#K6">}</a></span> <a href="hidden.html#R1">=</a>  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">c<sub>9</sub></font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>9</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="domain_1.html#K6">}</a></span> &amp; <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default">0,<font color="Maroon">c<sub>3</sub></font></span><a href="domain_1.html#K7">}</a></span> &amp; (  <a href="card_fin.html#K1" target="_self">Choose</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  implies  <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">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  ) )
 
<b>by </b><i><a class="ref" href="funcop_1.html#T19">FUNCOP_1:19</a>, <a class="ref" href="funcop_1.html#T20">FUNCOP_1:20</a>, <a class="ref" href="zfmisc_1.html#T12">ZFMISC_1:12</a>, <a class="ref" href="card_1.html#T47">CARD_1:47</a></i>;<br/>
<a NAME="E15:83"/><b>then </b>
( <font color="Maroon">c<sub>9</sub></font> <a href="relat_1.html#K10">"</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="domain_1.html#K6">}</a></span> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>8</sub></font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>9</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default">0,<font color="Maroon">c<sub>3</sub></font></span><a href="domain_1.html#K7">}</a></span> &amp;  <a href="relset_1.html#K4">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">(<span class="default"><a href="card_fin.html#K1" target="_self">Choose</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> )
 
<b>by </b><i><a class="txt" href="card_fin.html#E9:83"><i><font color="Green">E70</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E16:83"/><b>then </b>
(  <a href="stirl2_1.html#K7">Sum</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <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"><a href="card_1.html#K4">card</a> <span class="p3">(<span class="default"><a href="card_fin.html#K1" target="_self">Choose</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span></span>)</span> &amp;  <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">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <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>2</sub></font></span>)</span> <a href="newton.html#K9">choose</a> <font color="Maroon">c<sub>4</sub></font> )
 
<b>by </b><i><a class="txt" href="card_fin.html#E4:83"><i><font color="Green">E67</font></i></a>, <a class="ref" href="card_fin.html#T45" target="_self">Th45</a>, <a class="ref" href="card_fin.html#T18" target="_self">Th18</a></i>;<br/>
<b>hence </b><a NAME="E17:83"/>
 <a href="card_fin.html#K5" target="_self">Card_Intersection</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="binop_2.html#K24">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="newton.html#K9">choose</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="card_fin.html#E11:83"><i><font color="Green">E72</font></i></a>, <a class="ref" href="card_1.html#T66">CARD_1:66</a></i>;<br/>


</div>
