<?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="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:85"/><i><font color="Green">E68</font></i>: 
( not <font color="Maroon">c<sub>1</sub></font> is <a href="xboole_0.html#V1">empty</a> &amp; not <font color="Maroon">c<sub>2</sub></font> is <a href="xboole_0.html#V1">empty</a> )
 ;<br/>

<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> : not <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Olive">b<sub>1</sub></font> </span>}</span> ;<br/>
<a NAME="E2:85"/><i><font color="Green">E69</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> <font color="Maroon">c<sub>2</sub></font> holds <br/><font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>) <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> <span class="p1">(<span class="default"><a href="funct_2.html#K1">Funcs</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="85_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:85_1"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<a NAME="E2:85_1"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>3</sub></font>) <a href="tarski.html#R1">c=</a>  <a href="funct_2.html#K1">Funcs</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="85_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:85_1_1"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>3</sub></font>)
 ;<br/>

<a NAME="E2:85_1_1"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> st <br/>( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; not <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Olive">b<sub>1</sub></font> )
 
<b>by </b><i><a class="txt" href="card_fin.html#E1:85_1_1"><i><font color="Green">E71</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:85_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="funct_2.html#K1">Funcs</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="card_fin.html#E1:85_1"><i><font color="Green">E70</font></i></a>, <a class="ref" href="funct_2.html#T11">FUNCT_2:11</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:85_1"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>3</sub></font>) <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> <span class="p1">(<span class="default"><a href="funct_2.html#K1">Funcs</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">c<sub>3</sub></font> being   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c<sub>2</sub></font>, <a href="zfmisc_1.html#K1">bool</a> <span class="p1">(<span class="default"><a href="funct_2.html#K1">Funcs</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span><b> such that </b><br/><a NAME="E4:85"/><i><font color="Green">E70</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> <font color="Maroon">c<sub>2</sub></font> holds <br/><font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>)
 <b>from </b><i><a class="ref" href="funct_2.html#S2">FUNCT_2:sch 2</a>(<a class="txt" href="card_fin.html#E2:85"><i><font color="Green">E69</font></i></a>);<br/></i>
<a NAME="E5:85"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  holds <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> is <a href="finset_1.html#V1">finite</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="85_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="85_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:85_2_1"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font> or  not <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="85_2_1_1"><b>suppose </b></a><a NAME="E1:85_2_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><div class="add"><a NAME="E2:85_2_1_1"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>hence </b><a NAME="E3:85_2_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i><a class="ref" href="finset_1.html#T13">FINSET_1:13</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="85_2_1_2"><b>suppose </b></a><a NAME="E1:85_2_1_2"/>
not <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:85_2_1_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i><a class="ref" href="funct_1.html#D4">FUNCT_1:def 4</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>3</sub></font> as  <a href="card_fin.html#V1" target="_self">finite-yielding</a> <a href="funct_1.html#NM1">Function</a> <b>by </b><i><a class="ref" href="card_fin.html#D3" target="_self">Def3</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] means for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> holds <br/><font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="binop_2.html#K10">-</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="binop_2.html#K10">-</a> 1</span>)</span> <a href="newton.html#K3">|^</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/>
<a NAME="E7:85"/><i><font color="Green">E71</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="card_1.html#K4">card</a> <font color="Maroon">c<sub>2</sub></font> holds <br/>ex <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a>  &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="85_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:85_3"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<a NAME="E2:85_3"/>
 <a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>2</sub></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="stirl2_1.html#T8">STIRL2_1:8</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>5</sub></font> as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="txt" href="card_fin.html#E1:85_3"><i><font color="Green">E72</font></i></a></i>;<br/>
<a NAME="E4:85_3"/>
<font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="card_fin.html#E1:85_3"><i><font color="Green">E72</font></i></a>, <a class="ref" href="euler_1.html#T1">EULER_1:1</a></i>;<br/>
<a NAME="E5:85_3"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>7</sub></font> = <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="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span> as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="ref" href="nat_1.html#T61">NAT_1:61</a></i>;<br/>
<a NAME="E7:85_3"/>
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="newton.html#K3">|^</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>] &amp; <font color="Maroon">c<sub>7</sub></font> <a href="newton.html#K3">|^</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> is   <a href="nat_1.html#NM1">Nat</a> )
 
<b>by </b><i><a class="ref" href="newton.html#T99">NEWTON:99</a></i>;<br/>
<b>hence </b><a NAME="E8:85_3"/>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a>  &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>5</sub></font>,<font color="Olive">b<sub>1</sub></font>] )
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>2</sub></font>, <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E9:85"/><i><font color="Green">E72</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="card_1.html#K4">card</a> <font color="Maroon">c<sub>2</sub></font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S1">FUNCT_2:sch 1</a>(<a class="txt" href="card_fin.html#E7:85"><i><font color="Green">E71</font></i></a>);<br/></i>
<a NAME="E10:85"/><i><font color="Green">E73</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>2</sub></font> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K5">NAT</a>  )
 
<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">c<sub>6</sub></font> = <font color="Maroon">c<sub>5</sub></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/>
<b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>6</sub></font> as   <a href="afinsq_1.html#NM2">XFinSequence</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="txt" href="card_fin.html#E10:85"><i><font color="Green">E73</font></i></a>, <a class="ref" href="ordinal1.html#D8">ORDINAL1:def 8</a></i>;<br/>
<a NAME="E13:85"/><i><font color="Green">E74</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</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>7</sub></font> holds <br/>ex <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b<sub>3</sub></font> &amp; ( for <font color="Olive">b<sub>4</sub></font> being  <a href="funct_1.html#NM1">Function</a> st <font color="Olive">b<sub>4</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>,<span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span>,<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</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>4</sub></font>,<font color="Olive">b<sub>4</sub></font>,<font color="Olive">b<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="stirl2_1.html#K8">.</a> <font color="Olive">b<sub>1</sub></font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="85_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>8</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:85_4"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/>

<b>take </b>
0
;<br/>

<b>take </b>
1
;<br/>

<b>thus </b><a NAME="E2:85_4"/>
0 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1
 ;<br/>

<b>let </b><font color="Maroon">c<sub>9</sub></font> be   <a href="funct_1.html#NM1">Function</a>;<br/>

<b>assume </b><a NAME="E3:85_4"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">c<sub>9</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>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span>,0,1
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>10</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="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="E5:85_4"/><i><font color="Green">E77</font></i>: 
( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> &amp;  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></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> <font color="Maroon">c<sub>8</sub></font> <a href="binop_2.html#K23">+</a> 1 )
 <b>by </b><i><a class="txt" href="card_fin.html#E3:85_4"><i><font color="Green">E76</font></i></a>, <a class="ref" href="card_fin.html#D1" target="_self">Def1</a></i>;<br/>
<a NAME="E6:85_4"/><i><font color="Green">E78</font></i>: 
 <a href="card_fin.html#K2" target="_self">Intersection</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>10</sub></font>,0 <a href="tarski.html#R1">c=</a>  <a href="funct_2.html#K1">Funcs</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K6">\</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="relset_1.html#K11">"</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="85_4_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>11</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:85_4_1"/><i><font color="Green">E79</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="card_fin.html#K2" target="_self">Intersection</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>10</sub></font>,0
 ;<br/>

<a NAME="E2:85_4_1"/>
0 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>10</sub></font>
 
<b>by </b><i><a class="ref" href="funct_1.html#T142">FUNCT_1:142</a>, <a class="txt" href="card_fin.html#E5:85_4"><i><font color="Green">E77</font></i></a>, <a class="ref" href="card_1.html#T47">CARD_1:47</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>12</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:85_4_1"/><i><font color="Green">E80</font></i>: 
( <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font> )
 <b>by </b><i><a class="txt" href="card_fin.html#E1:85_4_1"><i><font color="Green">E79</font></i></a>, <a class="ref" href="card_fin.html#T24" target="_self">Th24</a></i>;<br/>
<a NAME="E5:85_4_1"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>12</sub></font>)
 
<b>by </b><i><a class="txt" href="card_fin.html#E4:85"><i><font color="Green">E70</font></i></a>, <a class="txt" href="card_fin.html#E4:85_4_1"><i><font color="Green">E80</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>13</sub></font> being   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E7:85_4_1"/><i><font color="Green">E81</font></i>: 
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> &amp; not <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>13</sub></font> )
 ;<br/>
<a NAME="E8:85_4_1"/><i><font color="Green">E82</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>13</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="85_4_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>14</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:85_4_1_1"/><i><font color="Green">E83</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>13</sub></font>
 ;<br/>

<b>assume </b><a NAME="E2:85_4_1_1"/><i><font color="Green">E84</font></i>: 
not <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></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>
 ;<br/>

<a NAME="E3:85_4_1_1"/>
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></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>
 
<b>by </b><i><a class="txt" href="card_fin.html#E1:85_4_1_1"><i><font color="Green">E83</font></i></a>, <a class="txt" href="card_fin.html#E2:85_4_1_1"><i><font color="Green">E84</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E4:85_4_1_1"/><b>then </b>
( <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font> <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="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/>
<a NAME="E5:85_4_1_1"/><b>then </b>
( <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="card_fin.html#K2" target="_self">Intersection</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>10</sub></font>,0 )
 
<b>by </b><i><a class="txt" href="card_fin.html#E1:85_4_1"><i><font color="Green">E79</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E6:85_4_1_1"/><b>then </b>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font>
 
<b>by </b><i><a class="txt" href="card_fin.html#E7:85_4_1"><i><font color="Green">E81</font></i></a>, <a class="ref" href="card_fin.html#D2" target="_self">Def2</a></i>;<br/>
<a NAME="E7:85_4_1_1"/><b>then </b>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>14</sub></font>)
 
<b>by </b><i><a class="txt" href="card_fin.html#E4:85"><i><font color="Green">E70</font></i></a>, <a class="txt" href="card_fin.html#E1:85_4_1_1"><i><font color="Green">E83</font></i></a></i>;<br/>
<a NAME="E8:85_4_1_1"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> st <br/>( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; not <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Olive">b<sub>1</sub></font> )
 
;<br/>
<b>hence </b><a NAME="E9:85_4_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="card_fin.html#E1:85_4_1_1"><i><font color="Green">E83</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E9:85_4_1"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="card_fin.html#E4:85_4_1"><i><font color="Green">E80</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<b>hence </b><a NAME="E10:85_4_1"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="funct_2.html#K1">Funcs</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K6">\</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="relset_1.html#K11">"</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="card_fin.html#E7:85_4_1"><i><font color="Green">E81</font></i></a>, <a class="txt" href="card_fin.html#E8:85_4_1"><i><font color="Green">E82</font></i></a>, <a class="ref" href="funct_2.html#D2">FUNCT_2:def 2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:85_4"/><i><font color="Green">E79</font></i>: 
 <a href="funct_2.html#K1">Funcs</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K6">\</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="relset_1.html#K11">"</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="card_fin.html#K2" target="_self">Intersection</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>10</sub></font>,0
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="85_4_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>11</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:85_4_2"/><i><font color="Green">E80</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="funct_2.html#K1">Funcs</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K6">\</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="relset_1.html#K11">"</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span>
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>12</sub></font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E3:85_4_2"/><i><font color="Green">E81</font></i>: 
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>12</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></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> )
 <b>by </b><i><a class="txt" href="card_fin.html#E1:85_4_2"><i><font color="Green">E80</font></i></a>, <a class="ref" href="funct_2.html#D2">FUNCT_2:def 2</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>13</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:85_4_2"/><i><font color="Green">E82</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></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>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a>, <a class="txt" href="card_fin.html#E5:85_4"><i><font color="Green">E77</font></i></a>, <a class="ref" href="card_1.html#T47">CARD_1:47</a></i>;<br/>
<a NAME="E6:85_4_2"/>
( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>10</sub></font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> &amp;  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="txt" href="card_fin.html#E5:85_4_2"><i><font color="Green">E82</font></i></a>, <a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E7:85_4_2"/><b>then </b><i><font color="Green">E83</font></i>: 
( <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>13</sub></font>) <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>3</sub></font> )
 
<b>by </b><i><a class="txt" href="card_fin.html#E4:85"><i><font color="Green">E70</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E8:85_4_2"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>12</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="card_fin.html#E3:85_4_2"><i><font color="Green">E81</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>14</sub></font> = <font color="Maroon">c<sub>12</sub></font> as   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="txt" href="card_fin.html#E3:85_4_2"><i><font color="Green">E81</font></i></a>, <a class="ref" href="funct_2.html#T4">FUNCT_2:4</a></i>;<br/>
<a NAME="E10:85_4_2"/>
not <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>12</sub></font>
 
<b>by </b><i><a class="txt" href="card_fin.html#E3:85_4_2"><i><font color="Green">E81</font></i></a>, <a class="txt" href="card_fin.html#E5:85_4_2"><i><font color="Green">E82</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E11:85_4_2"/><b>then </b>
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>13</sub></font>)
 
;<br/>
<a NAME="E12:85_4_2"/><b>then </b><i><font color="Green">E84</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</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">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="card_fin.html#E7:85_4_2"><i><font color="Green">E83</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<a NAME="E13:85_4_2"/>
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="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0 holds <br/><font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="85_4_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>15</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:85_4_2_1"/><i><font color="Green">E85</font></i>: 
( <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> 0 )
 ;<br/>

<a NAME="E2:85_4_2_1"/>
<font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font> <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="txt" href="card_fin.html#E1:85_4_2_1"><i><font color="Green">E85</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E3:85_4_2_1"/><b>then </b>
(  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></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> )
 
<b>by </b><i><a class="txt" href="card_fin.html#E1:85_4_2_1"><i><font color="Green">E85</font></i></a>, <a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E4:85_4_2_1"/><b>then </b>
( <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>15</sub></font>) &amp; not <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>14</sub></font> )
 
<b>by </b><i><a class="txt" href="card_fin.html#E4:85"><i><font color="Green">E70</font></i></a>, <a class="txt" href="card_fin.html#E3:85_4_2"><i><font color="Green">E81</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<b>hence </b><a NAME="E5:85_4_2_1"/>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E14:85_4_2"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="card_fin.html#K2" target="_self">Intersection</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>10</sub></font>,0
 <b>by </b><i><a class="txt" href="card_fin.html#E3:85_4_2"><i><font color="Green">E81</font></i></a>, <a class="txt" href="card_fin.html#E12:85_4_2"><i><font color="Green">E84</font></i></a>, <a class="ref" href="card_fin.html#D2" target="_self">Def2</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>11</sub></font> =  <a href="card_fin.html#K2" target="_self">Intersection</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>10</sub></font>,0 as  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i><a class="txt" href="card_fin.html#E6:85_4"><i><font color="Green">E78</font></i></a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>
<a NAME="E9:85_4"/><i><font color="Green">E80</font></i>: 
 <a href="funct_2.html#K1">Funcs</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K6">\</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="relset_1.html#K11">"</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_fin.html#K2" target="_self">Intersection</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>10</sub></font>,0
 
<b>by </b><i><a class="txt" href="card_fin.html#E6:85_4"><i><font color="Green">E78</font></i></a>, <a class="txt" href="card_fin.html#E7:85_4"><i><font color="Green">E79</font></i></a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>
<a NAME="E10:85_4"/>
 <a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i><a class="txt" href="card_fin.html#E1:85"><i><font color="Green">E68</font></i></a>, <a class="ref" href="card_2.html#T59">CARD_2:59</a></i>;<br/>
<a NAME="E11:85_4"/><b>then </b><i><font color="Green">E81</font></i>: 
(  <a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp;  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K6">\</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="relset_1.html#K11">"</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span>)</span></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="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span> )
 
<b>by </b><i><a class="txt" href="card_fin.html#E5:85_4"><i><font color="Green">E77</font></i></a>, <a class="ref" href="card_2.html#T63">CARD_2:63</a>, <a class="ref" href="nat_1.html#T39">NAT_1:39</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="85_4_3"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="85_4_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:85_4_3_1"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></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>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></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#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="85_4_3_1_1"><b>suppose </b></a><a NAME="E1:85_4_3_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></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>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:85_4_3_1_1"/><b>then </b>
(  <a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> 0 &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="binop_2.html#K10">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="binop_2.html#K10">-</a> 1</span>)</span> <a href="newton.html#K3">|^</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i><a class="txt" href="card_fin.html#E9:85_4"><i><font color="Green">E80</font></i></a>, <a class="ref" href="card_1.html#T47">CARD_1:47</a>, <a class="txt" href="card_fin.html#E11:85_4"><i><font color="Green">E81</font></i></a>, <a class="ref" href="funct_2.html#T14">FUNCT_2:14</a>, <a class="ref" href="newton.html#T16">NEWTON:16</a></i>;<br/><b>hence </b><a NAME="E3:85_4_3_1_1"/>
 <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>4</sub></font>,<font color="Maroon">c<sub>9</sub></font>,0</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="stirl2_1.html#K8">.</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="card_fin.html#E5:85_4"><i><font color="Green">E77</font></i></a>, <a class="txt" href="card_fin.html#E9:85"><i><font color="Green">E72</font></i></a>, <a class="txt" href="card_fin.html#E10:85"><i><font color="Green">E73</font></i></a>, <a class="txt" href="card_fin.html#E1:85_4"><i><font color="Green">E75</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="85_4_3_1_2"><b>suppose </b></a><a NAME="E1:85_4_3_1_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></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#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:85_4_3_1_2"/><b>then </b>
( <font color="Maroon">c<sub>7</sub></font> <a href="stirl2_1.html#K8">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="binop_2.html#K10">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="binop_2.html#K10">-</a> 1</span>)</span> <a href="newton.html#K3">|^</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K6">\</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="relset_1.html#K11">"</a> <span class="p4"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span></span>)</span> <a href="newton.html#K3">|^</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>11</sub></font> )
 <b>by </b><i><a class="txt" href="card_fin.html#E9:85"><i><font color="Green">E72</font></i></a>, <a class="txt" href="card_fin.html#E10:85"><i><font color="Green">E73</font></i></a>, <a class="txt" href="card_fin.html#E1:85_4"><i><font color="Green">E75</font></i></a>, <a class="txt" href="card_fin.html#E9:85_4"><i><font color="Green">E80</font></i></a>, <a class="ref" href="card_fin.html#T4" target="_self">Th4</a></i>;<br/><b>hence </b><a NAME="E3:85_4_3_1_2"/>
 <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>4</sub></font>,<font color="Maroon">c<sub>9</sub></font>,0</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="stirl2_1.html#K8">.</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="card_fin.html#E5:85_4"><i><font color="Green">E77</font></i></a>, <a class="txt" href="card_fin.html#E11:85_4"><i><font color="Green">E81</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E13:85_4"/>
 <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>4</sub></font>,<font color="Maroon">c<sub>9</sub></font>,0</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="stirl2_1.html#K8">.</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E14:85"/>
(  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>2</sub></font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</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#K4">INT</a> <b> such that </b><br/><a NAME="E16:85"/><i><font color="Green">E75</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> <font color="Maroon">c<sub>2</sub></font>
 <b>and </b><br/><a NAME="E17:85"/><i><font color="Green">E76</font></i>: 
 <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">c<sub>4</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_fin.html#K6" target="_self">Sum</a> <font color="Maroon">c<sub>8</sub></font>
 <b>and </b><br/><a NAME="E18:85"/><i><font color="Green">E77</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</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="card_fin.html#K7" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="binop_2.html#K7">-</a> 1</span>)</span> <a href="newton.html#K3">|^</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="stirl2_1.html#K8">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span>)</span> <a href="binop_2.html#K11">*</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> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="card_fin.html#E13:85"><i><font color="Green">E74</font></i></a>, <a class="ref" href="card_fin.html#T69" target="_self">Th69</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>9</sub></font> = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> : <font color="Olive">b<sub>1</sub></font> is <a href="funct_2.html#V2">onto</a> </span>}</span> ;<br/>
<a NAME="E19:85"/><i><font color="Green">E78</font></i>: 
<span class="p1">(<span class="default"><a href="funct_2.html#K1">Funcs</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K6">\</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">c<sub>4</sub></font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> : <font color="Olive">b<sub>1</sub></font> is <a href="funct_2.html#V2">onto</a> </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="85_5"><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> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:85_5"/><i><font color="Green">E79</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="funct_2.html#K1">Funcs</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K6">\</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">c<sub>4</sub></font></span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:85_5"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="funct_2.html#K1">Funcs</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="card_fin.html#E1:85_5"><i><font color="Green">E79</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>11</sub></font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E4:85_5"/><i><font color="Green">E80</font></i>: 
( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>11</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="ref" href="funct_2.html#D2">FUNCT_2:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>11</sub></font> as   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="txt" href="card_fin.html#E4:85_5"><i><font color="Green">E80</font></i></a>, <a class="ref" href="funct_2.html#T4">FUNCT_2:4</a></i>;<br/>
<b>assume </b><a NAME="E6:85_5"/>
not <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> : <font color="Olive">b<sub>1</sub></font> is <a href="funct_2.html#V2">onto</a> </span>}</span> 
 ;<br/>

<a NAME="E7:85_5"/><b>then </b>
not <font color="Maroon">c<sub>12</sub></font> is <a href="funct_2.html#V2">onto</a>
 
<b>by </b><i><a class="txt" href="card_fin.html#E4:85_5"><i><font color="Green">E80</font></i></a></i>;<br/>
<a NAME="E8:85_5"/><b>then </b>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D3">FUNCT_2:def 3</a></i>;<br/>
<a NAME="E9:85_5"/><b>then </b>
not <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>12</sub></font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>13</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E11:85_5"/><i><font color="Green">E81</font></i>: 
( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; not <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>12</sub></font> )
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<a NAME="E12:85_5"/>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="card_fin.html#E11:85_5"><i><font color="Green">E81</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E13:85_5"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E14:85_5"/><b>then </b>
( <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>13</sub></font>) <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>13</sub></font>) )
 
<b>by </b><i><a class="txt" href="card_fin.html#E4:85"><i><font color="Green">E70</font></i></a>, <a class="txt" href="card_fin.html#E11:85_5"><i><font color="Green">E81</font></i></a></i>;<br/>
<a NAME="E15:85_5"/><b>then </b>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</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">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>hence </b><a NAME="E16:85_5"/>
contradiction
 <b>by </b><i><a class="txt" href="card_fin.html#E1:85_5"><i><font color="Green">E79</font></i></a>, <a class="txt" href="card_fin.html#E4:85_5"><i><font color="Green">E80</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E20:85"/><i><font color="Green">E79</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> : <font color="Olive">b<sub>1</sub></font> is <a href="funct_2.html#V2">onto</a> </span>}</span>  <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="funct_2.html#K1">Funcs</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K6">\</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">c<sub>4</sub></font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="85_6"><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> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:85_6"/><i><font color="Green">E80</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> : <font color="Olive">b<sub>1</sub></font> is <a href="funct_2.html#V2">onto</a> </span>}</span> 
 ;<br/>

<b>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>1</sub></font>,<font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E3:85_6"/><i><font color="Green">E81</font></i>: 
( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> is <a href="funct_2.html#V2">onto</a> )
 <b>by </b><i><a class="txt" href="card_fin.html#E1:85_6"><i><font color="Green">E80</font></i></a></i>;<br/>
<b>assume </b><a NAME="E4:85_6"/><i><font color="Green">E82</font></i>: 
not <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="funct_2.html#K1">Funcs</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K6">\</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">c<sub>4</sub></font></span>)</span></span>)</span>
 ;<br/>

<a NAME="E5:85_6"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="funct_2.html#K1">Funcs</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="card_fin.html#E1:85"><i><font color="Green">E68</font></i></a>, <a class="ref" href="funct_2.html#T11">FUNCT_2:11</a></i>;<br/>
<a NAME="E6:85_6"/><b>then </b>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</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">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="card_fin.html#E3:85_6"><i><font color="Green">E81</font></i></a>, <a class="txt" href="card_fin.html#E4:85_6"><i><font color="Green">E82</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>12</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E8:85_6"/><i><font color="Green">E83</font></i>: 
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>13</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E10:85_6"/><i><font color="Green">E84</font></i>: 
( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> )
 <b>by </b><i><a class="txt" href="card_fin.html#E8:85_6"><i><font color="Green">E83</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E11:85_6"/>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="card_fin.html#E10:85_6"><i><font color="Green">E84</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E12:85_6"/><b>then </b>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>13</sub></font>)
 
<b>by </b><i><a class="txt" href="card_fin.html#E4:85"><i><font color="Green">E70</font></i></a>, <a class="txt" href="card_fin.html#E8:85_6"><i><font color="Green">E83</font></i></a>, <a class="txt" href="card_fin.html#E10:85_6"><i><font color="Green">E84</font></i></a></i>;<br/>
<a NAME="E13:85_6"/><b>then </b>
( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; ex <font color="Olive">b<sub>1</sub></font> being  <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> st <br/>( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; not <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Olive">b<sub>1</sub></font> ) )
 
<b>by </b><i><a class="txt" href="card_fin.html#E10:85_6"><i><font color="Green">E84</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<b>hence </b><a NAME="E14:85_6"/>
contradiction
 <b>by </b><i><a class="txt" href="card_fin.html#E3:85_6"><i><font color="Green">E81</font></i></a>, <a class="ref" href="funct_2.html#D3">FUNCT_2:def 3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E21:85"/>
 <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> <span class="p2">(<span class="default"><a href="funct_2.html#K1">Funcs</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T95">ZFMISC_1:95</a></i>;<br/>
<a NAME="E22:85"/><b>then </b><i><font color="Green">E80</font></i>: 
 <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="funct_2.html#K1">Funcs</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T99">ZFMISC_1:99</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>10</sub></font> =  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> as  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i><a class="ref" href="finset_1.html#T13">FINSET_1:13</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>8</sub></font>
;<br/>

<b>thus </b><a NAME="E24:85"/>
 <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> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="card_fin.html#E16:85"><i><font color="Green">E75</font></i></a></i>;<br/>

<a NAME="E25:85"/>
(  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_2.html#K1">Funcs</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><a href="funct_2.html#K1">Funcs</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> &amp;  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="funct_2.html#K1">Funcs</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">(<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#K3">|^</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="funct_2.html#K1">Funcs</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> : <font color="Olive">b<sub>1</sub></font> is <a href="funct_2.html#V2">onto</a> </span>}</span>  )
 
<b>by </b><i><a class="txt" href="card_fin.html#E1:85"><i><font color="Green">E68</font></i></a>, <a class="txt" href="card_fin.html#E19:85"><i><font color="Green">E78</font></i></a>, <a class="txt" href="card_fin.html#E20:85"><i><font color="Green">E79</font></i></a>, <a class="txt" href="card_fin.html#E22:85"><i><font color="Green">E80</font></i></a>, <a class="ref" href="card_2.html#T63">CARD_2:63</a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a>, <a class="ref" href="card_fin.html#T4" target="_self">Th4</a></i>;<br/>
<b>hence </b><a NAME="E26:85"/>
 <a href="card_1.html#K1">Card</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> : <font color="Olive">b<sub>1</sub></font> is <a href="funct_2.html#V2">onto</a> </span>}</span>  <a href="hidden.html#R1">=</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#K3">|^</a> <span class="p2">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="card_fin.html#K6" target="_self">Sum</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="card_fin.html#E17:85"><i><font color="Green">E76</font></i></a></i>;<br/>

<b>let </b><font color="Maroon">c<sub>11</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E27:85"/><i><font color="Green">E81</font></i>: 
<font color="Maroon">c<sub>11</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>
 ;<br/>

<a NAME="E28:85"/>
( <font color="Maroon">c<sub>8</sub></font> <a href="card_fin.html#K7" target="_self">.</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="binop_2.html#K7">-</a> 1</span>)</span> <a href="newton.html#K3">|^</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="stirl2_1.html#K8">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span>)</span> <a href="binop_2.html#K11">*</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> <span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="stirl2_1.html#K8">.</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="binop_2.html#K10">-</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="binop_2.html#K10">-</a> 1</span>)</span> <a href="newton.html#K3">|^</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> )
 
<b>by </b><i><a class="txt" href="card_fin.html#E9:85"><i><font color="Green">E72</font></i></a>, <a class="txt" href="card_fin.html#E16:85"><i><font color="Green">E75</font></i></a>, <a class="txt" href="card_fin.html#E18:85"><i><font color="Green">E77</font></i></a>, <a class="txt" href="card_fin.html#E27:85"><i><font color="Green">E81</font></i></a></i>;<br/>
<b>hence </b><a NAME="E29:85"/>
<font color="Maroon">c<sub>8</sub></font> <a href="card_fin.html#K7" target="_self">.</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="binop_2.html#K7">-</a> 1</span>)</span> <a href="newton.html#K3">|^</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><span class="p3">(<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> <span class="p3">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="binop_2.html#K10">-</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="binop_2.html#K10">-</a> 1</span>)</span> <a href="newton.html#K3">|^</a> <span class="p2">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
 ;<br/>


</div>
