<?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> be  non <a href="xboole_0.html#V1">empty</a> <a href="finset_1.html#V1">finite</a> <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:97"/><i><font color="Green">E56</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> is <a href="armstrng.html#V1" target="_self">(B1)</a> &amp; <font color="Maroon">c<sub>3</sub></font> is <a href="finsub_1.html#V2">(B2)</a> )
 <b>and </b><br/><a NAME="E2:97"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="armstrng.html#R5" target="_self">is_generator-set_of</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<a NAME="E3:97"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="setfam_1.html#K8">Intersect</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> </span>}</span> 
 
<b>by </b><i><a class="txt" href="armstrng.html#E2:97"><i><font color="Green">E57</font></i></a>, <a class="ref" href="armstrng.html#D26" target="_self">Def26</a></i>;<br/>
<a NAME="E4:97"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="armstrng.html#E2:97"><i><font color="Green">E57</font></i></a>, <a class="ref" href="armstrng.html#D26" target="_self">Def26</a></i>;<br/>
<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="E5:97"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="armstrng.html#K2" target="_self">/\-IRR</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<a NAME="E6:97"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 
;<br/>
<b>then consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E8:97"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">c<sub>5</sub></font>
 <b>and </b><br/><a NAME="E9:97"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">c<sub>5</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="armstrng.html#E3:97"><i><font color="Green">E58</font></i></a></i>;<br/>
<a NAME="E10:97"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="armstrng.html#E4:97"><i><font color="Green">E59</font></i></a>, <a class="txt" href="armstrng.html#E9:97"><i><font color="Green">E62</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E11:97"/>
<font color="Maroon">c<sub>2</sub></font> is <a href="finset_1.html#V1">finite</a>
 
<b>by </b><i><a class="txt" href="armstrng.html#E4:97"><i><font color="Green">E59</font></i></a>, <a class="ref" href="finset_1.html#T13">FINSET_1:13</a></i>;<br/>
<a NAME="E12:97"/><b>then </b><i><font color="Green">E64</font></i>: 
<font color="Maroon">c<sub>5</sub></font> is <a href="finset_1.html#V1">finite</a>
 
<b>by </b><i><a class="txt" href="armstrng.html#E9:97"><i><font color="Green">E62</font></i></a>, <a class="ref" href="finset_1.html#T13">FINSET_1:13</a></i>;<br/>
<b>assume </b><a NAME="E13:97"/><i><font color="Green">E65</font></i>: 
not <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K1">}</a></span>
 ;<br/>

<a NAME="E14:97"/><b>then </b>
( not <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; not <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K1">}</a></span> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E15:97"/><b>then </b><i><font color="Green">E66</font></i>: 
( not <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E16:97"/><i><font color="Green">E67</font></i>: 
not <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="armstrng.html#E9:97"><i><font color="Green">E62</font></i></a>, <a class="txt" href="armstrng.html#E13:97"><i><font color="Green">E65</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>4</sub></font> as    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c<sub>3</sub></font> <b>by </b><i><a class="txt" href="armstrng.html#E5:97"><i><font color="Green">E60</font></i></a></i>;<br/>
<a NAME="E18:97"/>
<font color="Maroon">c<sub>6</sub></font> <a href="armstrng.html#R1" target="_self">is_/\-irreducible_in</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="armstrng.html#E5:97"><i><font color="Green">E60</font></i></a>, <a class="ref" href="armstrng.html#D3" target="_self">Def3</a></i>;<br/>
<b>hence </b><a NAME="E19:97"/>
contradiction
 <b>by </b><i><a class="txt" href="armstrng.html#E1:97"><i><font color="Green">E56</font></i></a>, <a class="txt" href="armstrng.html#E8:97"><i><font color="Green">E61</font></i></a>, <a class="txt" href="armstrng.html#E10:97"><i><font color="Green">E63</font></i></a>, <a class="txt" href="armstrng.html#E12:97"><i><font color="Green">E64</font></i></a>, <a class="txt" href="armstrng.html#E15:97"><i><font color="Green">E66</font></i></a>, <a class="txt" href="armstrng.html#E16:97"><i><font color="Green">E67</font></i></a>, <a class="ref" href="armstrng.html#T4" target="_self">Th4</a></i>;<br/>


</div>
