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

<b>let </b><font color="Maroon">X</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">B</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">X</font>;<br/>



<b>assume </b><a NAME="E1:15"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">B</font> is <a href="finsub_1.html#V2">(B2)</a>
 ;<br/>

<b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">B</font>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E2:15"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">x</font> <a href="armstrng.html#R1" target="_self">is_/\-irreducible_in</a> <font color="Maroon">B</font>
 <b>and </b><br/><a NAME="E3:15"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">X</font>
 ;<br/>

<b>let </b><font color="Maroon">S</font> be  <a href="finset_1.html#V1">finite</a> <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">X</font>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E4:15"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">S</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font>
 <b>and </b><br/><a NAME="E5:15"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">S</font>
 <b>and </b><br/><a NAME="E6:15"/><i><font color="Green">E40</font></i>: 
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ] <b>means </b>(  ex <font color="Olive">a</font>, <font color="Olive">b</font> being   <a href="subset_1.html#M2">Element</a> of <font color="Maroon">B</font> st <br/>( <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">a</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">a</font> <a href="subset_1.html#K9">/\</a> <font color="Olive">b</font> ) or  ex <font color="Olive">f</font> being  <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">X</font> st <br/>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or ( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">f</font> &amp;  <a href="setfam_1.html#K8">Intersect</a> <font color="Olive">f</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font> &amp;  <a href="setfam_1.html#K8">Intersect</a> <font color="Olive">f</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> ) ) );<br/>
<a NAME="E7:15"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">S</font> is <a href="finset_1.html#V1">finite</a>
 
;<br/>
<a NAME="E8:15"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="xboole_0.html#K1">{}</a> ]
 
;<br/>
<div><i><font color="Green">E45</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">s</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">A</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><b>that </b><br/><a NAME="E1:15_1"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 <b>and </b><br/><a NAME="E2:15_1"/>
<font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">S</font>
 <b>and </b><br/><a NAME="E3:15_1"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">A</font>]
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:15_1_1"/>
(  ex <font color="Olive">a</font>, <font color="Olive">b</font> being   <a href="subset_1.html#M2">Element</a> of <font color="Maroon">B</font> st <br/>( <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">a</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">a</font> <a href="subset_1.html#K9">/\</a> <font color="Olive">b</font> ) or  ex <font color="Olive">f</font> being  <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">X</font> st <br/>( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or ( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Olive">f</font> &amp;  <a href="setfam_1.html#K8">Intersect</a> <font color="Olive">f</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font> &amp;  <a href="setfam_1.html#K8">Intersect</a> <font color="Olive">f</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> ) ) )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_1_1_1"><b>suppose </b></a><a NAME="E1:15_1_1_1"/>
 ex <font color="Olive">a</font>, <font color="Olive">b</font> being   <a href="subset_1.html#M2">Element</a> of <font color="Maroon">B</font> st <br/>( <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">a</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">a</font> <a href="subset_1.html#K9">/\</a> <font color="Olive">b</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:15_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">A</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">s</font></span><a href="tarski.html#K1">}</a></span>]
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_1_1_2"><b>suppose </b></a><a NAME="E1:15_1_1_2"/>
 ex <font color="Olive">f</font> being  <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">X</font> st <br/>( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or ( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Olive">f</font> &amp;  <a href="setfam_1.html#K8">Intersect</a> <font color="Olive">f</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font> &amp;  <a href="setfam_1.html#K8">Intersect</a> <font color="Olive">f</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> ) )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">f</font> being   <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E3:15_1_1_2"/><i><font color="Green">E74</font></i>: 
( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or ( <font color="Maroon">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> &amp;  <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">f</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font> &amp;  <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">f</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> ) )
 ;<br/><b>thus </b><a NAME="E4:15_1_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">A</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">s</font></span><a href="tarski.html#K1">}</a></span>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_1_2_1"><b>proof </b></a><div class="add">

<a NAME="E1:15_1_1_2_1"/>
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">s</font></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">X</font>
 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">sf</font> = <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">s</font></span><a href="tarski.html#K1">}</a></span> as   <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">X</font> ;<br/>
<a NAME="E3:15_1_1_2_1"/><i><font color="Green">E75</font></i>: 
 <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">sf</font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K6">meet</a> <font color="Maroon">sf</font>
 
<b>by </b><i><a class="ref" href="setfam_1.html#D10">SETFAM_1:def 10</a></i>;<br/>
<a NAME="E4:15_1_1_2_1"/><b>then </b><i><font color="Green">E79</font></i>: 
 <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">sf</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font>
 
<b>by </b><i><a class="ref" href="setfam_1.html#T11">SETFAM_1:11</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:15_1_1_2_1_1"/>
( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or ( <font color="Maroon">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> &amp;  <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">f</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font> &amp;  <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">f</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> ) )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:15_1_1_2_1_1_1"/>
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:15_1_1_2_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">A</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">s</font></span><a href="tarski.html#K1">}</a></span>]
 <b>by </b><i>, , , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:15_1_1_2_1_1_2"/><i><font color="Green">E80</font></i>: 
( <font color="Maroon">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> &amp;  <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">f</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font> &amp;  <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">f</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> )
 ;<br/><div class="add"><a NAME="E2:15_1_1_2_1_1_2"/><b>then </b>
<font color="Maroon">A</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">sf</font> <a href="tarski.html#R1">c=</a>  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T8">XBOOLE_1:8</a></i>;<br/><b>then reconsider </b><font color="Maroon">As</font> = <font color="Maroon">A</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">sf</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">X</font> <b>by </b><i><a class="ref" href="xboole_1.html#T15">XBOOLE_1:15</a></i>;<br/><a NAME="E4:15_1_1_2_1_1_2"/><i><font color="Green">E81</font></i>: 
 <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K6">meet</a> <font color="Maroon">f</font>
 <b>by </b><i>, <a class="ref" href="setfam_1.html#D10">SETFAM_1:def 10</a></i>;<br/><i><font color="Green">E82</font></i>:  <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">As</font> = 
 <a href="setfam_1.html#K6">meet</a> <font color="Maroon">As</font>
<b>by </b><i><a class="ref" href="setfam_1.html#D10">SETFAM_1:def 10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="setfam_1.html#K1">meet</a> <font color="Maroon">A</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K6">meet</a> <font color="Maroon">sf</font></span>)</span>
<b>by </b><i>, <a class="ref" href="setfam_1.html#T10">SETFAM_1:10</a></i>
;<br/><b>thus </b><a NAME="E6:15_1_1_2_1_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">A</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">s</font></span><a href="tarski.html#K1">}</a></span>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_1_2_1_1_2_2"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_1_1_2_1_1_2_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:15_1_1_2_1_1_2_2_1"/>
(  <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">As</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font> or  <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">As</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_1_1_2_1_1_2_2_1_1"><b>suppose </b></a><a NAME="E1:15_1_1_2_1_1_2_2_1_1"/><i><font color="Green">E83</font></i>: 
 <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">As</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font>
 ;<br/><div class="add"><a NAME="E2:15_1_1_2_1_1_2_2_1_1"/>
 <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">As</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 <b>by </b><i>, , , , , , , <a class="txt" href="armstrng.html#E1"><i><font color="Green">Lemma26</font></i></a>, <a class="ref" href="finsub_1.html#D2">FINSUB_1:def 2</a></i>;<br/><b>hence </b><a NAME="E3:15_1_1_2_1_1_2_2_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">A</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">s</font></span><a href="tarski.html#K1">}</a></span>]
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_1_1_2_1_1_2_2_1_2"><b>suppose </b></a><a NAME="E1:15_1_1_2_1_1_2_2_1_2"/>
 <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">As</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:15_1_1_2_1_1_2_2_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">A</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">s</font></span><a href="tarski.html#K1">}</a></span>]
 <b>by </b><i>, , , , , , , <a class="txt" href="armstrng.html#E1"><i><font color="Green">Lemma26</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E10:15"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">S</font>]
 
<b>from </b><i><a class="ref" href="finset_1.html#S2">FINSET_1:sch 2</a>(, , );<br/></i>
<b>then consider </b><font color="Maroon">f</font> being   <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E12:15"/><i><font color="Green">E84</font></i>: 
( <font color="Maroon">S</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or ( <font color="Maroon">S</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> &amp;  <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">f</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font> ) )
 <b>by </b><i>, </i>;<br/>
<b>thus </b><a NAME="E13:15"/>
contradiction
 <b>by </b><i>, , , <a class="ref" href="setfam_1.html#D10">SETFAM_1:def 10</a></i>;<br/>


</div>
