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<b>let </b><font color="Maroon">X</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</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">S</font> being non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</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="setfam_1.html#K1">meet</a> <font color="Olive">S</font> &amp; ( for <font color="Olive">s</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">s</font> <a href="hidden.html#R2">in</a> <font color="Olive">S</font> holds <br/><font color="Olive">s</font> <a href="armstrng.html#R1" target="_self">is_/\-irreducible_in</a> <font color="Maroon">X</font> ) );<br/>
<div><i><font color="Green">E33</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:11_1"/><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">X</font>
 ;<br/><b>thus </b><a NAME="E2:11_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">x</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:11_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 
<b>by </b><i>, </i>;<br/>
<b>then reconsider </b><font color="Maroon">S</font> = <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> as  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font> <b>by </b><i><a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/>
<b>take </b>
<font color="Maroon">S</font>
;<br/>

<b>thus </b><a NAME="E3:11_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K1">meet</a> <font color="Maroon">S</font>
 <b>by </b><i><a class="ref" href="setfam_1.html#T11">SETFAM_1:11</a></i>;<br/>

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

<b>assume </b><a NAME="E4:11_1_1"/>
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 ;<br/>

<b>hence </b><a NAME="E5:11_1_1"/>
<font color="Maroon">s</font> <a href="armstrng.html#R1" target="_self">is_/\-irreducible_in</a> <font color="Maroon">X</font>
 <b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<div><i><font color="Green">E35</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="11_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><b>that </b><br/><a NAME="E1:11_2"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> )
 <b>and </b><br/><a NAME="E2:11_2"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">x</font>]
 <b>and </b><br/><a NAME="E3:11_2"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">y</font>]
 ;<br/><b>consider </b><font color="Maroon">Sx</font> being  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E5:11_2"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K1">meet</a> <font color="Maroon">Sx</font>
 <b>and </b><br/><a NAME="E6:11_2"/><i><font color="Green">E43</font></i>: 
for <font color="Olive">s</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Sx</font> holds <br/><font color="Olive">s</font> <a href="armstrng.html#R1" target="_self">is_/\-irreducible_in</a> <font color="Maroon">X</font>
 <b>by </b><i/>;<br/><b>consider </b><font color="Maroon">Sy</font> being  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E8:11_2"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K1">meet</a> <font color="Maroon">Sy</font>
 <b>and </b><br/><a NAME="E9:11_2"/><i><font color="Green">E45</font></i>: 
for <font color="Olive">s</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Sy</font> holds <br/><font color="Olive">s</font> <a href="armstrng.html#R1" target="_self">is_/\-irreducible_in</a> <font color="Maroon">X</font>
 <b>by </b><i/>;<br/><b>reconsider </b><font color="Maroon">S</font> = <font color="Maroon">Sx</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">Sy</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font> ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_2_1"><b>now </b></a><div class="add"><b>take </b><font color="Maroon">S</font> = <font color="Maroon">S</font>;<br/><b>thus </b><a NAME="E1:11_2_1"/>
<font color="Maroon">x</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K1">meet</a> <font color="Maroon">S</font>
 <b>by </b><i>, , <a class="ref" href="setfam_1.html#T10">SETFAM_1:10</a></i>;<br/><b>let </b><font color="Maroon">s</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E2:11_2_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>
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="11_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:11_2_1_1"/>
( <font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Sx</font> or <font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Sy</font> )
 </span><b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="11_2_1_1_1"><b>suppose </b></a><a NAME="E1:11_2_1_1_1"/>
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Sx</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:11_2_1_1_1"/>
<font color="Maroon">s</font> <a href="armstrng.html#R1" target="_self">is_/\-irreducible_in</a> <font color="Maroon">X</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="11_2_1_1_2"><b>suppose </b></a><a NAME="E1:11_2_1_1_2"/>
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Sy</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:11_2_1_1_2"/>
<font color="Maroon">s</font> <a href="armstrng.html#R1" target="_self">is_/\-irreducible_in</a> <font color="Maroon">X</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E12:11_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">x</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">y</font>]
 ;<br/></div><b>end;</b></div>
<a NAME="E3:11"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>]
 
<b>from </b><i><a class="ref" href="armstrng.html#S2" target="_self">ARMSTRNG:sch 2</a>(, );<br/></i>
<b>hence </b><a NAME="E4:11"/>
 ex <font color="Olive">A</font> being non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font> st <br/>( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K1">meet</a> <font color="Olive">A</font> &amp; ( for <font color="Olive">s</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">s</font> <a href="hidden.html#R2">in</a> <font color="Olive">A</font> holds <br/><font color="Olive">s</font> <a href="armstrng.html#R1" target="_self">is_/\-irreducible_in</a> <font color="Maroon">X</font> ) )
 ;<br/>


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