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

<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">x</font> where <font color="Olive">x</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() : <font color="Maroon">a<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Olive">x</font> </span>}</span> ;<br/>
<b>given </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><a NAME="E1:10_1"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>1</sub></font>()
 <b>and </b><br/><a NAME="E2:10_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">x</font>]
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b> ex <font color="Olive">s</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() st <br/>(  <a href="card_1.html#K1">Card</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">s</font>) <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">s</font>] );<br/>
<a NAME="E3:10_1"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">x</font>) <a href="tarski.html#R1">c=</a> <font color="Maroon">F<sub>1</sub></font>()
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x1</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="E1:10_1_1"/>
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">x</font>)
 ;<br/>

<a NAME="E2:10_1_1"/><b>then </b>
 ex <font color="Olive">xx</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() st <br/>( <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Olive">xx</font> &amp; <font color="Maroon">x</font> <a href="tarski.html#R1">c=</a> <font color="Olive">xx</font> )
 
;<br/>
<b>hence </b><a NAME="E3:10_1_1"/>
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>1</sub></font>()
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">Ux</font> = <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">x</font>) 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/>
<a NAME="E5:10_1"/><i><font color="Green">E39</font></i>: 
 ex <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_2"><b>proof </b></a><div class="add">

<b>take </b><font color="Maroon">k</font> =  <a href="card_1.html#K4">card</a> <font color="Maroon">Ux</font>;<br/>

<b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() <b>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">x</font>
;<br/>

<b>thus </b><a NAME="E2:10_1_2"/>
 <a href="card_1.html#K1">Card</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">x</font>) <a href="hidden.html#R1">=</a> <font color="Maroon">k</font>
 ;<br/>

<b>thus </b><a NAME="E3:10_1_2"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">x</font>]
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">k</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E7:10_1"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font>]
 <b>and </b><br/><a NAME="E8:10_1"/><i><font color="Green">E43</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>] holds <br/><font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">n</font>
 <b>from </b><i><a class="ref" href="nat_1.html#S5">NAT_1:sch 5</a>();<br/></i>
<b>consider </b><font color="Maroon">s</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>()<b> such that </b><br/><a NAME="E10:10_1"/><i><font color="Green">E44</font></i>: 
 <a href="card_1.html#K1">Card</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">s</font>) <a href="hidden.html#R1">=</a> <font color="Maroon">k</font>
 <b>and </b><br/><a NAME="E11:10_1"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">s</font>]
 <b>by </b><i/>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_1_3"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:10_1_3"/>
( <font color="Maroon">s</font> <a href="armstrng.html#R1" target="_self">is_/\-irreducible_in</a> <font color="Maroon">F<sub>1</sub></font>() or <font color="Maroon">s</font> <a href="armstrng.html#NR2" target="_self">is_/\-reducible_in</a> <font color="Maroon">F<sub>1</sub></font>() )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_1_3_1"><b>suppose </b></a><a NAME="E1:10_1_3_1"/>
<font color="Maroon">s</font> <a href="armstrng.html#R1" target="_self">is_/\-irreducible_in</a> <font color="Maroon">F<sub>1</sub></font>()
 ;<br/><div class="add"><b>hence </b><a NAME="E2:10_1_3_1"/>
contradiction
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_1_3_2"><b>suppose </b></a><a NAME="E1:10_1_3_2"/>
<font color="Maroon">s</font> <a href="armstrng.html#NR2" target="_self">is_/\-reducible_in</a> <font color="Maroon">F<sub>1</sub></font>()
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">z</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:10_1_3_2"/><i><font color="Green">E46</font></i>: 
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>1</sub></font>() &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>1</sub></font>() )
 <b>and </b><br/><a NAME="E4:10_1_3_2"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">y</font>
 <b>and </b><br/><a NAME="E5:10_1_3_2"/><i><font color="Green">E74</font></i>: 
( <font color="Maroon">s</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">z</font> &amp; <font color="Maroon">s</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">y</font> )
 <b>by </b><i/>;<br/><a NAME="E6:10_1_3_2"/><i><font color="Green">E75</font></i>: 
( <font color="Maroon">s</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">z</font> &amp; <font color="Maroon">s</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">y</font> )
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/><a NAME="E7:10_1_3_2"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">s</font>) <a href="tarski.html#R1">c=</a> <font color="Maroon">F<sub>1</sub></font>()
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_3_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</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="E1:10_1_3_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">s</font>)
 ;<br/>

<a NAME="E2:10_1_3_2_1"/><b>then </b>
 ex <font color="Olive">xx</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() st <br/>( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">xx</font> &amp; <font color="Maroon">s</font> <a href="tarski.html#R1">c=</a> <font color="Olive">xx</font> )
 
;<br/>
<b>hence </b><a NAME="E3:10_1_3_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>1</sub></font>()
 ;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">Us</font> = <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">s</font>) 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/><a NAME="E9:10_1_3_2"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">y</font>) <a href="tarski.html#R1">c=</a> <font color="Maroon">F<sub>1</sub></font>()
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_3_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</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="E1:10_1_3_2_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">y</font>)
 ;<br/>

<a NAME="E2:10_1_3_2_2"/><b>then </b>
 ex <font color="Olive">xx</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() st <br/>( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">xx</font> &amp; <font color="Maroon">y</font> <a href="tarski.html#R1">c=</a> <font color="Olive">xx</font> )
 
;<br/>
<b>hence </b><a NAME="E3:10_1_3_2_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>1</sub></font>()
 ;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">Uy</font> = <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">y</font>) 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/><a NAME="E11:10_1_3_2"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">z</font>) <a href="tarski.html#R1">c=</a> <font color="Maroon">F<sub>1</sub></font>()
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_3_2_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</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="E1:10_1_3_2_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">z</font>)
 ;<br/>

<a NAME="E2:10_1_3_2_3"/><b>then </b>
 ex <font color="Olive">xx</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() st <br/>( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">xx</font> &amp; <font color="Maroon">z</font> <a href="tarski.html#R1">c=</a> <font color="Olive">xx</font> )
 
;<br/>
<b>hence </b><a NAME="E3:10_1_3_2_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>1</sub></font>()
 ;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">Uz</font> = <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">z</font>) 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>reconsider </b><font color="Maroon">y</font> = <font color="Maroon">y</font>, <font color="Maroon">z</font> = <font color="Maroon">z</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() <b>by </b><i/>;<br/><div><i><font color="Green">E79</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_3_2_4"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:10_1_3_2_4"/>
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Uz</font>
 ;<br/><a NAME="E2:10_1_3_2_4"/><b>then </b>
 ex <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() st <br/>( <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Olive">x</font> &amp; <font color="Maroon">z</font> <a href="tarski.html#R1">c=</a> <font color="Olive">x</font> )
 ;<br/><b>hence </b><a NAME="E3:10_1_3_2_4"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_3_2_5"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:10_1_3_2_5"/>
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Uy</font>
 ;<br/><a NAME="E2:10_1_3_2_5"/><b>then </b>
 ex <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() st <br/>( <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Olive">x</font> &amp; <font color="Maroon">y</font> <a href="tarski.html#R1">c=</a> <font color="Olive">x</font> )
 ;<br/><b>hence </b><a NAME="E3:10_1_3_2_5"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/></div><b>end;</b></div><a NAME="E16:10_1_3_2"/><b>then </b><i><font color="Green">E80</font></i>: 
( <font color="Maroon">Uz</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">Us</font> &amp; <font color="Maroon">Uy</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">Us</font> )
 <b>by </b><i/>;<br/><a NAME="E17:10_1_3_2"/><i><font color="Green">E81</font></i>: 
<font color="Maroon">Uz</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Us</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_3_2_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</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="E1:10_1_3_2_6"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Uz</font>
 ;<br/>

<b>then consider </b><font color="Maroon">xx</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>()<b> such that </b><br/><a NAME="E3:10_1_3_2_6"/><i><font color="Green">E82</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">xx</font>
 <b>and </b><br/><a NAME="E4:10_1_3_2_6"/><i><font color="Green">E83</font></i>: 
<font color="Maroon">z</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">xx</font>
 ;<br/>
<a NAME="E5:10_1_3_2_6"/>
<font color="Maroon">s</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">xx</font>
 
<b>by </b><i>, , <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>hence </b><a NAME="E6:10_1_3_2_6"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Us</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div><a NAME="E18:10_1_3_2"/>
<font color="Maroon">Uy</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Us</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_3_2_7"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</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="E1:10_1_3_2_7"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Uy</font>
 ;<br/>

<b>then consider </b><font color="Maroon">xx</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>()<b> such that </b><br/><a NAME="E3:10_1_3_2_7"/><i><font color="Green">E84</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">xx</font>
 <b>and </b><br/><a NAME="E4:10_1_3_2_7"/><i><font color="Green">E85</font></i>: 
<font color="Maroon">y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">xx</font>
 ;<br/>
<a NAME="E5:10_1_3_2_7"/>
<font color="Maroon">s</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">xx</font>
 
<b>by </b><i>, , <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>hence </b><a NAME="E6:10_1_3_2_7"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Us</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div><a NAME="E19:10_1_3_2"/><b>then </b>
( <font color="Maroon">Uz</font> <a href="xboole_0.html#R2">c&lt;</a> <font color="Maroon">Us</font> &amp; <font color="Maroon">Uy</font> <a href="xboole_0.html#R2">c&lt;</a> <font color="Maroon">Us</font> )
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#D8">XBOOLE_0:def 8</a></i>;<br/><a NAME="E20:10_1_3_2"/><b>then </b>
(  <a href="card_1.html#K4">card</a> <font color="Maroon">Us</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">Uz</font> &amp;  <a href="card_1.html#K4">card</a> <font color="Maroon">Us</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">Uy</font> )
 <b>by </b><i><a class="ref" href="trees_1.html#T24">TREES_1:24</a></i>;<br/><a NAME="E21:10_1_3_2"/><b>then </b>
( <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">z</font>] &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">y</font>] )
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E22:10_1_3_2"/>
contradiction
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
