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

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

<b>assume </b><b>that </b><br/><a NAME="E1:6"/><i><font color="Green">E30</font></i>: 
not <font color="Maroon">F</font> is <a href="xboole_0.html#V1">empty</a>
 <b>and </b><br/><a NAME="E2:6"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">F</font> is <a href="setfam_1.html#V1">without_zero</a>
 <b>and </b><br/><a NAME="E3:6"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">F</font> is <a href="ordinal1.html#V6">c=-linear</a>
 ;<br/>

<b>thus </b><a NAME="E4:6"/>
<font color="Maroon">F</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i/>; <i><font color="Red">:: according to </font></i><a class="ref" href="compts_1.html#D2">COMPTS_1:def 2</a><br/>

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

<b>assume </b><b>that </b><br/><a NAME="E5:6"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">G</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>and </b><br/><a NAME="E6:6"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">G</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">F</font>
 <b>and </b><br/><a NAME="E7:6"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">G</font> is <a href="finset_1.html#V1">finite</a>
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ] <b>means </b>( <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>  implies  ex <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a<sub>1</sub></font> &amp; ( for <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a<sub>1</sub></font> holds <br/><font color="Olive">x</font> <a href="tarski.html#R1">c=</a> <font color="Olive">y</font> ) ) );<br/>
<a NAME="E8:6"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="xboole_0.html#K1">{}</a> ]
 
;<br/>
<div><i><font color="Green">E41</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1"><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">B</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><b>that </b><br/><a NAME="E1:6_1"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font>
 <b>and </b><br/><a NAME="E2:6_1"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">B</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">G</font>
 <b>and </b><br/><a NAME="E3:6_1"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">B</font>]
 ;<br/><b>thus </b><a NAME="E4:6_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">B</font> <a href="xboole_0.html#K2">\/</a> <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>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:6_1_1_1"/>
( <font color="Maroon">B</font> is <a href="xboole_0.html#V1">empty</a> or  not <font color="Maroon">B</font> is <a href="xboole_0.html#V1">empty</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_1_1_1_1"><b>suppose </b></a><a NAME="E1:6_1_1_1_1"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">B</font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/><div class="add"><b>assume </b><a NAME="E2:6_1_1_1_1"/>
<font color="Maroon">B</font> <a href="xboole_0.html#K2">\/</a> <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> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><b>take </b><font color="Maroon">x'</font> = <font color="Maroon">x</font>;<br/><b>thus </b><a NAME="E3:6_1_1_1_1"/>
<font color="Maroon">x'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> <a href="xboole_0.html#K2">\/</a> <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>
 <b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>thus </b><a NAME="E4:6_1_1_1_1"/>
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> <a href="xboole_0.html#K2">\/</a> <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> implies <font color="Maroon">x'</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">y</font> )
 <b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_1_1_1_2"><b>suppose </b></a><a NAME="E1:6_1_1_1_2"/>
not <font color="Maroon">B</font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">z</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:6_1_1_1_2"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 <b>and </b><br/><a NAME="E4:6_1_1_1_2"/><i><font color="Green">E50</font></i>: 
for <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> holds <br/><font color="Maroon">z</font> <a href="tarski.html#R1">c=</a> <font color="Olive">y</font>
 <b>by </b><i/>;<br/><b>assume </b><a NAME="E5:6_1_1_1_2"/>
<font color="Maroon">B</font> <a href="xboole_0.html#K2">\/</a> <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> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><b>thus </b><a NAME="E6:6_1_1_1_2"/>
 ex <font color="Olive">x'</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">x'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> <a href="xboole_0.html#K2">\/</a> <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> &amp; ( for <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> <a href="xboole_0.html#K2">\/</a> <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> holds <br/><font color="Olive">x'</font> <a href="tarski.html#R1">c=</a> <font color="Olive">y</font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1_1_1_2_1"><b>proof </b></a><div class="add">

<a NAME="E1:6_1_1_1_2_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E2:6_1_1_1_2_1"/><b>then </b><i><font color="Green">E51</font></i>: 
<font color="Maroon">x</font>,<font color="Maroon">z</font> <a href="xboole_0.html#R3">are_c=-comparable</a> 
 
<b>by </b><i>, , , <a class="ref" href="ordinal1.html#D9">ORDINAL1:def 9</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_1_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="E1:6_1_1_1_2_1_1"/>
( <font color="Maroon">x</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">z</font> or <font color="Maroon">z</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">x</font> )
 </span><b>by </b><i>, <a class="ref" href="xboole_0.html#D9">XBOOLE_0:def 9</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_1_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:6_1_1_1_2_1_1_1"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">x</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">z</font>
 ;<br/><div class="add"><b>take </b><font color="Maroon">x'</font> = <font color="Maroon">x</font>;<br/><a NAME="E2:6_1_1_1_2_1_1_1"/>
<font color="Maroon">x'</font> <a href="hidden.html#R2">in</a> <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>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E3:6_1_1_1_2_1_1_1"/>
<font color="Maroon">x'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> <a href="xboole_0.html#K2">\/</a> <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>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E4:6_1_1_1_2_1_1_1"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> <a href="xboole_0.html#K2">\/</a> <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>
 ;<br/><b>thus </b><a NAME="E5:6_1_1_1_2_1_1_1"/>
<font color="Maroon">x'</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">y</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1_1_1_2_1_1_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_1_1_1_2_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:6_1_1_1_2_1_1_1_1_1"/>
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> or <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <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> )
 </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="6_1_1_1_2_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:6_1_1_1_2_1_1_1_1_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 ;<br/><div class="add"><a NAME="E2:6_1_1_1_2_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon">z</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">y</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:6_1_1_1_2_1_1_1_1_1_1"/>
<font color="Maroon">x'</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">y</font>
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_1_1_1_2_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:6_1_1_1_2_1_1_1_1_1_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <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>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:6_1_1_1_2_1_1_1_1_1_2"/>
<font color="Maroon">x'</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">y</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><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_1_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:6_1_1_1_2_1_1_2"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">z</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">x</font>
 ;<br/><div class="add"><b>take </b><font color="Maroon">x'</font> = <font color="Maroon">z</font>;<br/><b>thus </b><a NAME="E2:6_1_1_1_2_1_1_2"/>
<font color="Maroon">x'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> <a href="xboole_0.html#K2">\/</a> <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>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E3:6_1_1_1_2_1_1_2"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> <a href="xboole_0.html#K2">\/</a> <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>
 ;<br/><b>thus </b><a NAME="E4:6_1_1_1_2_1_1_2"/>
<font color="Maroon">x'</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">y</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1_1_1_2_1_1_2_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_1_1_1_2_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="E1:6_1_1_1_2_1_1_2_1_1"/>
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> or <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <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> )
 </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="6_1_1_1_2_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:6_1_1_1_2_1_1_2_1_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:6_1_1_1_2_1_1_2_1_1_1"/>
<font color="Maroon">x'</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">y</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_1_1_1_2_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:6_1_1_1_2_1_1_2_1_1_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <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>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:6_1_1_1_2_1_1_2_1_1_2"/>
<font color="Maroon">x'</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">y</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><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="E10:6"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">G</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">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E12:6"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font>
 <b>and </b><br/><a NAME="E13:6"/><i><font color="Green">E57</font></i>: 
for <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> holds <br/><font color="Maroon">x</font> <a href="tarski.html#R1">c=</a> <font color="Olive">y</font>
 <b>by </b><i/>;<br/>
<a NAME="E14:6"/>
<font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i>, , , <a class="ref" href="setfam_1.html#D9">SETFAM_1:def 9</a></i>;<br/>
<b>then consider </b><font color="Maroon">xe</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E16:6"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">xe</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:6_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font>
 ;<br/><a NAME="E2:6_2"/><b>then </b>
<font color="Maroon">x</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">y</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:6_2"/>
<font color="Maroon">xe</font> <a href="hidden.html#R2">in</a> <font color="Maroon">y</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E18:6"/>
 <a href="setfam_1.html#K1">meet</a> <font color="Maroon">G</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i>, <a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/>


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