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

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

<b>assume </b><a NAME="E1:25"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">A</font> is <a href="finset_1.html#V1" target="_self">finite</a>
 ;<br/>

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

<b>assume </b><a NAME="E2:25"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<b>consider </b><font color="Maroon">p</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E4:25"/><i><font color="Green">E28</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font>
 <b>and </b><br/><a NAME="E5:25"/><i><font color="Green">E35</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K5">omega</a> 
 <b>by </b><i>, </i>;<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">p</font> <a href="relat_1.html#K9">.:</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>;<br/>
<b>consider </b><font color="Maroon">G</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E7:25"/><i><font color="Green">E39</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> iff ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">p</font></span>)</span> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>] ) )
 <b>from </b><i><a class="ref" href="xboole_0.html#S1">XBOOLE_0:sch 1</a>();<br/></i>
<a NAME="E8:25"/>
<font color="Maroon">G</font> <a href="tarski.html#R1">c=</a>  <a href="zfmisc_1.html#K1">bool</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">p</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_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="E1:25_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font>
 ;<br/>

<b>hence </b><a NAME="E2:25_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">p</font></span>)</span>
 <b>by </b><i><a class="ref" href="finset_1.html#D1" target="_self">Def1</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">GX</font> = <font color="Maroon">G</font> as   <a href="setfam_1.html#NM1">Subset-Family</a> of <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">p</font></span>)</span> ;<br/>
<b>consider </b><font color="Maroon">x</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font>;<br/>
<a NAME="E11:25"/>
<font color="Maroon">x</font> is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">A</font>
 
<b>by </b><i>, <a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<a NAME="E12:25"/><b>then </b><i><font color="Green">E40</font></i>: 
<font color="Maroon">p</font> <a href="relat_1.html#K9">.:</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="relat_1.html#K10">"</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#T147">FUNCT_1:147</a></i>;<br/>
<a NAME="E13:25"/>
<font color="Maroon">p</font> <a href="relat_1.html#K10">"</a> <font color="Maroon">x</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">p</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T167">RELAT_1:167</a></i>;<br/>
<a NAME="E14:25"/><b>then </b><i><font color="Green">E41</font></i>: 
<font color="Maroon">GX</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="finset_1.html#D1" target="_self">Def1</a>, </i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1">set</a> ] <b>means </b>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K5">omega</a>  implies for <font color="Olive">X</font> being  <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">a<sub>1</sub></font>  st <font color="Olive">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  holds <br/> 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="Olive">X</font> &amp; ( for <font color="Olive">B</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">B</font> <a href="hidden.html#R2">in</a> <font color="Olive">X</font> &amp; <font color="Olive">x</font> <a href="tarski.html#R1">c=</a> <font color="Olive">B</font> holds <br/><font color="Olive">B</font> <a href="hidden.html#R1">=</a> <font color="Olive">x</font> ) ) );<br/>
<a NAME="E15:25"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[ <a href="xboole_0.html#K1">{}</a> ]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:25_2"/>
 <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K5">omega</a> 
 ;<br/>

<b>let </b><font color="Maroon">X</font> be   <a href="setfam_1.html#NM1">Subset-Family</a> of <a href="xboole_0.html#K1">{}</a> ;<br/>

<b>assume </b><a NAME="E2:25_2"/>
<font color="Maroon">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<a NAME="E3:25_2"/><b>then </b><i><font color="Green">E43</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T1">ZFMISC_1:1</a>, <a class="ref" href="zfmisc_1.html#T39">ZFMISC_1:39</a></i>;<br/>
<b>take </b>
 <a href="xboole_0.html#K1">{}</a> 
;<br/>

<b>thus </b><a NAME="E4:25_2"/>
(  <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; ( for <font color="Olive">B</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp;  <a href="xboole_0.html#K1">{}</a>  <a href="tarski.html#R1">c=</a> <font color="Olive">B</font> holds <br/><font color="Olive">B</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  ) )
 <b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E16:25"/><i><font color="Green">E44</font></i>: 
for <font color="Olive">O</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">O</font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[ <a href="ordinal1.html#K1">succ</a> <font color="Olive">O</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">O</font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/>

<b>assume </b><a NAME="E1:25_3"/><i><font color="Green">E45</font></i>: 
( <font color="Maroon">O</font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K5">omega</a>  implies for <font color="Olive">X</font> being  <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">O</font>  st <font color="Olive">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  holds <br/> 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="Olive">X</font> &amp; ( for <font color="Olive">B</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">B</font> <a href="hidden.html#R2">in</a> <font color="Olive">X</font> &amp; <font color="Olive">x</font> <a href="tarski.html#R1">c=</a> <font color="Olive">B</font> holds <br/><font color="Olive">B</font> <a href="hidden.html#R1">=</a> <font color="Olive">x</font> ) ) )
 ;<br/>

<b>thus </b><a NAME="E2:25_3"/>
(  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">O</font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K5">omega</a>  implies for <font color="Olive">X</font> being  <a href="setfam_1.html#NM1">Subset-Family</a> of <span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">O</font></span>)</span>  st <font color="Olive">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  holds <br/> 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="Olive">X</font> &amp; ( for <font color="Olive">B</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">B</font> <a href="hidden.html#R2">in</a> <font color="Olive">X</font> &amp; <font color="Olive">x</font> <a href="tarski.html#R1">c=</a> <font color="Olive">B</font> holds <br/><font color="Olive">B</font> <a href="hidden.html#R1">=</a> <font color="Olive">x</font> ) ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_3_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:25_3_1"/>
 <a href="ordinal1.html#K1">succ</a> <font color="Maroon">O</font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K5">omega</a> 
 ;<br/>

<a NAME="E2:25_3_1"/><b>then </b><i><font color="Green">E46</font></i>: 
 <a href="ordinal1.html#K1">succ</a> <font color="Maroon">O</font> <a href="ordinal1.html#R1">c=</a>  <a href="ordinal1.html#K5">omega</a> 
 
<b>by </b><i><a class="ref" href="ordinal1.html#D2">ORDINAL1:def 2</a></i>;<br/>
<b>let </b><font color="Maroon">X</font> be   <a href="setfam_1.html#NM1">Subset-Family</a> of <span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">O</font></span>)</span>;<br/>

<b>assume </b><a NAME="E3:25_3_1"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>3</sub></font>[   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">A</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">A</font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">O</font></span><a href="tarski.html#K1">}</a></span> );<br/>
<b>consider </b><font color="Maroon">X1</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:25_3_1"/><i><font color="Green">E48</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X1</font> iff ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">O</font> &amp; <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">x</font>] ) )
 <b>from </b><i><a class="ref" href="xboole_0.html#S1">XBOOLE_0:sch 1</a>();<br/></i>
<a NAME="E6:25_3_1"/>
<font color="Maroon">X1</font> <a href="tarski.html#R1">c=</a>  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">O</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_3_1_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="E1:25_3_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X1</font>
 ;<br/>

<b>hence </b><a NAME="E2:25_3_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">O</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">X2</font> = <font color="Maroon">X1</font> as   <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">O</font> ;<br/>
<b>consider </b><font color="Maroon">y</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font>;<br/>
<a NAME="E9:25_3_1"/><i><font color="Green">E49</font></i>: 
 <a href="ordinal1.html#K1">succ</a> <font color="Maroon">O</font> <a href="hidden.html#R1">=</a> <font color="Maroon">O</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">O</font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="ordinal1.html#D1">ORDINAL1:def 1</a></i>;<br/>
<a NAME="E10:25_3_1"/>
<font color="Maroon">y</font> is   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">O</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<a NAME="E11:25_3_1"/><b>then </b>
<font color="Maroon">y</font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">O</font></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><font color="Maroon">O</font> <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">O</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">O</font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T33">XBOOLE_1:33</a></i>;<br/>
<a NAME="E12:25_3_1"/><b>then </b><i><font color="Green">E50</font></i>: 
<font color="Maroon">y</font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">O</font></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">O</font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">O</font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T40">XBOOLE_1:40</a></i>;<br/>
<a NAME="E13:25_3_1"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">O</font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">O</font>
 
<b>by </b><i><a class="ref" href="ordinal1.html#T10">ORDINAL1:10</a></i>;<br/>
<a NAME="E14:25_3_1"/><i><font color="Green">E69</font></i>: 
not <font color="Maroon">O</font> <a href="hidden.html#R2">in</a> <font color="Maroon">O</font>
 
;<br/>
<a NAME="E15:25_3_1"/><b>then </b>
<font color="Maroon">y</font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">O</font></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">O</font>
 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T65">ZFMISC_1:65</a></i>;<br/>
<a NAME="E16:25_3_1"/><b>then </b>
<font color="Maroon">X2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i>, </i>;<br/>
<b>then consider </b><font color="Maroon">Z</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E18:25_3_1"/><i><font color="Green">E70</font></i>: 
( <font color="Maroon">Z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X2</font> &amp; ( for <font color="Olive">B</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X2</font> &amp; <font color="Maroon">Z</font> <a href="tarski.html#R1">c=</a> <font color="Olive">B</font> holds <br/><font color="Olive">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Z</font> ) )
 <b>by </b><i>, , </i>;<br/>
<b>consider </b><font color="Maroon">X1</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E20:25_3_1"/><i><font color="Green">E71</font></i>: 
( <font color="Maroon">X1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">Z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X1</font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">O</font></span><a href="tarski.html#K1">}</a></span> )
 <b>by </b><i>, <a class="txt" href="finset_1.html#E1:25"><i><font color="Green">E26</font></i></a></i>;<br/>
<a NAME="E21:25_3_1"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">O</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">O</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/>
<a NAME="E22:25_3_1"/><b>then </b><i><font color="Green">E73</font></i>: 
not <font color="Maroon">O</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font>
 
<b>by </b><i><a class="txt" href="finset_1.html#E2:25"><i><font color="Green">E27</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<div><i><font color="Green">E74</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="25_3_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:25_3_1_2"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">Z</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">O</font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/><b>take </b><font color="Maroon">A</font> = <font color="Maroon">Z</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">O</font></span><a href="tarski.html#K1">}</a></span>;<br/><a NAME="E2:25_3_1_2"/>
for <font color="Olive">B</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Olive">B</font> holds <br/><font color="Olive">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_3_1_2_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:25_3_1_2_1"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/>

<b>assume </b><a NAME="E2:25_3_1_2_1"/><i><font color="Green">E77</font></i>: 
<font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font>
 ;<br/>

<div><i><font color="Green">E78</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="25_3_1_2_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:25_3_1_2_1_1"/><i><font color="Green">E79</font></i>: 
<font color="Maroon">O</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 ;<br/><b>set </b><font color="Maroon">Y</font> = <font color="Maroon">B</font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">O</font></span><a href="tarski.html#K1">}</a></span>;<br/><a NAME="E2:25_3_1_2_1_1"/>
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">O</font></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="txt" href="finset_1.html#E5:25"><i><font color="Green">E35</font></i></a>, <a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/><a NAME="E3:25_3_1_2_1_1"/><b>then </b><i><font color="Green">E80</font></i>: 
<font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="subset_1.html#K6">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">O</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">O</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="xboole_1.html#T45">XBOOLE_1:45</a></i>;<br/><a NAME="E4:25_3_1_2_1_1"/>
<font color="Maroon">A</font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">O</font></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">O</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="finset_1.html#E3"><i><font color="Green">Lemma33</font></i></a>, <a class="ref" href="xboole_1.html#T33">XBOOLE_1:33</a></i>;<br/><a NAME="E5:25_3_1_2_1_1"/><b>then </b>
( <font color="Maroon">Z</font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">O</font></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">O</font></span><a href="tarski.html#K1">}</a></span> &amp; not <font color="Maroon">O</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> )
 <b>by </b><i><a class="txt" href="finset_1.html#E2:25"><i><font color="Green">E27</font></i></a>, <a class="txt" href="finset_1.html#E4:25"><i><font color="Green">E28</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a>, <a class="ref" href="xboole_1.html#T40">XBOOLE_1:40</a></i>;<br/><a NAME="E6:25_3_1_2_1_1"/><b>then </b><i><font color="Green">E81</font></i>: 
<font color="Maroon">Z</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">O</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T65">ZFMISC_1:65</a></i>;<br/><a NAME="E7:25_3_1_2_1_1"/>
<font color="Maroon">B</font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">O</font></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><font color="Maroon">O</font> <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">O</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">O</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i>, , <a class="ref" href="xboole_1.html#T33">XBOOLE_1:33</a></i>;<br/><a NAME="E8:25_3_1_2_1_1"/><b>then </b>
<font color="Maroon">B</font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">O</font></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">O</font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">O</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="xboole_1.html#T40">XBOOLE_1:40</a></i>;<br/><a NAME="E9:25_3_1_2_1_1"/><b>then </b>
<font color="Maroon">B</font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">O</font></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">O</font>
 <b>by </b><i><a class="txt" href="finset_1.html#E2"><i><font color="Green">Lemma25</font></i></a>, <a class="ref" href="zfmisc_1.html#T65">ZFMISC_1:65</a></i>;<br/><a NAME="E10:25_3_1_2_1_1"/><b>then </b>
<font color="Maroon">B</font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">O</font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">X2</font>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E11:25_3_1_2_1_1"/>
<font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="txt" href="finset_1.html#E1:25"><i><font color="Green">E26</font></i></a>, , </i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_3_1_2_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:25_3_1_2_1_2"/><i><font color="Green">E82</font></i>: 
not <font color="Maroon">O</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 ;<br/><a NAME="E2:25_3_1_2_1_2"/>
<font color="Maroon">O</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">O</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/><a NAME="E3:25_3_1_2_1_2"/><b>then </b>
<font color="Maroon">O</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><b>hence </b><a NAME="E4:25_3_1_2_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="finset_1.html#E3"><i><font color="Green">Lemma33</font></i></a>, <a class="txt" href="finset_1.html#E5:25"><i><font color="Green">E35</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:25_3_1_2_1"/>
<font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E3:25_3_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">X</font> &amp; ( for <font color="Olive">B</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Olive">x</font> <a href="tarski.html#R1">c=</a> <font color="Olive">B</font> holds <br/><font color="Olive">B</font> <a href="hidden.html#R1">=</a> <font color="Olive">x</font> ) )
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_3_1_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:25_3_1_3"/><i><font color="Green">E83</font></i>: 
not <font color="Maroon">Z</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">O</font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/><b>take </b><font color="Maroon">A</font> = <font color="Maroon">Z</font>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_3_1_3_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:25_3_1_3_1"/>
<font color="Maroon">O</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X1</font>
 ;<br/><b>then </b><font color="Maroon">X1</font> = 
<font color="Maroon">X1</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">O</font></span><a href="tarski.html#K1">}</a></span>
<b>by </b><i><a class="ref" href="zfmisc_1.html#T46">ZFMISC_1:46</a></i>
<br/>.= 
<font color="Maroon">Z</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">O</font></span><a href="tarski.html#K1">}</a></span>
<b>by </b><i><a class="txt" href="finset_1.html#E2:25"><i><font color="Green">E27</font></i></a>, <a class="ref" href="xboole_1.html#T39">XBOOLE_1:39</a></i>
;<br/><b>hence </b><a NAME="E3:25_3_1_3_1"/>
contradiction
 <b>by </b><i><a class="txt" href="finset_1.html#E2:25"><i><font color="Green">E27</font></i></a>, </i>;<br/></div><b>end;</b></div><a NAME="E3:25_3_1_3"/><b>then </b><i><font color="Green">E84</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="txt" href="finset_1.html#E2:25"><i><font color="Green">E27</font></i></a>, <a class="ref" href="zfmisc_1.html#T65">ZFMISC_1:65</a></i>;<br/><a NAME="E4:25_3_1_3"/>
for <font color="Olive">B</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Olive">B</font> holds <br/><font color="Olive">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_3_1_3_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:25_3_1_3_2"/><i><font color="Green">E85</font></i>: 
<font color="Maroon">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/>

<a NAME="E2:25_3_1_3_2"/><b>then </b>
<font color="Maroon">B</font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">O</font></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><font color="Maroon">O</font> <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">O</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">O</font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T33">XBOOLE_1:33</a></i>;<br/>
<a NAME="E3:25_3_1_3_2"/><b>then </b>
<font color="Maroon">B</font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">O</font></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">O</font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">O</font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T40">XBOOLE_1:40</a></i>;<br/>
<a NAME="E4:25_3_1_3_2"/><b>then </b>
<font color="Maroon">B</font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">O</font></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">O</font>
 
<b>by </b><i><a class="txt" href="finset_1.html#E2"><i><font color="Green">Lemma25</font></i></a>, <a class="ref" href="zfmisc_1.html#T65">ZFMISC_1:65</a></i>;<br/>
<a NAME="E5:25_3_1_3_2"/><b>then </b><i><font color="Green">E86</font></i>: 
<font color="Maroon">B</font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">O</font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">X2</font>
 
<b>by </b><i>, <a class="txt" href="finset_1.html#E3"><i><font color="Green">Lemma33</font></i></a></i>;<br/>
<b>assume </b><a NAME="E6:25_3_1_3_2"/><i><font color="Green">E87</font></i>: 
<font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font>
 ;<br/>

<a NAME="E7:25_3_1_3_2"/><b>then </b>
<font color="Maroon">A</font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">O</font></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">O</font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T33">XBOOLE_1:33</a></i>;<br/>
<a NAME="E8:25_3_1_3_2"/><b>then </b><i><font color="Green">E88</font></i>: 
<font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">O</font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T65">ZFMISC_1:65</a></i>;<br/>
<div><i><font color="Green">E89</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="25_3_1_3_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:25_3_1_3_2_1"/><i><font color="Green">E90</font></i>: 
<font color="Maroon">O</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 ;<br/><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">O</font></span><a href="tarski.html#K1">}</a></span> = 
<span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="subset_1.html#K6">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">O</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">O</font></span><a href="tarski.html#K1">}</a></span>
<b>by </b><i><a class="txt" href="finset_1.html#E1:25"><i><font color="Green">E26</font></i></a>, , </i>
<br/>.= 
<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">O</font></span><a href="tarski.html#K1">}</a></span>
<b>by </b><i><a class="ref" href="xboole_1.html#T39">XBOOLE_1:39</a></i>
<br/>.= 
<font color="Maroon">B</font>
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T46">ZFMISC_1:46</a></i>
;<br/><b>hence </b><a NAME="E3:25_3_1_3_2_1"/>
contradiction
 <b>by </b><i>, <a class="txt" href="finset_1.html#E3"><i><font color="Green">Lemma33</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E10:25_3_1_3_2"/>
<font color="Maroon">B</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">O</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_3_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="E1:25_3_1_3_2_2"/><i><font color="Green">E91</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 ;<br/>

<a NAME="E2:25_3_1_3_2_2"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">O</font> or <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">O</font></span><a href="tarski.html#K1">}</a></span> )
 
<b>by </b><i>, <a class="txt" href="finset_1.html#E3"><i><font color="Green">Lemma33</font></i></a>, , <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<b>hence </b><a NAME="E3:25_3_1_3_2_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">O</font>
 <b>by </b><i>, , <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E11:25_3_1_3_2"/><b>then </b>
( <font color="Maroon">B</font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">O</font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font> &amp; <font color="Maroon">B</font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">O</font> )
 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T65">ZFMISC_1:65</a></i>;<br/>
<a NAME="E12:25_3_1_3_2"/><b>then </b>
<font color="Maroon">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X2</font>
 
<b>by </b><i>, <a class="txt" href="finset_1.html#E3"><i><font color="Green">Lemma33</font></i></a></i>;<br/>
<b>hence </b><a NAME="E13:25_3_1_3_2"/>
<font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="txt" href="finset_1.html#E1:25"><i><font color="Green">E26</font></i></a>, <a class="txt" href="finset_1.html#E5:25"><i><font color="Green">E35</font></i></a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E5:25_3_1_3"/>
 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">X</font> &amp; ( for <font color="Olive">B</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Olive">x</font> <a href="tarski.html#R1">c=</a> <font color="Olive">B</font> holds <br/><font color="Olive">B</font> <a href="hidden.html#R1">=</a> <font color="Olive">x</font> ) )
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E25:25_3_1"/>
 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">X</font> &amp; ( for <font color="Olive">B</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Olive">x</font> <a href="tarski.html#R1">c=</a> <font color="Olive">B</font> holds <br/><font color="Olive">B</font> <a href="hidden.html#R1">=</a> <font color="Olive">x</font> ) )
 <b>by </b><i/>;<br/>


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


</div><b>end;</b></div>
<a NAME="E17:25"/><i><font color="Green">E92</font></i>: 
for <font color="Olive">O</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  st <font color="Olive">O</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Olive">O</font> <a href="ordinal1.html#NR2" target="_self">is_limit_ordinal</a>  &amp; ( for <font color="Olive">B</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  st <font color="Olive">B</font> <a href="hidden.html#R2">in</a> <font color="Olive">O</font> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">B</font>] ) holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">O</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">O</font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:25_4"/><i><font color="Green">E93</font></i>: 
<font color="Maroon">O</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>and </b><br/><a NAME="E2:25_4"/><i><font color="Green">E94</font></i>: 
<font color="Maroon">O</font> <a href="ordinal1.html#NR2" target="_self">is_limit_ordinal</a> 
 <b>and </b><br/><a NAME="E3:25_4"/>
for <font color="Olive">B</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  st <font color="Olive">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">O</font> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">B</font>]
 <b>and </b><br/><a NAME="E4:25_4"/><i><font color="Green">E95</font></i>: 
<font color="Maroon">O</font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K5">omega</a> 
 ;<br/>

<a NAME="E5:25_4"/>
 <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">O</font>
 
<b>by </b><i>, <a class="ref" href="ordinal1.html#T24">ORDINAL1:24</a></i>;<br/>
<a NAME="E6:25_4"/><b>then </b>
 <a href="ordinal1.html#K5">omega</a>  <a href="ordinal1.html#R1">c=</a> <font color="Maroon">O</font>
 
<b>by </b><i>, <a class="ref" href="ordinal1.html#D12">ORDINAL1:def 12</a></i>;<br/>
<a NAME="E7:25_4"/><b>then </b>
<font color="Maroon">O</font> <a href="hidden.html#R2">in</a> <font color="Maroon">O</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E8:25_4"/>
for <font color="Olive">X</font> being  <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">O</font>  st <font color="Olive">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  holds <br/> 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="Olive">X</font> &amp; ( for <font color="Olive">B</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">B</font> <a href="hidden.html#R2">in</a> <font color="Olive">X</font> &amp; <font color="Olive">x</font> <a href="tarski.html#R1">c=</a> <font color="Olive">B</font> holds <br/><font color="Olive">B</font> <a href="hidden.html#R1">=</a> <font color="Olive">x</font> ) )
 ;<br/>


</div><b>end;</b></div>
<a NAME="E18:25"/><i><font color="Green">E96</font></i>: 
for <font color="Olive">O</font> being  <a href="ordinal1.html#NM2">Ordinal</a> holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">O</font>]
 
<b>from </b><i><a class="ref" href="ordinal2.html#S1">ORDINAL2:sch 1</a>(, , );<br/></i>
<a NAME="E19:25"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">p</font> is   <a href="ordinal1.html#NM2">Ordinal</a>
 
<b>by </b><i>, <a class="ref" href="ordinal1.html#T23">ORDINAL1:23</a></i>;<br/>
<b>then consider </b><font color="Maroon">g</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E21:25"/><i><font color="Green">E98</font></i>: 
( <font color="Maroon">g</font> <a href="hidden.html#R2">in</a> <font color="Maroon">GX</font> &amp; ( for <font color="Olive">B</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">GX</font> &amp; <font color="Maroon">g</font> <a href="tarski.html#R1">c=</a> <font color="Olive">B</font> holds <br/><font color="Olive">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> ) )
 <b>by </b><i>, , </i>;<br/>
<b>take </b>
<font color="Maroon">p</font> <a href="relat_1.html#K9">.:</a> <font color="Maroon">g</font>
;<br/>

<a NAME="E22:25"/>
for <font color="Olive">B</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">p</font> <a href="relat_1.html#K9">.:</a> <font color="Maroon">g</font> <a href="tarski.html#R1">c=</a> <font color="Olive">B</font> holds <br/><font color="Olive">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="relat_1.html#K9">.:</a> <font color="Maroon">g</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_5"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:25_5"/><i><font color="Green">E99</font></i>: 
<font color="Maroon">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/>

<b>assume </b><a NAME="E2:25_5"/>
<font color="Maroon">p</font> <a href="relat_1.html#K9">.:</a> <font color="Maroon">g</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font>
 ;<br/>

<a NAME="E3:25_5"/><b>then </b><i><font color="Green">E100</font></i>: 
<font color="Maroon">p</font> <a href="relat_1.html#K10">"</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="relat_1.html#K9">.:</a> <font color="Maroon">g</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">p</font> <a href="relat_1.html#K10">"</a> <font color="Maroon">B</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T178">RELAT_1:178</a></i>;<br/>
<a NAME="E4:25_5"/>
<font color="Maroon">g</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">p</font> <a href="relat_1.html#K10">"</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="relat_1.html#K9">.:</a> <font color="Maroon">g</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#T146">FUNCT_1:146</a></i>;<br/>
<a NAME="E5:25_5"/><b>then </b><i><font color="Green">E101</font></i>: 
<font color="Maroon">g</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">p</font> <a href="relat_1.html#K10">"</a> <font color="Maroon">B</font>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E6:25_5"/><i><font color="Green">E102</font></i>: 
<font color="Maroon">p</font> <a href="relat_1.html#K9">.:</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="relat_1.html#K10">"</a> <font color="Maroon">B</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font>
 
<b>by </b><i>, , <a class="ref" href="funct_1.html#T147">FUNCT_1:147</a></i>;<br/>
<a NAME="E7:25_5"/>
<font color="Maroon">p</font> <a href="relat_1.html#K10">"</a> <font color="Maroon">B</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">p</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T167">RELAT_1:167</a></i>;<br/>
<a NAME="E8:25_5"/><b>then </b>
<font color="Maroon">p</font> <a href="relat_1.html#K10">"</a> <font color="Maroon">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">GX</font>
 
<b>by </b><i><a class="ref" href="finset_1.html#D1" target="_self">Def1</a>, , </i>;<br/>
<b>hence </b><a NAME="E9:25_5"/>
<font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="relat_1.html#K9">.:</a> <font color="Maroon">g</font>
 <b>by </b><i>, , </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E23:25"/>
( <font color="Maroon">p</font> <a href="relat_1.html#K9">.:</a> <font color="Maroon">g</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; ( for <font color="Olive">B</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">p</font> <a href="relat_1.html#K9">.:</a> <font color="Maroon">g</font> <a href="tarski.html#R1">c=</a> <font color="Olive">B</font> holds <br/><font color="Olive">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="relat_1.html#K9">.:</a> <font color="Maroon">g</font> ) )
 <b>by </b><i><a class="ref" href="finset_1.html#D1" target="_self">Def1</a>, </i>;<br/>


</div>
