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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:26"/><i><font color="Green">E9</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is <a href="finset_1.html#V1" target="_self">finite</a>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>assume </b><a NAME="E2:26"/><i><font color="Green">E10</font></i>: 
<font color="Maroon">c<sub>2</sub></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">c<sub>3</sub></font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E4:26"/><i><font color="Green">E11</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 <b>and </b><br/><a NAME="E5:26"/><i><font color="Green">E12</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K5">omega</a> 
 <b>by </b><i><a class="txt" href="finset_1.html#E1:26"><i><font color="Green">E9</font></i></a>, <a class="ref" href="finset_1.html#D1" target="_self">Def1</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ] means <font color="Maroon">c<sub>3</sub></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">c<sub>2</sub></font>;<br/>
<b>consider </b><font color="Maroon">c<sub>4</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E7:26"/><i><font color="Green">E13</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> iff ( <font color="Olive">b<sub>1</sub></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">c<sub>3</sub></font></span>)</span> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] ) )
 <b>from </b><i><a class="ref" href="xboole_0.html#S1">XBOOLE_0:sch 1</a>();<br/></i>
<a NAME="E8:26"/>
<font color="Maroon">c<sub>4</sub></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">c<sub>3</sub></font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</sub></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="E1:26_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<b>hence </b><a NAME="E2:26_1"/>
<font color="Maroon">c<sub>5</sub></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">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="finset_1.html#E7:26"><i><font color="Green">E13</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</sub></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">c<sub>3</sub></font></span>)</span> ;<br/>
<b>consider </b><font color="Maroon">c<sub>6</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>
<a NAME="E11:26"/>
<font color="Maroon">c<sub>6</sub></font> is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="finset_1.html#E2:26"><i><font color="Green">E10</font></i></a>, <a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<a NAME="E12:26"/><b>then </b><i><font color="Green">E14</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="relat_1.html#K9">.:</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="relat_1.html#K10">"</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="finset_1.html#E4:26"><i><font color="Green">E11</font></i></a>, <a class="ref" href="funct_1.html#T147">FUNCT_1:147</a></i>;<br/>
<a NAME="E13:26"/>
<font color="Maroon">c<sub>3</sub></font> <a href="relat_1.html#K10">"</a> <font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T167">RELAT_1:167</a></i>;<br/>
<a NAME="E14:26"/><b>then </b><i><font color="Green">E15</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="txt" href="finset_1.html#E2:26"><i><font color="Green">E10</font></i></a>, <a class="txt" href="finset_1.html#E7:26"><i><font color="Green">E13</font></i></a>, <a class="txt" href="finset_1.html#E12:26"><i><font color="Green">E14</font></i></a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1">set</a> ] means ( <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">b<sub>1</sub></font> being  <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">a<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  holds <br/>ex <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> &amp; ( for <font color="Olive">b<sub>3</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>3</sub></font> holds <br/><font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> ) ) );<br/>
<a NAME="E15:26"/><i><font color="Green">E16</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="26_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:26_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">c<sub>7</sub></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="E2:26_2"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<a NAME="E3:26_2"/><b>then </b><i><font color="Green">E17</font></i>: 
<font color="Maroon">c<sub>7</sub></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="E4:26_2"/>
(  <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> &amp;  <a href="xboole_0.html#K1">{}</a>  <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  ) )
 <b>by </b><i><a class="txt" href="finset_1.html#E3:26_2"><i><font color="Green">E17</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E16:26"/><i><font color="Green">E17</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[ <a href="ordinal1.html#K1">succ</a> <font color="Olive">b<sub>1</sub></font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>7</sub></font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/>

<b>assume </b><a NAME="E1:26_3"/><i><font color="Green">E18</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K5">omega</a>  implies for <font color="Olive">b<sub>1</sub></font> being  <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">c<sub>7</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  holds <br/>ex <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> &amp; ( for <font color="Olive">b<sub>3</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>3</sub></font> holds <br/><font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> ) ) )
 ;<br/>

<b>thus </b><a NAME="E2:26_3"/>
(  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K5">omega</a>  implies for <font color="Olive">b<sub>1</sub></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">c<sub>7</sub></font></span>)</span> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  holds <br/>ex <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> &amp; ( for <font color="Olive">b<sub>3</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>3</sub></font> holds <br/><font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> ) ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_3_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:26_3_1"/>
 <a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K5">omega</a> 
 ;<br/>

<a NAME="E2:26_3_1"/><b>then </b><i><font color="Green">E19</font></i>: 
 <a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>7</sub></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">c<sub>8</sub></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">c<sub>7</sub></font></span>)</span>;<br/>

<b>assume </b><a NAME="E3:26_3_1"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">c<sub>8</sub></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> ] means ex <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="tarski.html#K1">}</a></span> );<br/>
<b>consider </b><font color="Maroon">c<sub>9</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:26_3_1"/><i><font color="Green">E21</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font> iff ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">b<sub>1</sub></font>] ) )
 <b>from </b><i><a class="ref" href="xboole_0.html#S1">XBOOLE_0:sch 1</a>();<br/></i>
<a NAME="E6:26_3_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="tarski.html#R1">c=</a>  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">c<sub>7</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_3_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>10</sub></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="E1:26_3_1_1"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font>
 ;<br/>

<b>hence </b><a NAME="E2:26_3_1_1"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="finset_1.html#E5:26_3_1"><i><font color="Green">E21</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>9</sub></font> as   <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">c<sub>7</sub></font> ;<br/>
<b>consider </b><font color="Maroon">c<sub>11</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>8</sub></font>;<br/>
<a NAME="E9:26_3_1"/><i><font color="Green">E22</font></i>: 
 <a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>7</sub></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="E10:26_3_1"/>
<font color="Maroon">c<sub>11</sub></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">c<sub>7</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="finset_1.html#E3:26_3_1"><i><font color="Green">E20</font></i></a>, <a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<a NAME="E11:26_3_1"/><b>then </b>
<font color="Maroon">c<sub>11</sub></font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>7</sub></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">c<sub>7</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>7</sub></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">c<sub>7</sub></font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="txt" href="finset_1.html#E9:26_3_1"><i><font color="Green">E22</font></i></a>, <a class="ref" href="xboole_1.html#T33">XBOOLE_1:33</a></i>;<br/>
<a NAME="E12:26_3_1"/><b>then </b><i><font color="Green">E23</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>7</sub></font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>7</sub></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="E13:26_3_1"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="ref" href="ordinal1.html#T10">ORDINAL1:10</a></i>;<br/>
<a NAME="E14:26_3_1"/><i><font color="Green">E25</font></i>: 
not <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font>
 
;<br/>
<a NAME="E15:26_3_1"/><b>then </b>
<font color="Maroon">c<sub>11</sub></font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="finset_1.html#E12:26_3_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="zfmisc_1.html#T65">ZFMISC_1:65</a></i>;<br/>
<a NAME="E16:26_3_1"/><b>then </b>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="txt" href="finset_1.html#E3:26_3_1"><i><font color="Green">E20</font></i></a>, <a class="txt" href="finset_1.html#E5:26_3_1"><i><font color="Green">E21</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>12</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E18:26_3_1"/><i><font color="Green">E26</font></i>: 
( <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> ) )
 <b>by </b><i><a class="txt" href="finset_1.html#E1:26_3"><i><font color="Green">E18</font></i></a>, <a class="txt" href="finset_1.html#E2:26_3_1"><i><font color="Green">E19</font></i></a>, <a class="txt" href="finset_1.html#E13:26_3_1"><i><font color="Green">E24</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>13</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E20:26_3_1"/><i><font color="Green">E27</font></i>: 
( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="tarski.html#K1">}</a></span> )
 <b>by </b><i><a class="txt" href="finset_1.html#E5:26_3_1"><i><font color="Green">E21</font></i></a>, <a class="txt" href="finset_1.html#E18:26_3_1"><i><font color="Green">E26</font></i></a></i>;<br/>
<a NAME="E21:26_3_1"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>7</sub></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="E22:26_3_1"/><b>then </b><i><font color="Green">E29</font></i>: 
not <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>12</sub></font>
 
<b>by </b><i><a class="txt" href="finset_1.html#E20:26_3_1"><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">E30</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="26_3_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:26_3_1_2"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/><b>take </b><font color="Maroon">c<sub>14</sub></font> = <font color="Maroon">c<sub>12</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="tarski.html#K1">}</a></span>;<br/><a NAME="E2:26_3_1_2"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_3_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>15</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:26_3_1_2_1"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/>

<b>assume </b><a NAME="E2:26_3_1_2_1"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>15</sub></font>
 ;<br/>

<div><i><font color="Green">E34</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="26_3_1_2_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:26_3_1_2_1_1"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>15</sub></font>
 ;<br/><b>set </b><font color="Maroon">c<sub>16</sub></font> = <font color="Maroon">c<sub>15</sub></font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="tarski.html#K1">}</a></span>;<br/><a NAME="E2:26_3_1_2_1_1"/>
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>15</sub></font>
 <b>by </b><i><a class="txt" href="finset_1.html#E1:26_3_1_2_1_1"><i><font color="Green">E35</font></i></a>, <a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/><a NAME="E3:26_3_1_2_1_1"/><b>then </b><i><font color="Green">E36</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="subset_1.html#K6">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>7</sub></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">c<sub>7</sub></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="E4:26_3_1_2_1_1"/>
<font color="Maroon">c<sub>14</sub></font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>15</sub></font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="finset_1.html#E2:26_3_1_2_1"><i><font color="Green">E33</font></i></a>, <a class="ref" href="xboole_1.html#T33">XBOOLE_1:33</a></i>;<br/><a NAME="E5:26_3_1_2_1_1"/><b>then </b>
( <font color="Maroon">c<sub>12</sub></font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>15</sub></font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="tarski.html#K1">}</a></span> &amp; not <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>12</sub></font> )
 <b>by </b><i><a class="txt" href="finset_1.html#E20:26_3_1"><i><font color="Green">E27</font></i></a>, <a class="txt" href="finset_1.html#E21:26_3_1"><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="E6:26_3_1_2_1_1"/><b>then </b><i><font color="Green">E37</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>15</sub></font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>7</sub></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="E7:26_3_1_2_1_1"/>
<font color="Maroon">c<sub>15</sub></font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>7</sub></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">c<sub>7</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>7</sub></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">c<sub>7</sub></font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="finset_1.html#E9:26_3_1"><i><font color="Green">E22</font></i></a>, <a class="txt" href="finset_1.html#E1:26_3_1_2_1"><i><font color="Green">E32</font></i></a>, <a class="ref" href="xboole_1.html#T33">XBOOLE_1:33</a></i>;<br/><a NAME="E8:26_3_1_2_1_1"/><b>then </b>
<font color="Maroon">c<sub>15</sub></font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>7</sub></font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>7</sub></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="E9:26_3_1_2_1_1"/><b>then </b>
<font color="Maroon">c<sub>15</sub></font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="finset_1.html#E14:26_3_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="zfmisc_1.html#T65">ZFMISC_1:65</a></i>;<br/><a NAME="E10:26_3_1_2_1_1"/><b>then </b>
<font color="Maroon">c<sub>15</sub></font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="finset_1.html#E5:26_3_1"><i><font color="Green">E21</font></i></a>, <a class="txt" href="finset_1.html#E1:26_3_1_2_1"><i><font color="Green">E32</font></i></a></i>;<br/><b>hence </b><a NAME="E11:26_3_1_2_1_1"/>
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font>
 <b>by </b><i><a class="txt" href="finset_1.html#E18:26_3_1"><i><font color="Green">E26</font></i></a>, <a class="txt" href="finset_1.html#E3:26_3_1_2_1_1"><i><font color="Green">E36</font></i></a>, <a class="txt" href="finset_1.html#E6:26_3_1_2_1_1"><i><font color="Green">E37</font></i></a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_3_1_2_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:26_3_1_2_1_2"/><i><font color="Green">E35</font></i>: 
not <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>15</sub></font>
 ;<br/><a NAME="E2:26_3_1_2_1_2"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>7</sub></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="E3:26_3_1_2_1_2"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>14</sub></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="E4:26_3_1_2_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="finset_1.html#E2:26_3_1_2_1"><i><font color="Green">E33</font></i></a>, <a class="txt" href="finset_1.html#E1:26_3_1_2_1_2"><i><font color="Green">E35</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:26_3_1_2_1"/>
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font>
 <b>by </b><i><a class="txt" href="finset_1.html#E3:26_3_1_2_1"><i><font color="Green">E34</font></i></a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E3:26_3_1_2"/>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> &amp; ( for <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>2</sub></font> holds <br/><font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> ) )
 <b>by </b><i><a class="txt" href="finset_1.html#E1:26_3_1_2"><i><font color="Green">E31</font></i></a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_3_1_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:26_3_1_3"/><i><font color="Green">E31</font></i>: 
not <font color="Maroon">c<sub>12</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/><b>take </b><font color="Maroon">c<sub>14</sub></font> = <font color="Maroon">c<sub>12</sub></font>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_3_1_3_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:26_3_1_3_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>13</sub></font>
 ;<br/><b>then </b><font color="Maroon">c<sub>13</sub></font> = 
<font color="Maroon">c<sub>13</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>7</sub></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">c<sub>12</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="tarski.html#K1">}</a></span>
<b>by </b><i><a class="txt" href="finset_1.html#E20:26_3_1"><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="E3:26_3_1_3_1"/>
contradiction
 <b>by </b><i><a class="txt" href="finset_1.html#E20:26_3_1"><i><font color="Green">E27</font></i></a>, <a class="txt" href="finset_1.html#E1:26_3_1_3"><i><font color="Green">E31</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E3:26_3_1_3"/><b>then </b><i><font color="Green">E32</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="finset_1.html#E20:26_3_1"><i><font color="Green">E27</font></i></a>, <a class="ref" href="zfmisc_1.html#T65">ZFMISC_1:65</a></i>;<br/><a NAME="E4:26_3_1_3"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_3_1_3_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>15</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:26_3_1_3_2"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/>

<a NAME="E2:26_3_1_3_2"/><b>then </b>
<font color="Maroon">c<sub>15</sub></font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>7</sub></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">c<sub>7</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>7</sub></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">c<sub>7</sub></font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="txt" href="finset_1.html#E9:26_3_1"><i><font color="Green">E22</font></i></a>, <a class="ref" href="xboole_1.html#T33">XBOOLE_1:33</a></i>;<br/>
<a NAME="E3:26_3_1_3_2"/><b>then </b>
<font color="Maroon">c<sub>15</sub></font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>7</sub></font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>7</sub></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="E4:26_3_1_3_2"/><b>then </b>
<font color="Maroon">c<sub>15</sub></font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="finset_1.html#E14:26_3_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="zfmisc_1.html#T65">ZFMISC_1:65</a></i>;<br/>
<a NAME="E5:26_3_1_3_2"/><b>then </b><i><font color="Green">E34</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font>
 
<b>by </b><i><a class="txt" href="finset_1.html#E5:26_3_1"><i><font color="Green">E21</font></i></a>, <a class="txt" href="finset_1.html#E1:26_3_1_3_2"><i><font color="Green">E33</font></i></a></i>;<br/>
<b>assume </b><a NAME="E6:26_3_1_3_2"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>15</sub></font>
 ;<br/>

<a NAME="E7:26_3_1_3_2"/><b>then </b>
<font color="Maroon">c<sub>14</sub></font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>15</sub></font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>7</sub></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="E8:26_3_1_3_2"/><b>then </b><i><font color="Green">E36</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>15</sub></font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="txt" href="finset_1.html#E22:26_3_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="zfmisc_1.html#T65">ZFMISC_1:65</a></i>;<br/>
<div><i><font color="Green">E37</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="26_3_1_3_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:26_3_1_3_2_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>15</sub></font>
 ;<br/><font color="Maroon">c<sub>14</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="tarski.html#K1">}</a></span> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="subset_1.html#K6">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>7</sub></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">c<sub>7</sub></font></span><a href="tarski.html#K1">}</a></span>
<b>by </b><i><a class="txt" href="finset_1.html#E18:26_3_1"><i><font color="Green">E26</font></i></a>, <a class="txt" href="finset_1.html#E5:26_3_1_3_2"><i><font color="Green">E34</font></i></a>, <a class="txt" href="finset_1.html#E8:26_3_1_3_2"><i><font color="Green">E36</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>15</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>7</sub></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">c<sub>15</sub></font>
<b>by </b><i><a class="txt" href="finset_1.html#E1:26_3_1_3_2_1"><i><font color="Green">E38</font></i></a>, <a class="ref" href="zfmisc_1.html#T46">ZFMISC_1:46</a></i>
;<br/><b>hence </b><a NAME="E3:26_3_1_3_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="finset_1.html#E1:26_3_1_3"><i><font color="Green">E31</font></i></a>, <a class="txt" href="finset_1.html#E1:26_3_1_3_2"><i><font color="Green">E33</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E10:26_3_1_3_2"/>
<font color="Maroon">c<sub>15</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>7</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_3_1_3_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>16</sub></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="E1:26_3_1_3_2_2"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>15</sub></font>
 ;<br/>

<a NAME="E2:26_3_1_3_2_2"/>
( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> or <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="tarski.html#K1">}</a></span> )
 
<b>by </b><i><a class="txt" href="finset_1.html#E9:26_3_1"><i><font color="Green">E22</font></i></a>, <a class="txt" href="finset_1.html#E1:26_3_1_3_2"><i><font color="Green">E33</font></i></a>, <a class="txt" href="finset_1.html#E1:26_3_1_3_2_2"><i><font color="Green">E38</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<b>hence </b><a NAME="E3:26_3_1_3_2_2"/>
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="finset_1.html#E9:26_3_1_3_2"><i><font color="Green">E37</font></i></a>, <a class="txt" href="finset_1.html#E1:26_3_1_3_2_2"><i><font color="Green">E38</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E11:26_3_1_3_2"/><b>then </b>
( <font color="Maroon">c<sub>15</sub></font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">c<sub>7</sub></font> )
 
<b>by </b><i><a class="txt" href="finset_1.html#E9:26_3_1_3_2"><i><font color="Green">E37</font></i></a>, <a class="ref" href="zfmisc_1.html#T65">ZFMISC_1:65</a></i>;<br/>
<a NAME="E12:26_3_1_3_2"/><b>then </b>
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font>
 
<b>by </b><i><a class="txt" href="finset_1.html#E5:26_3_1"><i><font color="Green">E21</font></i></a>, <a class="txt" href="finset_1.html#E1:26_3_1_3_2"><i><font color="Green">E33</font></i></a></i>;<br/>
<b>hence </b><a NAME="E13:26_3_1_3_2"/>
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font>
 <b>by </b><i><a class="txt" href="finset_1.html#E18:26_3_1"><i><font color="Green">E26</font></i></a>, <a class="txt" href="finset_1.html#E6:26_3_1_3_2"><i><font color="Green">E35</font></i></a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E5:26_3_1_3"/>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> &amp; ( for <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>2</sub></font> holds <br/><font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> ) )
 <b>by </b><i><a class="txt" href="finset_1.html#E3:26_3_1_3"><i><font color="Green">E32</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E25:26_3_1"/>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> &amp; ( for <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>2</sub></font> holds <br/><font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> ) )
 <b>by </b><i><a class="txt" href="finset_1.html#E23:26_3_1"><i><font color="Green">E30</font></i></a></i>;<br/>


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


</div><b>end;</b></div>
<a NAME="E17:26"/><i><font color="Green">E18</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Olive">b<sub>1</sub></font> <a href="ordinal1.html#NR2" target="_self">is_limit_ordinal</a>  &amp; ( for <font color="Olive">b<sub>2</sub></font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>2</sub></font>] ) holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>7</sub></font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:26_4"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>and </b><br/><a NAME="E2:26_4"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="ordinal1.html#NR2" target="_self">is_limit_ordinal</a> 
 <b>and </b><br/><a NAME="E3:26_4"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 <b>and </b><br/><a NAME="E4:26_4"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K5">omega</a> 
 ;<br/>

<a NAME="E5:26_4"/>
 <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="finset_1.html#E1:26_4"><i><font color="Green">E19</font></i></a>, <a class="ref" href="ordinal1.html#T24">ORDINAL1:24</a></i>;<br/>
<a NAME="E6:26_4"/><b>then </b>
 <a href="ordinal1.html#K5">omega</a>  <a href="ordinal1.html#R1">c=</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="finset_1.html#E2:26_4"><i><font color="Green">E20</font></i></a>, <a class="ref" href="ordinal1.html#D12">ORDINAL1:def 12</a></i>;<br/>
<a NAME="E7:26_4"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="finset_1.html#E4:26_4"><i><font color="Green">E21</font></i></a></i>;<br/>
<b>hence </b><a NAME="E8:26_4"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">c<sub>7</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  holds <br/>ex <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> &amp; ( for <font color="Olive">b<sub>3</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>3</sub></font> holds <br/><font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> ) )
 ;<br/>


</div><b>end;</b></div>
<a NAME="E18:26"/><i><font color="Green">E19</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="ordinal1.html#NM2">Ordinal</a> holds <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 
<b>from </b><i><a class="ref" href="ordinal2.html#S1">ORDINAL2:sch 1</a>(<a class="txt" href="finset_1.html#E15:26"><i><font color="Green">E16</font></i></a>, <a class="txt" href="finset_1.html#E16:26"><i><font color="Green">E17</font></i></a>, <a class="txt" href="finset_1.html#E17:26"><i><font color="Green">E18</font></i></a>);<br/></i>
<a NAME="E19:26"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>3</sub></font> is   <a href="ordinal1.html#NM2">Ordinal</a>
 
<b>by </b><i><a class="txt" href="finset_1.html#E5:26"><i><font color="Green">E12</font></i></a>, <a class="ref" href="ordinal1.html#T23">ORDINAL1:23</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>7</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E21:26"/><i><font color="Green">E20</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> ) )
 <b>by </b><i><a class="txt" href="finset_1.html#E5:26"><i><font color="Green">E12</font></i></a>, <a class="txt" href="finset_1.html#E14:26"><i><font color="Green">E15</font></i></a>, <a class="txt" href="finset_1.html#E18:26"><i><font color="Green">E19</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>3</sub></font> <a href="relat_1.html#K9">.:</a> <font color="Maroon">c<sub>7</sub></font>
;<br/>

<a NAME="E22:26"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="relat_1.html#K9">.:</a> <font color="Maroon">c<sub>7</sub></font> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="relat_1.html#K9">.:</a> <font color="Maroon">c<sub>7</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>8</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:26_5"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<b>assume </b><a NAME="E2:26_5"/>
<font color="Maroon">c<sub>3</sub></font> <a href="relat_1.html#K9">.:</a> <font color="Maroon">c<sub>7</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/>

<a NAME="E3:26_5"/><b>then </b><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="relat_1.html#K10">"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="relat_1.html#K9">.:</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font> <a href="relat_1.html#K10">"</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T178">RELAT_1:178</a></i>;<br/>
<a NAME="E4:26_5"/>
<font color="Maroon">c<sub>7</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font> <a href="relat_1.html#K10">"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="relat_1.html#K9">.:</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="finset_1.html#E21:26"><i><font color="Green">E20</font></i></a>, <a class="ref" href="funct_1.html#T146">FUNCT_1:146</a></i>;<br/>
<a NAME="E5:26_5"/><b>then </b><i><font color="Green">E23</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font> <a href="relat_1.html#K10">"</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="txt" href="finset_1.html#E3:26_5"><i><font color="Green">E22</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E6:26_5"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="relat_1.html#K9">.:</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="relat_1.html#K10">"</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="txt" href="finset_1.html#E4:26"><i><font color="Green">E11</font></i></a>, <a class="txt" href="finset_1.html#E1:26_5"><i><font color="Green">E21</font></i></a>, <a class="ref" href="funct_1.html#T147">FUNCT_1:147</a></i>;<br/>
<a NAME="E7:26_5"/>
<font color="Maroon">c<sub>3</sub></font> <a href="relat_1.html#K10">"</a> <font color="Maroon">c<sub>8</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T167">RELAT_1:167</a></i>;<br/>
<a NAME="E8:26_5"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="relat_1.html#K10">"</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="finset_1.html#E7:26"><i><font color="Green">E13</font></i></a>, <a class="txt" href="finset_1.html#E1:26_5"><i><font color="Green">E21</font></i></a>, <a class="txt" href="finset_1.html#E6:26_5"><i><font color="Green">E24</font></i></a></i>;<br/>
<b>hence </b><a NAME="E9:26_5"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="relat_1.html#K9">.:</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="finset_1.html#E21:26"><i><font color="Green">E20</font></i></a>, <a class="txt" href="finset_1.html#E5:26_5"><i><font color="Green">E23</font></i></a>, <a class="txt" href="finset_1.html#E6:26_5"><i><font color="Green">E24</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E23:26"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="relat_1.html#K9">.:</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="relat_1.html#K9">.:</a> <font color="Maroon">c<sub>7</sub></font> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="relat_1.html#K9">.:</a> <font color="Maroon">c<sub>7</sub></font> ) )
 <b>by </b><i><a class="txt" href="finset_1.html#E7:26"><i><font color="Green">E13</font></i></a>, <a class="txt" href="finset_1.html#E21:26"><i><font color="Green">E20</font></i></a></i>;<br/>


</div>
