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<b>let </b><font color="Maroon">a</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">A</font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/>



<b>assume </b><a NAME="E1:22"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 ;<br/>

<a NAME="E2:22"/><b>then </b><i><font color="Green">E21</font></i>: 
<font color="Maroon">a</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font>
 
<b>by </b><i/>;<br/>
<a NAME="E3:22"/><b>then </b>
for <font color="Olive">y</font>, <font color="Olive">z</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a</font> &amp; <font color="Olive">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a</font> &amp; not <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Olive">z</font> &amp; not <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">z</font> holds <br/><font color="Olive">z</font> <a href="hidden.html#R2">in</a> <font color="Olive">y</font>
 
<b>by </b><i/>;<br/>
<a NAME="E4:22"/><b>then </b><i><font color="Green">E22</font></i>: 
<font color="Maroon">a</font> is <a href="ordinal1.html#V2" target="_self">epsilon-connected</a>
 
<b>by </b><i/>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:22_1"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a</font>
 ;<br/><b>assume </b><a NAME="E2:22_1"/>
not <font color="Maroon">y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">a</font>
 ;<br/><b>then consider </b><font color="Maroon">b</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:22_1"/><i><font color="Green">E24</font></i>: 
( <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">y</font> &amp; not <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><a NAME="E5:22_1"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">y</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">a</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></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="E7:22_1"/><i><font color="Green">E25</font></i>: 
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">y</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">a</font> &amp; ( for <font color="Olive">c</font> being   <a href="hidden.html#M1">set</a>   holds <br/>(  not <font color="Olive">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">y</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">a</font> or  not <font color="Olive">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">z</font> ) ) )
 <b>by </b><i><a class="ref" href="tarski.html#T7">TARSKI:7</a></i>;<br/><a NAME="E8:22_1"/><i><font color="Green">E26</font></i>: 
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">y</font> &amp; not <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a</font> )
 <b>by </b><i><a class="ref" href="ordinal1.html#T3" target="_self">Th3</a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E9:22_1"/>
<font color="Maroon">y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font>
 <b>by </b><i>, , </i>;<br/><a NAME="E10:22_1"/><b>then </b>
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a</font> or <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> or <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">z</font> )
 <b>by </b><i>, <a class="txt" href="ordinal1.html#E1:22"><i><font color="Green">E19</font></i></a>, </i>;<br/><b>hence </b><a NAME="E11:22_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="ordinal1.html#T3" target="_self">Th3</a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a>, </i>;<br/></div><b>end;</b></div>
<a NAME="E6:22"/><b>then </b>
<font color="Maroon">a</font> is <a href="ordinal1.html#V1" target="_self">epsilon-transitive</a>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E7:22"/>
<font color="Maroon">a</font> is   <a href="ordinal1.html#NM2">Ordinal</a>
 <b>by </b><i>, </i>;<br/>


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