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

<b>assume </b><a NAME="E1:30"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">x</font> is   <a href="ordinal1.html#NM2">Ordinal</a>
 ;<br/>

<div><i><font color="Green">E21</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="30_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="E1:30_1"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K1" target="_self">succ</a> <font color="Maroon">x</font>
 ;<br/><a NAME="E2:30_1"/>
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> implies <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E3:30_1"/>
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font> or <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="txt" href="ordinal1.html#E1:30"><i><font color="Green">E19</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:30_2"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K1" target="_self">succ</a> <font color="Maroon">x</font>
 ;<br/><div><i><font color="Green">E24</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="30_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:30_2_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font>
 ;<br/><a NAME="E2:30_2_1"/><b>then </b><i><font color="Green">E25</font></i>: 
<font color="Maroon">y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">x</font>
 <b>by </b><i>, </i>;<br/><a NAME="E3:30_2_1"/>
<font color="Maroon">x</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">x</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/><b>hence </b><a NAME="E4:30_2_1"/>
<font color="Maroon">y</font> <a href="tarski.html#R1">c=</a>  <a href="ordinal1.html#K1" target="_self">succ</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="txt" href="ordinal1.html#E2:30"><i><font color="Green">E21</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/></div><b>end;</b></div><a NAME="E3:30_2"/>
( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> implies <font color="Maroon">y</font> <a href="tarski.html#R1">c=</a>  <a href="ordinal1.html#K1" target="_self">succ</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/><b>hence </b><a NAME="E4:30_2"/>
<font color="Maroon">y</font> <a href="tarski.html#R1">c=</a>  <a href="ordinal1.html#K1" target="_self">succ</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="ordinal1.html#T3" target="_self">Th3</a>, <a class="txt" href="ordinal1.html#E1:30"><i><font color="Green">E19</font></i></a>, </i>;<br/></div><b>end;</b></div>
<a NAME="E4:30"/><b>then </b><i><font color="Green">E26</font></i>: 
 <a href="ordinal1.html#K1" target="_self">succ</a> <font color="Maroon">x</font> is <a href="ordinal1.html#V1" target="_self">epsilon-transitive</a>
 
<b>by </b><i/>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_3"><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>let </b><font color="Maroon">z</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:30_3"/><i><font color="Green">E27</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K1" target="_self">succ</a> <font color="Maroon">x</font> &amp; <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K1" target="_self">succ</a> <font color="Maroon">x</font> )
 ;<br/><a NAME="E2:30_3"/><b>then </b><i><font color="Green">E28</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font> or <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="ordinal1.html#T3" target="_self">Th3</a></i>;<br/><a NAME="E3:30_3"/>
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font> or <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="ordinal1.html#T3" target="_self">Th3</a>, </i>;<br/><b>hence </b><a NAME="E4:30_3"/>
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">z</font> or <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font> or <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">y</font> )
 <b>by </b><i>, <a class="txt" href="ordinal1.html#E2:30"><i><font color="Green">E21</font></i></a>, </i>;<br/></div><b>end;</b></div>
<a NAME="E6:30"/><b>then </b>
 <a href="ordinal1.html#K1" target="_self">succ</a> <font color="Maroon">x</font> is <a href="ordinal1.html#V2" target="_self">epsilon-connected</a>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E7:30"/>
 <a href="ordinal1.html#K1" target="_self">succ</a> <font color="Maroon">x</font> is   <a href="ordinal1.html#NM2">Ordinal</a>
 <b>by </b><i><a class="txt" href="ordinal1.html#E1:30"><i><font color="Green">E19</font></i></a>, </i>;<br/>


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