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<b>let </b><font color="Maroon" title="c1">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<a NAME="E1:69"/>
 <a href="classes1.html#K5" title="CLASSES1:func.5">the_transitive-closure_of</a> <font color="Maroon" title="c1">X</font> is <a href="ordinal1.html#V1" title="ORDINAL1:attr.1">epsilon-transitive</a>
 
<b>by </b><i><a class="ref" href="classes1.html#T58" target="_self" title="CLASSES1:th.58">Th58</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c2">A</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a><b> such that </b><br/><a NAME="E3:69"/><i><font color="Green" title="E51">A1</font></i>: 
 <a href="classes1.html#K1" title="CLASSES1:func.1">Tarski-Class</a> <span class="p1">(<span class="default"><a href="classes1.html#K5" title="CLASSES1:func.5">the_transitive-closure_of</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="classes1.html#K4" title="CLASSES1:func.4">Rank</a> <font color="Maroon" title="c2">A</font>
 <b>by </b><i><a class="ref" href="classes1.html#T53" target="_self" title="CLASSES1:th.53">Th53</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c2">A</font>
;<br/>

<a NAME="E4:69"/>
(  <a href="classes1.html#K5" title="CLASSES1:func.5">the_transitive-closure_of</a> <font color="Maroon" title="c1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="classes1.html#K1" title="CLASSES1:func.1">Tarski-Class</a> <span class="p1">(<span class="default"><a href="classes1.html#K5" title="CLASSES1:func.5">the_transitive-closure_of</a> <font color="Maroon" title="c1">X</font></span>)</span> &amp; <font color="Maroon" title="c1">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="classes1.html#K5" title="CLASSES1:func.5">the_transitive-closure_of</a> <font color="Maroon" title="c1">X</font> )
 
<b>by </b><i><a class="ref" href="classes1.html#T5" target="_self" title="CLASSES1:th.5">Th5</a>, <a class="ref" href="classes1.html#T59" target="_self" title="CLASSES1:th.59">Th59</a></i>;<br/>
<a NAME="E5:69"/><b>then </b>
<font color="Maroon" title="c1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="classes1.html#K1" title="CLASSES1:func.1">Tarski-Class</a> <span class="p1">(<span class="default"><a href="classes1.html#K5" title="CLASSES1:func.5">the_transitive-closure_of</a> <font color="Maroon" title="c1">X</font></span>)</span>
 
<b>by </b><i><a class="ref" href="classes1.html#T6" target="_self" title="CLASSES1:th.6">Th6</a></i>;<br/>
<a NAME="E6:69"/><b>then </b>
( <font color="Maroon" title="c1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="classes1.html#K4" title="CLASSES1:func.4">Rank</a> <font color="Maroon" title="c2">A</font> &amp;  <a href="classes1.html#K4" title="CLASSES1:func.4">Rank</a> <font color="Maroon" title="c2">A</font> is <a href="ordinal1.html#V1" title="ORDINAL1:attr.1">epsilon-transitive</a> )
 
<b>by </b><i><a class="txt" href="classes1.html#E3:69"><i><font color="Green" title="E51">A1</font></i></a>, <a class="ref" href="classes1.html#T37" target="_self" title="CLASSES1:th.37">Th37</a></i>;<br/>
<b>hence </b><a NAME="E7:69"/>
<font color="Maroon" title="c1">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="classes1.html#K4" title="CLASSES1:func.4">Rank</a> <font color="Maroon" title="c2">A</font>
 <b>by </b><i><a class="ref" href="ordinal1.html#D2" title="ORDINAL1:def.2">ORDINAL1:def 2</a></i>;<br/>


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