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<span class="kw">let </span><font color="Maroon" title="c1">W</font> be   <a href="classes2.html#NM1" title="CLASSES2:NM.1">Universe</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">W</font> implies  ex <font color="Olive" title="b1">a</font> being   <a href="ordinal4.html#NM1" title="ORDINAL4:NM.1">Ordinal</a> of <font color="Maroon" title="c1">W</font> ex <font color="Olive" title="b2">M</font> being   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">a</font> <a href="ordinal2.html#R2" title="ORDINAL2:pred.2">is_cofinal_with</a>  <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a>  &amp; <font color="Olive" title="b2">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="classes1.html#K4" title="CLASSES1:func.4">Rank</a> <font color="Olive" title="b1">a</font> &amp; <font color="Olive" title="b2">M</font> <a href="zfrefle1.html#R2" title="ZFREFLE1:pred.2">&lt;==&gt;</a> <font color="Maroon" title="c1">W</font> ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:37"/>
 <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">W</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">a</font> being   <a href="ordinal4.html#NM1" title="ORDINAL4:NM.1">Ordinal</a> of <font color="Maroon" title="c1">W</font> ex <font color="Olive" title="b2">M</font> being   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">a</font> <a href="ordinal2.html#R2" title="ORDINAL2:pred.2">is_cofinal_with</a>  <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a>  &amp; <font color="Olive" title="b2">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="classes1.html#K4" title="CLASSES1:func.4">Rank</a> <font color="Olive" title="b1">a</font> &amp; <font color="Olive" title="b2">M</font> <a href="zfrefle1.html#R2" title="ZFREFLE1:pred.2">&lt;==&gt;</a> <font color="Maroon" title="c1">W</font> )</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c2">b</font> being   <a href="ordinal4.html#NM1" title="ORDINAL4:NM.1">Ordinal</a> of <font color="Maroon" title="c1">W</font>, <font color="Maroon" title="c3">M</font> being   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:37"/><span class="lab"><font color="Green" title="E27">A1</font></span>: 
( <font color="Maroon" title="c2">b</font> <a href="ordinal2.html#R2" title="ORDINAL2:pred.2">is_cofinal_with</a>  <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a>  &amp; <font color="Maroon" title="c3">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="classes1.html#K4" title="CLASSES1:func.4">Rank</a> <font color="Maroon" title="c2">b</font> &amp; <font color="Maroon" title="c3">M</font> <a href="zfrefle1.html#R3" title="ZFREFLE1:pred.3">is_elementary_subsystem_of</a> <font color="Maroon" title="c1">W</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zfrefle1.html#T35" target="_self" title="ZFREFLE1:th.35">Th37</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c2">b</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">M</font> being   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Maroon" title="c2">b</font> <a href="ordinal2.html#R2" title="ORDINAL2:pred.2">is_cofinal_with</a>  <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a>  &amp; <font color="Olive" title="b1">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="classes1.html#K4" title="CLASSES1:func.4">Rank</a> <font color="Maroon" title="c2">b</font> &amp; <font color="Olive" title="b1">M</font> <a href="zfrefle1.html#R2" title="ZFREFLE1:pred.2">&lt;==&gt;</a> <font color="Maroon" title="c1">W</font> )</span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c3">M</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">b</font> <a href="ordinal2.html#R2" title="ORDINAL2:pred.2">is_cofinal_with</a>  <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a>  &amp; <font color="Maroon" title="c3">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="classes1.html#K4" title="CLASSES1:func.4">Rank</a> <font color="Maroon" title="c2">b</font> &amp; <font color="Maroon" title="c3">M</font> <a href="zfrefle1.html#R2" title="ZFREFLE1:pred.2">&lt;==&gt;</a> <font color="Maroon" title="c1">W</font> )</span><br/>

<span class="kw">thus </span><a NAME="E4:37"/>
( <font color="Maroon" title="c2">b</font> <a href="ordinal2.html#R2" title="ORDINAL2:pred.2">is_cofinal_with</a>  <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a>  &amp; <font color="Maroon" title="c3">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="classes1.html#K4" title="CLASSES1:func.4">Rank</a> <font color="Maroon" title="c2">b</font> &amp; <font color="Maroon" title="c3">M</font> <a href="zfrefle1.html#R2" title="ZFREFLE1:pred.2">&lt;==&gt;</a> <font color="Maroon" title="c1">W</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:37"><span class="lab"><font color="Green" title="E27">A1</font></span></a>, <a class="ref" href="zfrefle1.html#T9" target="_self" title="ZFREFLE1:th.9">Th9</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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