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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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="classes2.html#K13" title="CLASSES2:func.13">FinSETS</a>  implies <font color="Maroon" title="c1">x</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> )</span><br/>

<b>assume </b><a NAME="E1:4"/><i><font color="Green" title="E4">A1</font></i>: 
<font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="classes2.html#K13" title="CLASSES2:func.13">FinSETS</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">x</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a></span><br/>

<a NAME="E2:4"/>
 <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  <a href="ordinal1.html#NR2" title="ORDINAL1:NR.2">is_limit_ordinal</a> 
 
<b>by </b><i><a class="ref" href="ordinal1.html#D12" title="ORDINAL1:def.12">ORDINAL1:def 12</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c2">n</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a><b> such that </b><br/><a NAME="E4:4"/><i><font color="Green" title="E5">A2</font></i>: 
( <font color="Maroon" title="c2">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  &amp; <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">n</font> )
 <b>by </b><i><a class="txt" href="neckla_2.html#E1:4"><i><font color="Green" title="E4">A1</font></i></a>, <a class="ref" href="classes1.html#T35" title="CLASSES1:th.35">CLASSES1:35</a>, <a class="ref" href="classes2.html#T71" title="CLASSES2:th.71">CLASSES2:71</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c3">n</font> = <font color="Maroon" title="c2">n</font> as  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  <b>by </b><i><a class="txt" href="neckla_2.html#E4:4"><i><font color="Green" title="E5">A2</font></i></a>, <a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E6:4"/>
<font color="Maroon" title="c1">x</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="classes1.html#K4" title="CLASSES1:func.4">Rank</a> <font color="Maroon" title="c3">n</font>
 
<b>by </b><i><a class="txt" href="neckla_2.html#E4:4"><i><font color="Green" title="E5">A2</font></i></a></i>;<br/>
<b>hence </b><a NAME="E7:4"/>
<font color="Maroon" title="c1">x</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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