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<span class="kw">let </span><font color="Maroon" title="c1">e</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>; <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="c1">e</font> is  <a href="ordinal5.html#V4" title="ORDINAL5:attr.4">epsilon</a>  implies (  not <font color="Maroon" title="c1">e</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  &amp; <font color="Maroon" title="c1">e</font> is  <a href="ordinal1.html#V4" title="ORDINAL1:attr.4">limit_ordinal</a>  ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:55_1_1"/><span class="lab"><font color="Green" title="E34">A1</font></span>: 
<font color="Maroon" title="c1">e</font> is  <a href="ordinal5.html#V4" title="ORDINAL5:attr.4">epsilon</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c1">e</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  &amp; <font color="Maroon" title="c1">e</font> is  <a href="ordinal1.html#V4" title="ORDINAL1:attr.4">limit_ordinal</a>  )</span><br/>

<a NAME="E2:55_1_1"/>
 <a href="ordinal2.html#K12" title="ORDINAL2:func.12">exp</a> (<a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal2.html#T43" title="ORDINAL2:th.43">ORDINAL2:43</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:55_1_1"/>
 not <font color="Maroon" title="c1">e</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:55_1_1"><span class="lab"><font color="Green" title="E34">A1</font></span></a>, <a class="ref" href="ordinal5.html#D5" target="_self" title="ORDINAL5:def.5">Def5</a></span>; <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="c1">e</font> is  <a href="ordinal1.html#V4" title="ORDINAL1:attr.4">limit_ordinal</a> </span><br/>

<a NAME="E4:55_1_1"/><span class="kw">then </span>
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">e</font> &amp;  <a href="ordinal2.html#K12" title="ORDINAL2:func.12">exp</a> (<a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a>,<font color="Maroon" title="c1">e</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">e</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:55_1_1"><span class="lab"><font color="Green" title="E34">A1</font></span></a>, <a class="ref" href="ordinal5.html#D5" target="_self" title="ORDINAL5:def.5">Def5</a>, <a class="ref" href="ordinal3.html#T8" title="ORDINAL3:th.8">ORDINAL3:8</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:55_1_1"/>
<font color="Maroon" title="c1">e</font> is  <a href="ordinal1.html#V4" title="ORDINAL1:attr.4">limit_ordinal</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal5.html#T9" target="_self" title="ORDINAL5: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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