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<span class="kw">let </span><font color="Maroon" title="c1">e</font> be   <a href="ordinal5.html#V4" title="ORDINAL5:attr.4">epsilon</a>  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">e</font></span><br/>

<a NAME="E1:54"/><span class="lab"><font color="Green" title="E34">A0</font></span>: 
 <a href="ordinal2.html#K12" title="ORDINAL2:func.12">exp</a> <a href="ordinal1.html#K5" title="ORDINAL1:func.5">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="ref" href="ordinal5.html#D5" target="_self" title="ORDINAL5:def.5">EPSILON</a></span>;<br/>
<a NAME="E2:54"/><span class="lab"><font color="Green" title="E35">A1</font></span>: 
(  <a href="ordinal2.html#K12" title="ORDINAL2:func.12">exp</a> <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> ,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp;  <a href="ordinal2.html#K12" title="ORDINAL2:func.12">exp</a> <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> ,1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  &amp; 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal2.html#T60" title="ORDINAL2:th.60">ORDINAL2:60</a>, <a class="ref" href="ordinal2.html#T63" title="ORDINAL2:th.63">ORDINAL2:63</a></span>;<br/>
<a NAME="E3:54"/><span class="kw">then </span><span class="lab"><font color="Green" title="E36">A2</font></span>: 
( <font color="Maroon" title="c1">e</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c1">e</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 &amp;  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp;  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:54"><span class="lab"><font color="Green" title="E34">A0</font></span></a></span>;<br/>
<a NAME="E4:54"/><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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal3.html#T10" title="ORDINAL3:th.10">ORDINAL3:10</a></span>;<br/>
<a NAME="E5:54"/><span class="kw">then </span>
1 <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c1">e</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:54"><span class="lab"><font color="Green" title="E36">A2</font></span></a>, <a class="ref" href="ordinal1.html#T33" title="ORDINAL1:th.33">ORDINAL1:33</a></span>;<br/>
<a NAME="E6:54"/><span class="kw">then </span>
1 <a href="xboole_0.html#R2" title="XBOOLE_0:pred.2">c&lt;</a> <font color="Maroon" title="c1">e</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:54"><span class="lab"><font color="Green" title="E36">A2</font></span></a>, <a class="ref" href="xboole_0.html#D8" title="XBOOLE_0:def.8">XBOOLE_0:def 8</a></span>;<br/>
<a NAME="E7:54"/><span class="kw">then </span>
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">e</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#T21" title="ORDINAL1:th.21">ORDINAL1:21</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:54"/>
 <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">e</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:54"><span class="lab"><font color="Green" title="E34">A0</font></span></a>, <a class="txt" href="#E2:54"><span class="lab"><font color="Green" title="E35">A1</font></span></a>, <a class="ref" href="ordinal4.html#T24" title="ORDINAL4:th.24">ORDINAL4:24</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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