<?xml version="1.0"?>
<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">a</font>, <font color="Maroon" title="c2">b</font> be   <a href="ordinal1.html#V3" title="ORDINAL1:attr.3">ordinal</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</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">a</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c2">b</font> iff  <a href="ordinal5.html#K3" title="ORDINAL5:func.3">epsilon_</a> <font color="Maroon" title="c1">a</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="ordinal5.html#K3" title="ORDINAL5:func.3">epsilon_</a> <font color="Maroon" title="c2">b</font> )</span><br/>



<div><span class="kw">hereby </span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="ordinal5.html#K3" title="ORDINAL5:func.3">epsilon_</a> <font color="Maroon" title="c1">a</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="ordinal5.html#K3" title="ORDINAL5:func.3">epsilon_</a> <font color="Maroon" title="c2">b</font> implies <font color="Maroon" title="c1">a</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c2">b</font> )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:16_1"/>
<font color="Maroon" title="c1">a</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <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">  <a href="ordinal5.html#K3" title="ORDINAL5:func.3">epsilon_</a> <font color="Maroon" title="c1">a</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="ordinal5.html#K3" title="ORDINAL5:func.3">epsilon_</a> <font color="Maroon" title="c2">b</font></span><br/><a NAME="E2:16_1"/><span class="kw">then </span>
( <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">b</font> or <font color="Maroon" title="c1">a</font> <a href="xboole_0.html#R2" title="XBOOLE_0:pred.2">c&lt;</a> <font color="Maroon" title="c2">b</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D8" title="XBOOLE_0:def.8">XBOOLE_0:def 8</a></span>;<br/><a NAME="E3:16_1"/><span class="kw">then </span>
( <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">b</font> or  <a href="ordinal5.html#K3" title="ORDINAL5:func.3">epsilon_</a> <font color="Maroon" title="c1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal5.html#K3" title="ORDINAL5:func.3">epsilon_</a> <font color="Maroon" title="c2">b</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#T11" title="ORDINAL1:th.11">ORDINAL1:11</a>, <a class="ref" href="ordinal5.html#T44" title="ORDINAL5:th.44">ORDINAL5:44</a></span>;<br/><span class="kw">hence </span><a NAME="E4:16_1"/>
 <a href="ordinal5.html#K3" title="ORDINAL5:func.3">epsilon_</a> <font color="Maroon" title="c1">a</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="ordinal5.html#K3" title="ORDINAL5:func.3">epsilon_</a> <font color="Maroon" title="c2">b</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D2" title="ORDINAL1:def.2">ORDINAL1:def 2</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/></div>
<span class="kw">end;</span></div>

<span class="kw">assume </span><a NAME="E2:16"/><span class="lab"><font color="Green" title="E10">Z1</font></span>: 
 <a href="ordinal5.html#K3" title="ORDINAL5:func.3">epsilon_</a> <font color="Maroon" title="c1">a</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="ordinal5.html#K3" title="ORDINAL5:func.3">epsilon_</a> <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"> <font color="Maroon" title="c1">a</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c2">b</font></span><br/>

<span class="kw">assume </span><a NAME="E3:16"/>
<font color="Maroon" title="c1">a</font> <a href="matroid0.html#NR1" title="MATROID0:NR.1">c/=</a> <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"> contradiction</span><br/>

<a NAME="E4:16"/><span class="kw">then </span>
 <a href="ordinal5.html#K3" title="ORDINAL5:func.3">epsilon_</a> <font color="Maroon" title="c2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal5.html#K3" title="ORDINAL5:func.3">epsilon_</a> <font color="Maroon" title="c1">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#T16" title="ORDINAL1:th.16">ORDINAL1:16</a>, <a class="ref" href="ordinal5.html#T44" title="ORDINAL5:th.44">ORDINAL5:44</a></span>;<br/>
<a NAME="E5:16"/><span class="kw">then </span>
 <a href="ordinal5.html#K3" title="ORDINAL5:func.3">epsilon_</a> <font color="Maroon" title="c2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal5.html#K3" title="ORDINAL5:func.3">epsilon_</a> <font color="Maroon" title="c2">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:16"><span class="lab"><font color="Green" title="E10">Z1</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:16"/>
contradiction
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
