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

<b>let </b><font color="Maroon" title="c4">p</font>, <font color="Maroon" title="c5">q</font> be   <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">n</font>;<br/>

<b>given </b><font color="Maroon" title="c6">k</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a><b> such that </b><a NAME="E2:93_1_1"/><i><font color="Green" title="E47">A1</font></i>: 
<font color="Maroon" title="c4">p</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c6">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">q</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c6">k</font>
 <b>and </b><br/><a NAME="E3:93_1_1"/><i><font color="Green" title="E48">A2</font></i>: 
for <font color="Olive" title="b1">l</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <font color="Olive" title="b1">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">k</font> holds <br/><font color="Maroon" title="c4">p</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">q</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">l</font>
 ;<br/>

<b>given </b><font color="Maroon" title="c7">k1</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a><b> such that </b><a NAME="E5:93_1_1"/><i><font color="Green" title="E49">A3</font></i>: 
<font color="Maroon" title="c5">q</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c7">k1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">p</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c7">k1</font>
 <b>and </b><br/><a NAME="E6:93_1_1"/><i><font color="Green" title="E50">A4</font></i>: 
for <font color="Olive" title="b1">l</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <font color="Olive" title="b1">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">k1</font> holds <br/><font color="Maroon" title="c5">q</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">l</font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="93_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:93_1_1_1"/>
( <font color="Maroon" title="c6">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">k1</font> or <font color="Maroon" title="c7">k1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">k</font> or <font color="Maroon" title="c7">k1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">k</font> )
 </span><b>by </b><i><a class="ref" href="ordinal1.html#T24" title="ORDINAL1:th.24">ORDINAL1:24</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="93_1_1_1_1"><b>suppose </b></a><a NAME="E1:93_1_1_1_1"/>
<font color="Maroon" title="c6">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">k1</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:93_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="polynom1.html#E2:93_1_1"><i><font color="Green" title="E47">A1</font></i></a>, <a class="txt" href="polynom1.html#E6:93_1_1"><i><font color="Green" title="E50">A4</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="93_1_1_1_2"><b>suppose </b></a><a NAME="E1:93_1_1_1_2"/>
<font color="Maroon" title="c7">k1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">k</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:93_1_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="polynom1.html#E3:93_1_1"><i><font color="Green" title="E48">A2</font></i></a>, <a class="txt" href="polynom1.html#E5:93_1_1"><i><font color="Green" title="E49">A3</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="93_1_1_1_3"><b>suppose </b></a><a NAME="E1:93_1_1_1_3"/>
<font color="Maroon" title="c7">k1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">k</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:93_1_1_1_3"/>
contradiction
 <b>by </b><i><a class="txt" href="polynom1.html#E2:93_1_1"><i><font color="Green" title="E47">A1</font></i></a>, <a class="txt" href="polynom1.html#E5:93_1_1"><i><font color="Green" title="E49">A3</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
