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

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">p</font> be    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">n</font> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K5">NAT</a> , <font color="Maroon">q</font> be    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">n</font> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K5">NAT</a> ;<br/>



<b>assume </b><a NAME="E1:17"/><i><font color="Green">E33</font></i>: 
not <font color="Maroon">p</font> <a href="polynom3.html#R2" target="_self">&lt;=</a> <font color="Maroon">q</font>
 ;<br/>

<a NAME="E2:17"/><b>then </b><i><font color="Green">E35</font></i>: 
( not <font color="Maroon">p</font> <a href="polynom3.html#R1" target="_self">&lt;</a> <font color="Maroon">q</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">q</font> )
 
<b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">i</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E4:17"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font>
 <b>and </b><br/><a NAME="E5:17"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">p</font> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">q</font> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">i</font>
 <b>and </b><br/><a NAME="E6:17"/><i><font color="Green">E44</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i</font> holds <br/><font color="Maroon">p</font> <a href="wsierp_1.html#K3">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> <a href="wsierp_1.html#K3">.</a> <font color="Olive">k</font>
 <b>by </b><i>, </i>;<br/>
<b>take </b>
<font color="Maroon">i</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="polynom3.html#D1" target="_self">POLYNOM3:def 1</a><br/>

<b>thus </b><a NAME="E7:17"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font>
 <b>by </b><i/>;<br/>

<a NAME="E8:17"/>
<font color="Maroon">p</font> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">i</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">q</font> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">i</font>
 
<b>by </b><i>, , , </i>;<br/>
<b>hence </b><a NAME="E9:17"/>
<font color="Maroon">q</font> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">p</font> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/>

<b>thus </b><a NAME="E10:17"/>
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i</font> holds <br/><font color="Maroon">q</font> <a href="wsierp_1.html#K3">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="wsierp_1.html#K3">.</a> <font color="Olive">k</font>
 <b>by </b><i/>;<br/>


</div>
