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

<b>let </b><font color="Maroon">k</font> be  <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>let </b><font color="Maroon">n</font> be  <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>



<b>thus </b><a NAME="E1:12"/>
( <font color="Maroon">k</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">k</font> implies <font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:12_1"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">k</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">k</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">x</font> = <font color="Maroon">k</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">n</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/>
<b>assume </b><a NAME="E3:12_1"/>
not <font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font>
 ;<br/>

<a NAME="E4:12_1"/><b>then </b>
<font color="Maroon">k</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/>
<a NAME="E5:12_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 0
 
;<br/>
<b>hence </b><a NAME="E6:12_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E2:12"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font>
 ;<br/>

<a NAME="E3:12"/><b>then </b>
<font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
;<br/>
<a NAME="E4:12"/><b>then </b>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">k</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T5">FINSEQ_1:5</a></i>;<br/>
<b>hence </b><a NAME="E5:12"/>
<font color="Maroon">k</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">k</font>
 <b>by </b><i>, </i>;<br/>


</div>
