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

<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be  <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:11"/><i><font color="Green">E9</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font>
 <b>and </b><br/><a NAME="E2:11"/><i><font color="Green">E10</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>4</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E4:11"/><i><font color="Green">E11</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="finseq_3.html#E2:11"><i><font color="Green">E10</font></i></a>, <a class="ref" href="nat_1.html#T28">NAT_1:28</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>3</sub></font> as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="txt" href="finseq_3.html#E4:11"><i><font color="Green">E11</font></i></a></i>;<br/>
<div><i><font color="Green">E12</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:11_1"/>
not 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/><a NAME="E2:11_1"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="nat_1.html#T39">NAT_1:39</a></i>;<br/><b>hence </b><a NAME="E3:11_1"/>
contradiction
 <b>by </b><i><a class="txt" href="finseq_3.html#E2:11"><i><font color="Green">E10</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E7:11"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="txt" href="finseq_3.html#E1:11"><i><font color="Green">E9</font></i></a>, <a class="txt" href="finseq_3.html#E4:11"><i><font color="Green">E11</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a>, <a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/>
<a NAME="E8:11"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="finseq_3.html#E4:11"><i><font color="Green">E11</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E9:11"/>
<font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="finseq_3.html#E6:11"><i><font color="Green">E12</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>


</div>
