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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="afinsq_1.html#NM1">XFinSequence</a>;<br/>



<b>thus </b><a NAME="E1:61"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="afinsq_1.html#K6" target="_self">&lt;%</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="afinsq_1.html#K6" target="_self">%&gt;</a></span> implies (  <a href="afinsq_1.html#K2" target="_self">dom</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> 1 &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K1">}</a></span> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:61_1"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="afinsq_1.html#K6" target="_self">&lt;%</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="afinsq_1.html#K6" target="_self">%&gt;</a></span>
 ;<br/>

<b>hence </b><a NAME="E2:61_1"/>
 <a href="afinsq_1.html#K2" target="_self">dom</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="ref" href="afinsq_1.html#D5" target="_self">Def5</a></i>;<br/>

<a NAME="E3:61_1"/>
 <a href="afinsq_1.html#K2" target="_self">dom</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="txt" href="afinsq_1.html#E1:61_1"><i><font color="Green">E27</font></i></a>, <a class="ref" href="afinsq_1.html#D5" target="_self">Def5</a>, <a class="ref" href="card_5.html#T1">CARD_5:1</a></i>;<br/>
<a NAME="E4:61_1"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> 0</span>)</span></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="funct_1.html#T14">FUNCT_1:14</a></i>;<br/>
<b>hence </b><a NAME="E5:61_1"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="afinsq_1.html#E1:61_1"><i><font color="Green">E27</font></i></a>, <a class="ref" href="afinsq_1.html#D5" target="_self">Def5</a></i>;<br/>


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

<b>assume </b><a NAME="E2:61"/><i><font color="Green">E27</font></i>: 
(  <a href="afinsq_1.html#K2" target="_self">dom</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> 1 &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K1">}</a></span> )
 ;<br/>

<a NAME="E3:61"/>
1 <a href="hidden.html#R1">=</a> 0 <a href="nat_1.html#K1">+</a> 1
 
;<br/>
<a NAME="E4:61"/><b>then </b>
0 <a href="hidden.html#R2">in</a>  <a href="afinsq_1.html#K2" target="_self">dom</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="afinsq_1.html#E2:61"><i><font color="Green">E27</font></i></a>, <a class="ref" href="afinsq_1.html#T1" target="_self">Th1</a></i>;<br/>
<a NAME="E5:61"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="txt" href="afinsq_1.html#E2:61"><i><font color="Green">E27</font></i></a>, <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/>
<a NAME="E6:61"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E7:61"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="afinsq_1.html#K6" target="_self">&lt;%</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="afinsq_1.html#K6" target="_self">%&gt;</a></span>
 <b>by </b><i><a class="txt" href="afinsq_1.html#E2:61"><i><font color="Green">E27</font></i></a>, <a class="ref" href="afinsq_1.html#D5" target="_self">Def5</a></i>;<br/>


</div>
