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

<b>let </b><font color="Maroon">f</font> be   <a href="funct_1.html#NM1">Function</a>;<br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:54_1_1_1"/>
<font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="afinsq_1.html#K3" target="_self">&lt;%</a><span class="default"><font color="Maroon">x</font></span><a href="afinsq_1.html#K3" target="_self">%&gt;</a></span>
 ;<br/><a NAME="E2:54_1_1_1"/><b>then </b><i><font color="Green">E31</font></i>: 
<font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default">0,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span>
 ;<br/><b>hence </b><a NAME="E3:54_1_1_1"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="ref" href="card_5.html#T1">CARD_5:1</a>, <a class="ref" href="relat_1.html#T23">RELAT_1:23</a></i>;<br/><a NAME="E4:54_1_1_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font>
 <b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E5:54_1_1_1"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:54_1_1"/><i><font color="Green">E33</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> )
 ;<br/>

<b>reconsider </b><font color="Maroon">g</font> = <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default">0,<span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 0</span>)</span></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span> as   <a href="funct_1.html#NM1">Function</a> <b>by </b><i><a class="ref" href="grfunc_1.html#T15">GRFUNC_1:15</a></i>;<br/>
<a NAME="E4:54_1_1"/>
<font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default">0,<span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 0</span>)</span></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_1_1_2"><b>proof </b></a><div class="add">

<a NAME="E1:54_1_1_2"/>
for <font color="Olive">y</font>, <font color="Olive">z</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">y</font>,<font color="Olive">z</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> iff <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">y</font>,<font color="Olive">z</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">g</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">z</font> be    <a href="hidden.html#M1">set</a> ;<br/>



<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:54_1_1_2_1_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font>
 ;<br/><a NAME="E2:54_1_1_2_1_1"/><b>then </b>
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</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">f</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="card_5.html#T1">CARD_5:1</a>, <a class="ref" href="funct_1.html#T14">FUNCT_1:14</a>, <a class="ref" href="relat_1.html#D4">RELAT_1:def 4</a>, <a class="ref" href="relat_1.html#D5">RELAT_1:def 5</a></i>;<br/><a NAME="E3:54_1_1_2_1_1"/><b>then </b>
( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 0 )
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E4:54_1_1_2_1_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">g</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:54_1_1_2_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">g</font>
 ;<br/>

<a NAME="E3:54_1_1_2_1"/><b>then </b>
( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 0 &amp; 0 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> )
 
<b>by </b><i>, , <a class="ref" href="card_5.html#T1">CARD_5:1</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E4:54_1_1_2_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D4">FUNCT_1:def 4</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:54_1_1_2"/>
<font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default">0,<span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 0</span>)</span></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="relat_1.html#D2">RELAT_1:def 2</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:54_1_1"/>
<font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="afinsq_1.html#K3" target="_self">&lt;%</a><span class="default"><font color="Maroon">x</font></span><a href="afinsq_1.html#K3" target="_self">%&gt;</a></span>
 <b>by </b><i/>;<br/>


</div>
