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

<b>let </b><font color="Maroon" title="c1">g</font>, <font color="Maroon" title="c2">f</font> be   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>;<br/>



<b>assume </b><a NAME="E1:29"/><i><font color="Green" title="E12">A1</font></i>: 
<font color="Maroon" title="c1">g</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">f</font>
 ;<br/>

<a NAME="E2:29"/><b>then </b>
<span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c1">g</font></span>)</span> <a href="relat_1.html#K8" title="RELAT_1:func.8">|</a> <font color="Maroon" title="c1">g</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c1">g</font></span>)</span> <a href="relat_1.html#K8" title="RELAT_1:func.8">|</a> <font color="Maroon" title="c2">f</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T132" title="RELAT_1:th.132">RELAT_1:132</a></i>;<br/>
<a NAME="E3:29"/><b>then </b><i><font color="Green" title="E13">A2</font></i>: 
<font color="Maroon" title="c1">g</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c1">g</font></span>)</span> <a href="relat_1.html#K8" title="RELAT_1:func.8">|</a> <font color="Maroon" title="c2">f</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T125" title="RELAT_1:th.125">RELAT_1:125</a></i>;<br/>
<b>assume </b><a NAME="E4:29"/><i><font color="Green" title="E14">A3</font></i>: 
<font color="Maroon" title="c2">f</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 ;<br/>

<a NAME="E5:29"/>
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c1">g</font></span>)</span> <a href="relat_1.html#K8" title="RELAT_1:func.8">|</a> <font color="Maroon" title="c2">f</font> holds <br/><span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">g</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">x</font>, <font color="Maroon" title="c4">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>



<b>assume </b><a NAME="E1:29_1"/><i><font color="Green" title="E15">A4</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c1">g</font></span>)</span> <a href="relat_1.html#K8" title="RELAT_1:func.8">|</a> <font color="Maroon" title="c2">f</font>
 ;<br/>

<a NAME="E2:29_1"/><b>then </b>
<font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c1">g</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#D12" title="RELAT_1:def.12">RELAT_1:def 12</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c5">x1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:29_1"/><i><font color="Green" title="E16">A5</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">x1</font>,<font color="Maroon" title="c4">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">g</font>
 <b>by </b><i><a class="ref" href="relat_1.html#D5" title="RELAT_1:def.5">RELAT_1:def 5</a></i>;<br/>
<a NAME="E5:29_1"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">x1</font>,<font color="Maroon" title="c4">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">f</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">f</font> )
 
<b>by </b><i><a class="txt" href="grfunc_1.html#E1:29"><i><font color="Green" title="E12">A1</font></i></a>, <a class="txt" href="grfunc_1.html#E1:29_1"><i><font color="Green" title="E15">A4</font></i></a>, <a class="txt" href="grfunc_1.html#E4:29_1"><i><font color="Green" title="E16">A5</font></i></a>, <a class="ref" href="relat_1.html#D12" title="RELAT_1:def.12">RELAT_1:def 12</a></i>;<br/>
<b>hence </b><a NAME="E6:29_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">g</font>
 <b>by </b><i><a class="txt" href="grfunc_1.html#E4:29"><i><font color="Green" title="E14">A3</font></i></a>, <a class="txt" href="grfunc_1.html#E4:29_1"><i><font color="Green" title="E16">A5</font></i></a>, <a class="ref" href="grfunc_1.html#T25" target="_self" title="GRFUNC_1:th.25">Th25</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:29"/><b>then </b>
<span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c1">g</font></span>)</span> <a href="relat_1.html#K8" title="RELAT_1:func.8">|</a> <font color="Maroon" title="c2">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">g</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#D3" title="RELAT_1:def.3">RELAT_1:def 3</a></i>;<br/>
<b>hence </b><a NAME="E7:29"/>
<span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c1">g</font></span>)</span> <a href="relat_1.html#K8" title="RELAT_1:func.8">|</a> <font color="Maroon" title="c2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">g</font>
 <b>by </b><i><a class="txt" href="grfunc_1.html#E3:29"><i><font color="Green" title="E13">A2</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/>


</div>
