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

<a NAME="E1:130_1_1"/><i><font color="Green">E40</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="mcart_1.html#K15">pr1</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="ref" href="mcart_1.html#D12">MCART_1:def 12</a></i>;<br/>
<a NAME="E2:130_1_1"/><b>then </b><i><font color="Green">E41</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="mcart_1.html#K15">pr1</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" target="_self">Def1</a></i>;<br/>
<a NAME="E3:130_1_1"/>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="mcart_1.html#K15">pr1</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="130_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:130_1_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="mcart_1.html#K15">pr1</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>6</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:130_1_1_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="mcart_1.html#K15">pr1</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E4:130_1_1_1"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="mcart_1.html#K15">pr1</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E5:130_1_1_1"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="mcart_1.html#K1">`1</a> 
 
<b>by </b><i><a class="txt" href="funct_2.html#E1:130_1_1"><i><font color="Green">E40</font></i></a>, <a class="txt" href="funct_2.html#E3:130_1_1_1"><i><font color="Green">E42</font></i></a>, <a class="txt" href="funct_2.html#E4:130_1_1_1"><i><font color="Green">E43</font></i></a>, <a class="ref" href="mcart_1.html#D12">MCART_1:def 12</a></i>;<br/>
<a NAME="E6:130_1_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="txt" href="funct_2.html#E1:130_1_1"><i><font color="Green">E40</font></i></a>, <a class="txt" href="funct_2.html#E3:130_1_1_1"><i><font color="Green">E42</font></i></a>, <a class="ref" href="funct_2.html#T7" target="_self">Th7</a></i>;<br/>
<b>hence </b><a NAME="E7:130_1_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="funct_2.html#E5:130_1_1_1"><i><font color="Green">E44</font></i></a>, <a class="ref" href="mcart_1.html#T10">MCART_1:10</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:130_1_1"/>
 <a href="mcart_1.html#K15">pr1</a> <font color="Maroon">c<sub>4</sub></font> is   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="funct_2.html#E2:130_1_1"><i><font color="Green">E41</font></i></a>, <a class="ref" href="funct_2.html#T4" target="_self">Th4</a></i>;<br/>


</div>
