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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="funct_1.html#NM1">Function</a>;<br/>

<b>thus </b><a NAME="E1:6"/>
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>1</sub></font> is <a href="finset_1.html#V1">finite</a> implies <font color="Maroon">c<sub>1</sub></font> is <a href="finset_1.html#V1">finite</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:6_1"/><i><font color="Green">E3</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>1</sub></font> is <a href="finset_1.html#V1">finite</a>
 ;<br/>

<a NAME="E2:6_1"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>1</sub></font> is <a href="finset_1.html#V1">finite</a>
 
<b>by </b><i><a class="ref" href="finset_1.html#T26">FINSET_1:26</a></i>;<br/>
<a NAME="E3:6_1"/><b>then </b><i><font color="Green">E4</font></i>: 
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> is <a href="finset_1.html#V1">finite</a>
 
<b>by </b><i><a class="txt" href="prelamb.html#E1:6_1"><i><font color="Green">E3</font></i></a>, <a class="ref" href="finset_1.html#T19">FINSET_1:19</a></i>;<br/>
<a NAME="E4:6_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="relat_1.html#T21">RELAT_1:21</a></i>;<br/>
<b>hence </b><a NAME="E5:6_1"/>
<font color="Maroon">c<sub>1</sub></font> is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i><a class="txt" href="prelamb.html#E3:6_1"><i><font color="Green">E4</font></i></a>, <a class="ref" href="finset_1.html#T13">FINSET_1:13</a></i>;<br/>


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

<a NAME="E2:6"/>
<span class="p1">(<span class="default"><a href="funct_3.html#K9">pr1</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="relat_1.html#K9">.:</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="prelamb.html#E4"><i><font color="Green">Lemma1</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:6"/>
( <font color="Maroon">c<sub>1</sub></font> is <a href="finset_1.html#V1">finite</a> implies  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>1</sub></font> is <a href="finset_1.html#V1">finite</a> )
 <b>by </b><i><a class="ref" href="finset_1.html#T17">FINSET_1:17</a></i>;<br/>


</div>
