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<a NAME="E1:20_1_1"/>
for <font color="Olive">j</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">j</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <font color="Maroon">f</font></span>)</span> holds <br/> not <span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">j</font> is <a href="xboole_0.html#V1">empty</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_1_1"><b>proof </b></a><div class="add">

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

<b>assume </b><b>that </b><br/><a NAME="E1:20_1_1_1"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <font color="Maroon">f</font></span>)</span>
 <b>and </b><br/><a NAME="E2:20_1_1_1"/><i><font color="Green">E36</font></i>: 
<span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/>

<a NAME="E3:20_1_1_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relat_1.html#K10">"</a> <span class="p1">(<span class="default"><a href="funct_6.html#K1">SubFuncs</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="funct_6.html#D2">FUNCT_6:def 2</a></i>;<br/>
<a NAME="E4:20_1_1_1"/><b>then </b><i><font color="Green">E39</font></i>: 
( <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> is   <a href="funct_1.html#NM1">Function</a> )
 
<b>by </b><i><a class="ref" href="funct_6.html#T28">FUNCT_6:28</a></i>;<br/>
<b>reconsider </b><font color="Maroon">fj</font> = <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> as   <a href="funct_1.html#NM1">Function</a> <b>by </b><i>, <a class="ref" href="funct_6.html#T28">FUNCT_6:28</a></i>;<br/>
<a NAME="E6:20_1_1_1"/>
 <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">fj</font>
 
<b>by </b><i>, , <a class="ref" href="funct_6.html#T31">FUNCT_6:31</a></i>;<br/>
<a NAME="E7:20_1_1_1"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="relat_1.html#T64">RELAT_1:64</a></i>;<br/>
<b>hence </b><a NAME="E8:20_1_1_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="funct_1.html#D15">FUNCT_1:def 15</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:20_1_1"/>
 <a href="funct_6.html#K2">doms</a> <font color="Maroon">f</font> is <a href="relat_1.html#V2">non-empty</a>
 <b>by </b><i><a class="ref" href="funct_1.html#D15">FUNCT_1:def 15</a></i>;<br/>


</div>
