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<a NAME="E1:20_1_1"/>
for <font color="Olive" title="b1">j</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2" title="FUNCT_6:func.2">doms</a> <font color="Maroon" title="c1">f</font></span>)</span> holds <br/> not <span class="p1">(<span class="default"><a href="funct_6.html#K2" title="FUNCT_6:func.2">doms</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">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" title="c2">j</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2" title="FUNCT_6:func.2">doms</a> <font color="Maroon" title="c1">f</font></span>)</span> implies  not <span class="p1">(<span class="default"><a href="funct_6.html#K2" title="FUNCT_6:func.2">doms</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">j</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> )</span><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:20_1_1_1"/><i><font color="Green" title="E12">A1</font></i>: 
<font color="Maroon" title="c2">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2" title="FUNCT_6:func.2">doms</a> <font color="Maroon" title="c1">f</font></span>)</span>
 <b>and </b><br/><a NAME="E2:20_1_1_1"/><i><font color="Green" title="E13">A2</font></i>: 
<span class="p1">(<span class="default"><a href="funct_6.html#K2" title="FUNCT_6:func.2">doms</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">j</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E3:20_1_1_1"/><i><font color="Green" title="E14">A3</font></i>: 
<font color="Maroon" title="c2">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">f</font> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <span class="p1">(<span class="default"><a href="funct_6.html#K1" title="FUNCT_6:func.1">SubFuncs</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c1">f</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel_5.html#E1:20_1_1_1"><i><font color="Green" title="E12">A1</font></i></a>, <a class="ref" href="funct_6.html#D2" title="FUNCT_6:def.2">FUNCT_6:def 2</a></i>;<br/>
<a NAME="E4:20_1_1_1"/><b>then </b><i><font color="Green" title="E15">A4</font></i>: 
( <font color="Maroon" title="c2">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">f</font> &amp; <font color="Maroon" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">j</font> is   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> )
 
<b>by </b><i><a class="ref" href="funct_6.html#T28" title="FUNCT_6:th.28">FUNCT_6:28</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c3">fj</font> = <font color="Maroon" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">j</font> as   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> <b>by </b><i><a class="txt" href="waybel_5.html#E3:20_1_1_1"><i><font color="Green" title="E14">A3</font></i></a>, <a class="ref" href="funct_6.html#T28" title="FUNCT_6:th.28">FUNCT_6:28</a></i>;<br/>
<a NAME="E6:20_1_1_1"/>
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">fj</font>
 
<b>by </b><i><a class="txt" href="waybel_5.html#E2:20_1_1_1"><i><font color="Green" title="E13">A2</font></i></a>, <a class="txt" href="waybel_5.html#E4:20_1_1_1"><i><font color="Green" title="E15">A4</font></i></a>, <a class="ref" href="funct_6.html#T31" title="FUNCT_6:th.31">FUNCT_6:31</a></i>;<br/>
<a NAME="E7:20_1_1_1"/><b>then </b>
<font color="Maroon" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="ref" href="relat_1.html#T64" title="RELAT_1:th.64">RELAT_1:64</a></i>;<br/>
<b>hence </b><a NAME="E8:20_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="waybel_5.html#E4:20_1_1_1"><i><font color="Green" title="E15">A4</font></i></a>, <a class="ref" href="funct_1.html#D15" title="FUNCT_1:def.15">FUNCT_1:def 15</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:20_1_1"/>
 <a href="funct_6.html#K2" title="FUNCT_6:func.2">doms</a> <font color="Maroon" title="c1">f</font> is <a href="relat_1.html#V2" title="RELAT_1:attr.2">non-empty</a>
 <b>by </b><i><a class="ref" href="funct_1.html#D15" title="FUNCT_1:def.15">FUNCT_1:def 15</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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