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<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:111_1_1_1"/>
<font color="Maroon" title="c1">F</font> is <a href="relat_1.html#V2" title="RELAT_1:attr.2">non-empty</a>
 ;<br/><a NAME="E2:111_1_1_1"/><b>then </b><i><font color="Green" title="E44">A1</font></i>: 
not  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <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">F</font>
 <b>by </b><i><a class="ref" href="relat_1.html#D9" title="RELAT_1:def.9">RELAT_1:def 9</a></i>;<br/><b>let </b><font color="Maroon" title="c2">i</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E3:111_1_1_1"/>
<font color="Maroon" title="c2">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c1">F</font>
 ;<br/><b>hence </b><a NAME="E4:111_1_1_1"/>
not <font color="Maroon" title="c1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">i</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 <b>by </b><i><a class="txt" href="funct_1.html#E2:111_1_1_1"><i><font color="Green" title="E44">A1</font></i></a>, <a class="ref" href="funct_1.html#D5" target="_self" title="FUNCT_1:def.5">Def5</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:111_1_1"/><i><font color="Green" title="E44">A2</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c1">F</font> holds <br/> not <font color="Maroon" title="c1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ;<br/>

<b>assume </b><a NAME="E3:111_1_1"/>
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <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">F</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="relat_1.html#D9" title="RELAT_1:def.9">RELAT_1:def 9</a><br/>

<a NAME="E4:111_1_1"/><b>then </b>
 ex <font color="Olive" title="b1">i</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">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="Olive" title="b1">i</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="funct_1.html#D5" target="_self" title="FUNCT_1:def.5">Def5</a></i>;<br/>
<b>hence </b><a NAME="E5:111_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="funct_1.html#E2:111_1_1"><i><font color="Green" title="E44">A2</font></i></a></i>;<br/>


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