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<div class="add">

<b>let </b><font color="Maroon" title="c1">N</font> be   <a href="net_1.html#NM1" title="NET_1:NM.1">Pnet</a>;<br/><b>let </b><font color="Maroon" title="c2">x</font>, <font color="Maroon" title="c3">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>





<b>assume </b><a NAME="E1:34"/><i><font color="Green" title="E25">A1</font></i>: 
(  <a href="net_1.html#K1" title="NET_1:func.1">Elements</a> <font color="Maroon" title="c1">N</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c3">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="net_1.html#K1" title="NET_1:func.1">Elements</a> <font color="Maroon" title="c1">N</font> )
 ;<br/>

<a NAME="E2:34"/><i><font color="Green" title="E26">A2</font></i>: 
( <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="net_1.html#K11" title="NET_1:func.11">Input</a> <font color="Maroon" title="c1">N</font>,<font color="Maroon" title="c3">X</font> implies  ex <font color="Olive" title="b1">y</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="net_1.html#K1" title="NET_1:func.1">Elements</a> <font color="Maroon" title="c1">N</font> st <br/>( <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> &amp; <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="net_1.html#K4" title="NET_1:func.4">enter</a> <font color="Maroon" title="c1">N</font>,<font color="Olive" title="b1">y</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:34_1"/>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="net_1.html#K11" title="NET_1:func.11">Input</a> <font color="Maroon" title="c1">N</font>,<font color="Maroon" title="c3">X</font>
 ;<br/>

<a NAME="E2:34_1"/><b>then </b>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="net_1.html#K9" title="NET_1:func.9">Entr</a> <font color="Maroon" title="c1">N</font>,<font color="Maroon" title="c3">X</font></span>)</span>
 
;<br/>
<b>then consider </b><font color="Maroon" title="c4">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:34_1"/><i><font color="Green" title="E27">A3</font></i>: 
( <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">y</font> &amp; <font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="net_1.html#K9" title="NET_1:func.9">Entr</a> <font color="Maroon" title="c1">N</font>,<font color="Maroon" title="c3">X</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<a NAME="E5:34_1"/>
 ex <font color="Olive" title="b1">z</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="net_1.html#K1" title="NET_1:func.1">Elements</a> <font color="Maroon" title="c1">N</font> st <br/>( <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> &amp; <font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="net_1.html#K4" title="NET_1:func.4">enter</a> <font color="Maroon" title="c1">N</font>,<font color="Olive" title="b1">z</font> )
 
<b>by </b><i><a class="txt" href="net_1.html#E4:34_1"><i><font color="Green" title="E27">A3</font></i></a>, <a class="ref" href="net_1.html#D14" target="_self" title="NET_1:def.14">Def14</a></i>;<br/>
<b>hence </b><a NAME="E6:34_1"/>
 ex <font color="Olive" title="b1">y</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="net_1.html#K1" title="NET_1:func.1">Elements</a> <font color="Maroon" title="c1">N</font> st <br/>( <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> &amp; <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="net_1.html#K4" title="NET_1:func.4">enter</a> <font color="Maroon" title="c1">N</font>,<font color="Olive" title="b1">y</font> )
 <b>by </b><i><a class="txt" href="net_1.html#E4:34_1"><i><font color="Green" title="E27">A3</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:34"/>
(  ex <font color="Olive" title="b1">y</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="net_1.html#K1" title="NET_1:func.1">Elements</a> <font color="Maroon" title="c1">N</font> st <br/>( <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> &amp; <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="net_1.html#K4" title="NET_1:func.4">enter</a> <font color="Maroon" title="c1">N</font>,<font color="Olive" title="b1">y</font> ) implies <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="net_1.html#K11" title="NET_1:func.11">Input</a> <font color="Maroon" title="c1">N</font>,<font color="Maroon" title="c3">X</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_2"><b>proof </b></a><div class="add">

<b>given </b><font color="Maroon" title="c4">y</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="net_1.html#K1" title="NET_1:func.1">Elements</a> <font color="Maroon" title="c1">N</font><b> such that </b><a NAME="E1:34_2"/><i><font color="Green" title="E27">A4</font></i>: 
( <font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> &amp; <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="net_1.html#K4" title="NET_1:func.4">enter</a> <font color="Maroon" title="c1">N</font>,<font color="Maroon" title="c4">y</font> )
 ;<br/>

<a NAME="E2:34_2"/>
 <a href="net_1.html#K4" title="NET_1:func.4">enter</a> <font color="Maroon" title="c1">N</font>,<font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="net_1.html#K9" title="NET_1:func.9">Entr</a> <font color="Maroon" title="c1">N</font>,<font color="Maroon" title="c3">X</font>
 
<b>by </b><i><a class="txt" href="net_1.html#E1:34"><i><font color="Green" title="E25">A1</font></i></a>, <a class="txt" href="net_1.html#E1:34_2"><i><font color="Green" title="E27">A4</font></i></a>, <a class="ref" href="net_1.html#T28" target="_self" title="NET_1:th.28">Th28</a></i>;<br/>
<a NAME="E3:34_2"/><b>then </b>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="net_1.html#K9" title="NET_1:func.9">Entr</a> <font color="Maroon" title="c1">N</font>,<font color="Maroon" title="c3">X</font></span>)</span>
 
<b>by </b><i><a class="txt" href="net_1.html#E1:34_2"><i><font color="Green" title="E27">A4</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<b>hence </b><a NAME="E4:34_2"/>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="net_1.html#K11" title="NET_1:func.11">Input</a> <font color="Maroon" title="c1">N</font>,<font color="Maroon" title="c3">X</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:34"/>
( <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="net_1.html#K11" title="NET_1:func.11">Input</a> <font color="Maroon" title="c1">N</font>,<font color="Maroon" title="c3">X</font> iff  ex <font color="Olive" title="b1">y</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="net_1.html#K1" title="NET_1:func.1">Elements</a> <font color="Maroon" title="c1">N</font> st <br/>( <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> &amp; <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="net_1.html#K4" title="NET_1:func.4">enter</a> <font color="Maroon" title="c1">N</font>,<font color="Olive" title="b1">y</font> ) )
 <b>by </b><i><a class="txt" href="net_1.html#E2:34"><i><font color="Green" title="E26">A2</font></i></a></i>;<br/>


</div>
