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<span class="kw">let </span><font color="Maroon" title="c1">N</font> be   <a href="net_1.html#NM1" title="NET_1:NM.1">Pnet</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">x</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  <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>  holds <br/> <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">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></span><br/><span class="kw">let </span><font color="Maroon" title="c2">x</font> be    <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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <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>  implies  <a href="net_1.html#K4" title="NET_1:func.4">enter</a> <font color="Maroon" title="c1">N</font>,<font color="Maroon" title="c2">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> )</span><br/>



<span class="kw">assume </span><a NAME="E1:30"/><span class="lab"><font color="Green" title="E15">A1</font></span>: 
 <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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="net_1.html#K4" title="NET_1:func.4">enter</a> <font color="Maroon" title="c1">N</font>,<font color="Maroon" title="c2">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></span><br/>

<a NAME="E2:30"/><span class="lab"><font color="Green" title="E16">A2</font></span>: 
(  <a href="net_1.html#K4" title="NET_1:func.4">enter</a> <font color="Maroon" title="c1">N</font>,<font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> implies  <a href="net_1.html#K4" title="NET_1:func.4">enter</a> <font color="Maroon" title="c1">N</font>,<font color="Maroon" title="c2">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> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:30_1"/>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="net_1.html#K1" title="NET_1:func.1">Elements</a> <font color="Maroon" title="c1">N</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:30"><span class="lab"><font color="Green" title="E15">A1</font></span></a></span>;<br/>
<a NAME="E2:30_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E17">A3</font></span>: 
for <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> holds <br/><font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="net_1.html#K1" title="NET_1:func.1">Elements</a> <font color="Maroon" title="c1">N</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<span class="kw">assume </span><a NAME="E3:30_1"/>
 <a href="net_1.html#K4" title="NET_1:func.4">enter</a> <font color="Maroon" title="c1">N</font>,<font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="net_1.html#K4" title="NET_1:func.4">enter</a> <font color="Maroon" title="c1">N</font>,<font color="Maroon" title="c2">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></span><br/>

<span class="kw">hence </span><a NAME="E4:30_1"/>
 <a href="net_1.html#K4" title="NET_1:func.4">enter</a> <font color="Maroon" title="c1">N</font>,<font color="Maroon" title="c2">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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:30_1"><span class="lab"><font color="Green" title="E17">A3</font></span></a>, <a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E3:30"/>
(  <a href="net_1.html#K4" title="NET_1:func.4">enter</a> <font color="Maroon" title="c1">N</font>,<font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> or  <a href="net_1.html#K4" title="NET_1:func.4">enter</a> <font color="Maroon" title="c1">N</font>,<font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="net_1.html#K2" title="NET_1:func.2">Pre</a> <font color="Maroon" title="c1">N</font>,<font color="Maroon" title="c2">x</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:30"><span class="lab"><font color="Green" title="E15">A1</font></span></a>, <a class="ref" href="net_1.html#T22" target="_self" title="NET_1:th.22">Th22</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:30"/>
 <a href="net_1.html#K4" title="NET_1:func.4">enter</a> <font color="Maroon" title="c1">N</font>,<font color="Maroon" title="c2">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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:30"><span class="lab"><font color="Green" title="E16">A2</font></span></a>, <a class="ref" href="net_1.html#T16" target="_self" title="NET_1:th.16">Th16</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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