<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="e_siec.html#NM2">e_net</a>;<br/>

<a NAME="E1:40"/><i><font color="Green">E17</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1">set</a>  holds not <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default">the <a href="e_siec.html#U2" target="_self">escape</a> of <font color="Maroon">c<sub>1</sub></font> <a href="xboole_0.html#K4">\</a> <span class="p2">(<span class="default"><a href="relat_1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="relat_1.html#K5">*</a> <span class="p1">(<span class="default"><a href="relat_1.html#K6">id</a> <span class="p2">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <a href="xboole_0.html#K4">\</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> the <a href="e_siec.html#U2" target="_self">escape</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>



<b>assume </b><a NAME="E1:40_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default">the <a href="e_siec.html#U2" target="_self">escape</a> of <font color="Maroon">c<sub>1</sub></font> <a href="xboole_0.html#K4">\</a> <span class="p2">(<span class="default"><a href="relat_1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="relat_1.html#K5">*</a> <span class="p1">(<span class="default"><a href="relat_1.html#K6">id</a> <span class="p2">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <a href="xboole_0.html#K4">\</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> the <a href="e_siec.html#U2" target="_self">escape</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>4</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:40_1"/><i><font color="Green">E18</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="e_siec.html#U2" target="_self">escape</a> of <font color="Maroon">c<sub>1</sub></font> <a href="xboole_0.html#K4">\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K6">id</a> <span class="p1">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <a href="xboole_0.html#K4">\</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> the <a href="e_siec.html#U2" target="_self">escape</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="relat_1.html#D8">RELAT_1:def 8</a></i>;<br/>
<a NAME="E4:40_1"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="e_siec.html#U2" target="_self">escape</a> of <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <a href="xboole_0.html#K4">\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> the <a href="e_siec.html#U2" target="_self">escape</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span> )
 
<b>by </b><i><a class="txt" href="e_siec.html#E3:40_1"><i><font color="Green">E18</font></i></a>, <a class="ref" href="relat_1.html#D10">RELAT_1:def 10</a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E5:40_1"/><b>then </b>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="e_siec.html#U2" target="_self">escape</a> of <font color="Maroon">c<sub>1</sub></font> &amp; not <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> the <a href="e_siec.html#U2" target="_self">escape</a> of <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<b>hence </b><a NAME="E6:40_1"/>
contradiction
 <b>by </b><i><a class="ref" href="relat_1.html#D5">RELAT_1:def 5</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:40"/>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1">set</a>  holds not <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default">the <a href="e_siec.html#U1" target="_self">entrance</a> of <font color="Maroon">c<sub>1</sub></font> <a href="xboole_0.html#K4">\</a> <span class="p2">(<span class="default"><a href="relat_1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="relat_1.html#K5">*</a> <span class="p1">(<span class="default"><a href="relat_1.html#K6">id</a> <span class="p2">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <a href="xboole_0.html#K4">\</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> the <a href="e_siec.html#U1" target="_self">entrance</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>



<b>assume </b><a NAME="E1:40_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default">the <a href="e_siec.html#U1" target="_self">entrance</a> of <font color="Maroon">c<sub>1</sub></font> <a href="xboole_0.html#K4">\</a> <span class="p2">(<span class="default"><a href="relat_1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="relat_1.html#K5">*</a> <span class="p1">(<span class="default"><a href="relat_1.html#K6">id</a> <span class="p2">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <a href="xboole_0.html#K4">\</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> the <a href="e_siec.html#U1" target="_self">entrance</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>4</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:40_2"/><i><font color="Green">E18</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="e_siec.html#U1" target="_self">entrance</a> of <font color="Maroon">c<sub>1</sub></font> <a href="xboole_0.html#K4">\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K6">id</a> <span class="p1">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <a href="xboole_0.html#K4">\</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> the <a href="e_siec.html#U1" target="_self">entrance</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="relat_1.html#D8">RELAT_1:def 8</a></i>;<br/>
<a NAME="E4:40_2"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="e_siec.html#U1" target="_self">entrance</a> of <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <a href="xboole_0.html#K4">\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> the <a href="e_siec.html#U1" target="_self">entrance</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span> )
 
<b>by </b><i><a class="txt" href="e_siec.html#E3:40_2"><i><font color="Green">E18</font></i></a>, <a class="ref" href="relat_1.html#D10">RELAT_1:def 10</a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E5:40_2"/><b>then </b>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="e_siec.html#U1" target="_self">entrance</a> of <font color="Maroon">c<sub>1</sub></font> &amp; not <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> the <a href="e_siec.html#U1" target="_self">entrance</a> of <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<b>hence </b><a NAME="E6:40_2"/>
contradiction
 <b>by </b><i><a class="ref" href="relat_1.html#D5">RELAT_1:def 5</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:40"/>
( <span class="p1">(<span class="default">the <a href="e_siec.html#U2" target="_self">escape</a> of <font color="Maroon">c<sub>1</sub></font> <a href="xboole_0.html#K4">\</a> <span class="p2">(<span class="default"><a href="relat_1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="relat_1.html#K5">*</a> <span class="p1">(<span class="default"><a href="relat_1.html#K6">id</a> <span class="p2">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <a href="xboole_0.html#K4">\</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> the <a href="e_siec.html#U2" target="_self">escape</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <span class="p1">(<span class="default">the <a href="e_siec.html#U1" target="_self">entrance</a> of <font color="Maroon">c<sub>1</sub></font> <a href="xboole_0.html#K4">\</a> <span class="p2">(<span class="default"><a href="relat_1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="relat_1.html#K5">*</a> <span class="p1">(<span class="default"><a href="relat_1.html#K6">id</a> <span class="p2">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <a href="xboole_0.html#K4">\</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> the <a href="e_siec.html#U1" target="_self">entrance</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 <b>by </b><i><a class="txt" href="e_siec.html#E1:40"><i><font color="Green">E17</font></i></a>, <a class="ref" href="relat_1.html#T56">RELAT_1:56</a></i>;<br/>


</div>
