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

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b>for <font color="Olive">i</font>, <font color="Olive">j</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">i</font>,<font color="Olive">j</font></span><a href="tarski.html#K4">]</a></span> holds <br/><font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">i</font></span><a href="tarski.html#K1">}</a></span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="fintopo4.html#K1">Nbdl1</a> <font color="Maroon">m</font></span>)</span> <a href="relset_1.html#K10">.:</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">j</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="fintopo4.html#K1">Nbdl1</a> <font color="Maroon">n</font></span>)</span> <a href="relset_1.html#K10">.:</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">i</font></span><a href="tarski.html#K1">}</a></span></span>)</span>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">j</font></span><a href="tarski.html#K1">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span>;<br/>
<a NAME="E1:18_1_1"/><i><font color="Green">E24</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> holds <br/> ex <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:18_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 ;<br/>

<b>then consider </b><font color="Maroon">u</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:18_1_1_1"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">u</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">u</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon">i</font> = <font color="Maroon">u</font>, <font color="Maroon">j</font> = <font color="Maroon">y</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i/>;<br/>
<b>set </b><font color="Maroon">y3</font> = <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">i</font></span><a href="tarski.html#K1">}</a></span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="fintopo4.html#K1">Nbdl1</a> <font color="Maroon">m</font></span>)</span> <a href="relset_1.html#K10">.:</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">j</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="fintopo4.html#K1">Nbdl1</a> <font color="Maroon">n</font></span>)</span> <a href="relset_1.html#K10">.:</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">i</font></span><a href="tarski.html#K1">}</a></span></span>)</span>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">j</font></span><a href="tarski.html#K1">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span>;<br/>
<a NAME="E5:18_1_1_1"/>
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">i</font></span><a href="tarski.html#K1">}</a></span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="fintopo4.html#K1">Nbdl1</a> <font color="Maroon">m</font></span>)</span> <a href="relset_1.html#K10">.:</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">j</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="fintopo4.html#K1">Nbdl1</a> <font color="Maroon">n</font></span>)</span> <a href="relset_1.html#K10">.:</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">i</font></span><a href="tarski.html#K1">}</a></span></span>)</span>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">j</font></span><a href="tarski.html#K1">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">z</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:18_1_1_1_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">i</font></span><a href="tarski.html#K1">}</a></span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="fintopo4.html#K1">Nbdl1</a> <font color="Maroon">m</font></span>)</span> <a href="relset_1.html#K10">.:</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">j</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="fintopo4.html#K1">Nbdl1</a> <font color="Maroon">n</font></span>)</span> <a href="relset_1.html#K10">.:</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">i</font></span><a href="tarski.html#K1">}</a></span></span>)</span>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">j</font></span><a href="tarski.html#K1">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span>
 ;<br/>

<a NAME="E2:18_1_1_1_1"/><b>then </b><i><font color="Green">E35</font></i>: 
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">i</font></span><a href="tarski.html#K1">}</a></span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="fintopo4.html#K1">Nbdl1</a> <font color="Maroon">m</font></span>)</span> <a href="relset_1.html#K10">.:</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">j</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> or <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="fintopo4.html#K1">Nbdl1</a> <font color="Maroon">n</font></span>)</span> <a href="relset_1.html#K10">.:</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">i</font></span><a href="tarski.html#K1">}</a></span></span>)</span>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">j</font></span><a href="tarski.html#K1">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:18_1_1_1_1_1"/>
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">i</font></span><a href="tarski.html#K1">}</a></span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="fintopo4.html#K1">Nbdl1</a> <font color="Maroon">m</font></span>)</span> <a href="relset_1.html#K10">.:</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">j</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> or <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="fintopo4.html#K1">Nbdl1</a> <font color="Maroon">n</font></span>)</span> <a href="relset_1.html#K10">.:</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">i</font></span><a href="tarski.html#K1">}</a></span></span>)</span>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">j</font></span><a href="tarski.html#K1">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span> )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:18_1_1_1_1_1_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">i</font></span><a href="tarski.html#K1">}</a></span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="fintopo4.html#K1">Nbdl1</a> <font color="Maroon">m</font></span>)</span> <a href="relset_1.html#K10">.:</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">j</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">x4</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">y4</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:18_1_1_1_1_1_1"/><i><font color="Green">E36</font></i>: 
( <font color="Maroon">x4</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">i</font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">y4</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="fintopo4.html#K1">Nbdl1</a> <font color="Maroon">m</font></span>)</span> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">j</font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x4</font>,<font color="Maroon">y4</font></span><a href="tarski.html#K4">]</a></span> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/><a NAME="E4:18_1_1_1_1_1_1"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">x4</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a>, </i>;<br/><a NAME="E5:18_1_1_1_1_1_1"/>
( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font> &amp; <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m</font> )
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E6:18_1_1_1_1_1_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:18_1_1_1_1_1_2"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="fintopo4.html#K1">Nbdl1</a> <font color="Maroon">n</font></span>)</span> <a href="relset_1.html#K10">.:</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">i</font></span><a href="tarski.html#K1">}</a></span></span>)</span>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">j</font></span><a href="tarski.html#K1">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">x4</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">y4</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:18_1_1_1_1_1_2"/><i><font color="Green">E42</font></i>: 
( <font color="Maroon">x4</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="fintopo4.html#K1">Nbdl1</a> <font color="Maroon">n</font></span>)</span> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">i</font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">y4</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">j</font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x4</font>,<font color="Maroon">y4</font></span><a href="tarski.html#K4">]</a></span> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/><a NAME="E4:18_1_1_1_1_1_2"/>
<font color="Maroon">y4</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m</font>
 <b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a>, </i>;<br/><b>hence </b><a NAME="E5:18_1_1_1_1_1_2"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E6:18_1_1_1"/><b>then </b><i><font color="Green">E45</font></i>: 
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">i</font></span><a href="tarski.html#K1">}</a></span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="fintopo4.html#K1">Nbdl1</a> <font color="Maroon">m</font></span>)</span> <a href="relset_1.html#K10">.:</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">j</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="fintopo4.html#K1">Nbdl1</a> <font color="Maroon">n</font></span>)</span> <a href="relset_1.html#K10">.:</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">i</font></span><a href="tarski.html#K1">}</a></span></span>)</span>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">j</font></span><a href="tarski.html#K1">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 
;<br/>
<a NAME="E7:18_1_1_1"/>
for <font color="Olive">i4</font>, <font color="Olive">j4</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">i4</font>,<font color="Olive">j4</font></span><a href="tarski.html#K4">]</a></span> holds <br/><span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">i</font></span><a href="tarski.html#K1">}</a></span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="fintopo4.html#K1">Nbdl1</a> <font color="Maroon">m</font></span>)</span> <a href="relset_1.html#K10">.:</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">j</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="fintopo4.html#K1">Nbdl1</a> <font color="Maroon">n</font></span>)</span> <a href="relset_1.html#K10">.:</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">i</font></span><a href="tarski.html#K1">}</a></span></span>)</span>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">j</font></span><a href="tarski.html#K1">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">i4</font></span><a href="tarski.html#K1">}</a></span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="fintopo4.html#K1">Nbdl1</a> <font color="Maroon">m</font></span>)</span> <a href="relset_1.html#K10">.:</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">j4</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="fintopo4.html#K1">Nbdl1</a> <font color="Maroon">n</font></span>)</span> <a href="relset_1.html#K10">.:</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">i4</font></span><a href="tarski.html#K1">}</a></span></span>)</span>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">j4</font></span><a href="tarski.html#K1">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i4</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">j4</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:18_1_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i4</font>,<font color="Maroon">j4</font></span><a href="tarski.html#K4">]</a></span>
 ;<br/>

<a NAME="E2:18_1_1_1_2"/><b>then </b>
( <font color="Maroon">i4</font> <a href="hidden.html#R1">=</a> <font color="Maroon">u</font> &amp; <font color="Maroon">j4</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> )
 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>
<b>hence </b><a NAME="E3:18_1_1_1_2"/>
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">i</font></span><a href="tarski.html#K1">}</a></span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="fintopo4.html#K1">Nbdl1</a> <font color="Maroon">m</font></span>)</span> <a href="relset_1.html#K10">.:</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">j</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="fintopo4.html#K1">Nbdl1</a> <font color="Maroon">n</font></span>)</span> <a href="relset_1.html#K10">.:</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">i</font></span><a href="tarski.html#K1">}</a></span></span>)</span>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">j</font></span><a href="tarski.html#K1">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">i4</font></span><a href="tarski.html#K1">}</a></span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="fintopo4.html#K1">Nbdl1</a> <font color="Maroon">m</font></span>)</span> <a href="relset_1.html#K10">.:</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">j4</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="fintopo4.html#K1">Nbdl1</a> <font color="Maroon">n</font></span>)</span> <a href="relset_1.html#K10">.:</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">i4</font></span><a href="tarski.html#K1">}</a></span></span>)</span>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">j4</font></span><a href="tarski.html#K1">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E8:18_1_1_1"/>
 ex <font color="Olive">y2</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">y2</font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">x</font>,<font color="Olive">y2</font>] )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">f</font> being   <a href="funct_2.html#NM1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>, <a href="zfmisc_1.html#K1">bool</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span><b> such that </b><br/><a NAME="E3:18_1_1"/><i><font color="Green">E46</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S1">FUNCT_2:sch 1</a>();<br/></i>
<b>consider </b><font color="Maroon">R</font> being   <a href="relset_1.html#NM3">Relation</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span><b> such that </b><br/><a NAME="E5:18_1_1"/><i><font color="Green">E47</font></i>: 
for <font color="Olive">i</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> holds <br/><font color="Maroon">R</font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">i</font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T170">FUNCT_2:170</a></i>;<br/>
<b>take </b>
<font color="Maroon">R</font>
;<br/>

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E6:18_1_1"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 ;<br/>

<b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">j</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E7:18_1_1"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i</font>,<font color="Maroon">j</font></span><a href="tarski.html#K4">]</a></span>
 ;<br/>

<b>thus </b><font color="Maroon">R</font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> = 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">i</font></span><a href="tarski.html#K1">}</a></span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="fintopo4.html#K1">Nbdl1</a> <font color="Maroon">m</font></span>)</span> <a href="relset_1.html#K10">.:</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">j</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="fintopo4.html#K1">Nbdl1</a> <font color="Maroon">n</font></span>)</span> <a href="relset_1.html#K10">.:</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">i</font></span><a href="tarski.html#K1">}</a></span></span>)</span>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">j</font></span><a href="tarski.html#K1">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span>
<b>by </b><i>, , </i>
;<br/>


</div>
