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<b>let </b><font color="Maroon">n</font> be  non <a href="xboole_0.html#V1">zero</a>  <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">m</font> be  non <a href="xboole_0.html#V1">zero</a>  <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<a NAME="E1:16"/>
for <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="fintopo5.html#K2" target="_self">FTSL2</a> <font color="Maroon">n</font>,<font color="Maroon">m</font></span>)</span>  st <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="fin_topo.html#K1">U_FT</a> <font color="Olive">x</font> holds <br/><font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="fin_topo.html#K1">U_FT</a> <font color="Olive">y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="fintopo5.html#K2" target="_self">FTSL2</a> <font color="Maroon">n</font>,<font color="Maroon">m</font></span>)</span>, <font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="fintopo5.html#K2" target="_self">FTSL2</a> <font color="Maroon">n</font>,<font color="Maroon">m</font></span>)</span>;<br/>

<b>assume </b><a NAME="E1:16_1"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="fin_topo.html#K1">U_FT</a> <font color="Maroon">x</font>
 ;<br/>

<b>consider </b><font color="Maroon">xu</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">xv</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:16_1"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">xu</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">xv</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">xu</font>,<font color="Maroon">xv</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">xu</font>, <font color="Maroon">j</font> = <font color="Maroon">xv</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i/>;<br/>
<a NAME="E5:16_1"/><i><font color="Green">E35</font></i>: 
 <a href="fintopo4.html#K2">FTSL1</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a>  <a href="orders_2.html#G1">RelStr</a>(# <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m</font></span>)</span>,<span class="p1">(<span class="default"><a href="fintopo4.html#K1">Nbdl1</a> <font color="Maroon">m</font></span>)</span> #)
 
<b>by </b><i><a class="ref" href="fintopo4.html#D4">FINTOPO4:def 4</a></i>;<br/>
<a NAME="E6:16_1"/><i><font color="Green">E36</font></i>: 
 <a href="fintopo4.html#K2">FTSL1</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a>  <a href="orders_2.html#G1">RelStr</a>(# <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="fintopo4.html#K1">Nbdl1</a> <font color="Maroon">n</font></span>)</span> #)
 
<b>by </b><i><a class="ref" href="fintopo4.html#D4">FINTOPO4:def 4</a></i>;<br/>
<b>reconsider </b><font color="Maroon">pi</font> = <font color="Maroon">i</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="fintopo4.html#K2">FTSL1</a> <font color="Maroon">n</font></span>)</span> <b>by </b><i>, </i>;<br/>
<b>reconsider </b><font color="Maroon">pj</font> = <font color="Maroon">j</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="fintopo4.html#K2">FTSL1</a> <font color="Maroon">m</font></span>)</span> <b>by </b><i>, </i>;<br/>
<b>consider </b><font color="Maroon">yu</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">yv</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E10:16_1"/><i><font color="Green">E41</font></i>: 
( <font color="Maroon">yu</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">yv</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">y</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">yu</font>,<font color="Maroon">yv</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">i2</font> = <font color="Maroon">yu</font>, <font color="Maroon">j2</font> = <font color="Maroon">yv</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">pi2</font> = <font color="Maroon">i2</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="fintopo4.html#K2">FTSL1</a> <font color="Maroon">n</font></span>)</span> <b>by </b><i>, </i>;<br/>
<b>reconsider </b><font color="Maroon">pj2</font> = <font color="Maroon">j2</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="fintopo4.html#K2">FTSL1</a> <font color="Maroon">m</font></span>)</span> <b>by </b><i>, </i>;<br/>
<a NAME="E14:16_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="fintopo5.html#K2" target="_self">FTSL2</a> <font color="Maroon">n</font>,<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">x</font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="fin_topo.html#D1">FIN_TOPO:def 1</a>, </i>;<br/>
<a NAME="E15:16_1"/><b>then </b>
<font color="Maroon">y</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">(<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>
 
<b>by </b><i>, </i>;<br/>
<b>then consider </b><font color="Maroon">y1</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">y2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E17:16_1"/><i><font color="Green">E42</font></i>: 
( <font color="Maroon">y1</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">y2</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">y</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y1</font>,<font color="Maroon">y2</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="E18:16_1"/><i><font color="Green">E45</font></i>: 
( <font color="Maroon">y1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">yu</font> &amp; <font color="Maroon">y2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">yv</font> )
 
<b>by </b><i>, , <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>
<a NAME="E19:16_1"/><b>then </b><i><font color="Green">E46</font></i>: 
<font color="Maroon">pi2</font> <a href="hidden.html#R2">in</a>  <a href="fin_topo.html#K1">U_FT</a> <font color="Maroon">pi</font>
 
<b>by </b><i><a class="ref" href="fin_topo.html#D1">FIN_TOPO:def 1</a>, , </i>;<br/>
<a NAME="E20:16_1"/>
 <a href="fintopo4.html#K2">FTSL1</a> <font color="Maroon">n</font> is <a href="fin_topo.html#V2">symmetric</a>
 
<b>by </b><i><a class="ref" href="fintopo4.html#T19">FINTOPO4:19</a></i>;<br/>
<a NAME="E21:16_1"/><b>then </b>
<font color="Maroon">pi</font> <a href="hidden.html#R2">in</a>  <a href="fin_topo.html#K1">U_FT</a> <font color="Maroon">pi2</font>
 
<b>by </b><i>, <a class="ref" href="fin_topo.html#D15">FIN_TOPO:def 15</a></i>;<br/>
<a NAME="E22:16_1"/><b>then </b><i><font color="Green">E47</font></i>: 
<font color="Maroon">i</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">i2</font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i>, <a class="ref" href="fin_topo.html#D1">FIN_TOPO:def 1</a></i>;<br/>
<a NAME="E23:16_1"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">j2</font> <a href="hidden.html#R2">in</a>  <a href="fin_topo.html#K1">U_FT</a> <font color="Maroon">pj</font>
 
<b>by </b><i><a class="ref" href="fin_topo.html#D1">FIN_TOPO:def 1</a>, , , </i>;<br/>
<a NAME="E24:16_1"/>
 <a href="fintopo4.html#K2">FTSL1</a> <font color="Maroon">m</font> is <a href="fin_topo.html#V2">symmetric</a>
 
<b>by </b><i><a class="ref" href="fintopo4.html#T19">FINTOPO4:19</a></i>;<br/>
<a NAME="E25:16_1"/><b>then </b>
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="fin_topo.html#K1">U_FT</a> <font color="Maroon">pj2</font>
 
<b>by </b><i>, <a class="ref" href="fin_topo.html#D15">FIN_TOPO:def 15</a></i>;<br/>
<a NAME="E26:16_1"/><b>then </b>
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="fintopo4.html#K2">FTSL1</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">j2</font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="fin_topo.html#D1">FIN_TOPO:def 1</a></i>;<br/>
<a NAME="E27:16_1"/><b>then </b>
<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"><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">i2</font></span><a href="tarski.html#K1">}</a></span></span>)</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">j2</font></span><a href="tarski.html#K1">}</a></span></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/>
<a NAME="E28:16_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="fintopo5.html#K1" target="_self">Nbdl2</a> <font color="Maroon">n</font>,<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">y</font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E29:16_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="fin_topo.html#K1">U_FT</a> <font color="Maroon">y</font>
 <b>by </b><i><a class="ref" href="fin_topo.html#D1">FIN_TOPO:def 1</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:16"/>
 <a href="fintopo5.html#K2" target="_self">FTSL2</a> <font color="Maroon">n</font>,<font color="Maroon">m</font> is <a href="fin_topo.html#V2">symmetric</a>
 <b>by </b><i><a class="ref" href="fin_topo.html#D15">FIN_TOPO:def 15</a></i>;<br/>


</div>
