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

<span class="kw">let </span><font color="Maroon" title="c1">n</font>, <font color="Maroon" title="c2">m</font> be   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">zero</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="fintopo5.html#K3" title="FINTOPO5:func.3">FTSL2</a> (<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">m</font>) is  <a href="fin_topo.html#V1" title="FIN_TOPO:attr.1">symmetric</a> </span><br/>

<a NAME="E1:16"/>
 for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="fintopo5.html#K3" title="FINTOPO5:func.3">FTSL2</a> (<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">m</font>)</span>)</span>  st <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fin_topo.html#K1" title="FIN_TOPO:func.1">U_FT</a> <font color="Olive" title="b1">x</font> holds <br/><font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fin_topo.html#K1" title="FIN_TOPO:func.1">U_FT</a> <font color="Olive" title="b2">y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:16_1"/><span class="lab"><font color="Green" title="E9">A1</font></span>: 
 <a href="fintopo4.html#K2" title="FINTOPO4:func.2">FTSL1</a> <font color="Maroon" title="c2">m</font> is  <a href="fin_topo.html#V1" title="FIN_TOPO:attr.1">symmetric</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="fintopo4.html#T19" title="FINTOPO4:th.19">FINTOPO4:19</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c3">x</font>, <font color="Maroon" title="c4">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="fintopo5.html#K3" title="FINTOPO5:func.3">FTSL2</a> (<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">m</font>)</span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fin_topo.html#K1" title="FIN_TOPO:func.1">U_FT</a> <font color="Maroon" title="c3">x</font> implies <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fin_topo.html#K1" title="FIN_TOPO:func.1">U_FT</a> <font color="Maroon" title="c4">y</font> )</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c5">xu</font>, <font color="Maroon" title="c6">xv</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:16_1"/><span class="lab"><font color="Green" title="E10">A2</font></span>: 
<font color="Maroon" title="c5">xu</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font>
 <span class="kw">and </span><br/><a NAME="E4:16_1"/><span class="lab"><font color="Green" title="E11">A3</font></span>: 
<font color="Maroon" title="c6">xv</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">m</font>
 <span class="kw">and </span><br/><a NAME="E5:16_1"/><span class="lab"><font color="Green" title="E12">A4</font></span>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">xu</font>,<font color="Maroon" title="c6">xv</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c7">i</font> = <font color="Maroon" title="c5">xu</font>, <font color="Maroon" title="c8">j</font> = <font color="Maroon" title="c6">xv</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:16_1"><span class="lab"><font color="Green" title="E10">A2</font></span></a>, <a class="txt" href="#E4:16_1"><span class="lab"><font color="Green" title="E11">A3</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c9">yu</font>, <font color="Maroon" title="c10">yv</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E8:16_1"/><span class="lab"><font color="Green" title="E13">A5</font></span>: 
<font color="Maroon" title="c9">yu</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font>
 <span class="kw">and </span><br/><a NAME="E9:16_1"/><span class="lab"><font color="Green" title="E14">A6</font></span>: 
<font color="Maroon" title="c10">yv</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">m</font>
 <span class="kw">and </span><br/><a NAME="E10:16_1"/><span class="lab"><font color="Green" title="E15">A7</font></span>: 
<font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">yu</font>,<font color="Maroon" title="c10">yv</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c11">i2</font> = <font color="Maroon" title="c9">yu</font>, <font color="Maroon" title="c12">j2</font> = <font color="Maroon" title="c10">yv</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:16_1"><span class="lab"><font color="Green" title="E13">A5</font></span></a>, <a class="txt" href="#E9:16_1"><span class="lab"><font color="Green" title="E14">A6</font></span></a></span>;<br/>
<a NAME="E12:16_1"/><span class="lab"><font color="Green" title="E16">A8</font></span>: 
 <a href="fintopo4.html#K2" title="FINTOPO4:func.2">FTSL1</a> <font color="Maroon" title="c2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">m</font></span>)</span>,<span class="p1">(<span class="default"><a href="fintopo4.html#K1" title="FINTOPO4:func.1">Nbdl1</a> <font color="Maroon" title="c2">m</font></span>)</span> #)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="fintopo4.html#D4" title="FINTOPO4:def.4">FINTOPO4:def 4</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c13">pj</font> = <font color="Maroon" title="c8">j</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="fintopo4.html#K2" title="FINTOPO4:func.2">FTSL1</a> <font color="Maroon" title="c2">m</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:16_1"><span class="lab"><font color="Green" title="E11">A3</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c14">pj2</font> = <font color="Maroon" title="c12">j2</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="fintopo4.html#K2" title="FINTOPO4:func.2">FTSL1</a> <font color="Maroon" title="c2">m</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:16_1"><span class="lab"><font color="Green" title="E16">A8</font></span></a>, <a class="txt" href="#E9:16_1"><span class="lab"><font color="Green" title="E14">A6</font></span></a></span>;<br/>
<span class="kw">assume </span><a NAME="E15:16_1"/>
<font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fin_topo.html#K1" title="FIN_TOPO:func.1">U_FT</a> <font color="Maroon" title="c3">x</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fin_topo.html#K1" title="FIN_TOPO:func.1">U_FT</a> <font color="Maroon" title="c4">y</font></span><br/>

<a NAME="E16:16_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="fintopo5.html#K1" title="FINTOPO5:func.1">Im</a> (<span class="p3">(<span class="default"><a href="fintopo4.html#K1" title="FINTOPO4:func.1">Nbdl1</a> <font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c7">i</font>)</span>)</span>,<span class="p2">(<span class="default"><a href="fintopo5.html#K1" title="FINTOPO5:func.1">Im</a> (<span class="p3">(<span class="default"><a href="fintopo4.html#K1" title="FINTOPO4:func.1">Nbdl1</a> <font color="Maroon" title="c2">m</font></span>)</span>,<font color="Maroon" title="c8">j</font>)</span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:16_1"><span class="lab"><font color="Green" title="E12">A4</font></span></a>, <a class="ref" href="fintopo5.html#D2" target="_self" title="FINTOPO5:def.2">Def2</a></span>;<br/>
<a NAME="E17:16_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E17">A9</font></span>: 
 ex <font color="Olive" title="b1">y1</font>, <font color="Olive" title="b2">y2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> (<span class="p1">(<span class="default"><a href="fintopo4.html#K1" title="FINTOPO4:func.1">Nbdl1</a> <font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c7">i</font>) &amp; <font color="Olive" title="b2">y2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> (<span class="p1">(<span class="default"><a href="fintopo4.html#K1" title="FINTOPO4:func.1">Nbdl1</a> <font color="Maroon" title="c2">m</font></span>)</span>,<font color="Maroon" title="c8">j</font>) &amp; <font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">y1</font>,<font color="Olive" title="b2">y2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></span>;<br/>
<a NAME="E18:16_1"/><span class="kw">then </span>
<font color="Maroon" title="c12">j2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fin_topo.html#K1" title="FIN_TOPO:func.1">U_FT</a> <font color="Maroon" title="c13">pj</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:16_1"><span class="lab"><font color="Green" title="E16">A8</font></span></a>, <a class="txt" href="#E10:16_1"><span class="lab"><font color="Green" title="E15">A7</font></span></a>, <a class="ref" href="zfmisc_1.html#T27" title="ZFMISC_1:th.27">ZFMISC_1:27</a></span>;<br/>
<a NAME="E19:16_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E18">A10</font></span>: 
<font color="Maroon" title="c8">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fin_topo.html#K1" title="FIN_TOPO:func.1">U_FT</a> <font color="Maroon" title="c14">pj2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:16_1"><span class="lab"><font color="Green" title="E9">A1</font></span></a>, <a class="ref" href="fin_topo.html#D13" title="FIN_TOPO:def.13">FIN_TOPO:def 13</a></span>;<br/>
<a NAME="E20:16_1"/><span class="lab"><font color="Green" title="E19">A11</font></span>: 
 <a href="fintopo4.html#K2" title="FINTOPO4:func.2">FTSL1</a> <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="fintopo4.html#K1" title="FINTOPO4:func.1">Nbdl1</a> <font color="Maroon" title="c1">n</font></span>)</span> #)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="fintopo4.html#D4" title="FINTOPO4:def.4">FINTOPO4:def 4</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c15">pi</font> = <font color="Maroon" title="c7">i</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="fintopo4.html#K2" title="FINTOPO4:func.2">FTSL1</a> <font color="Maroon" title="c1">n</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:16_1"><span class="lab"><font color="Green" title="E10">A2</font></span></a></span>;<br/>
<a NAME="E22:16_1"/><span class="lab"><font color="Green" title="E20">A12</font></span>: 
 <a href="fintopo4.html#K2" title="FINTOPO4:func.2">FTSL1</a> <font color="Maroon" title="c1">n</font> is  <a href="fin_topo.html#V1" title="FIN_TOPO:attr.1">symmetric</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="fintopo4.html#T19" title="FINTOPO4:th.19">FINTOPO4:19</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c16">pi2</font> = <font color="Maroon" title="c11">i2</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="fintopo4.html#K2" title="FINTOPO4:func.2">FTSL1</a> <font color="Maroon" title="c1">n</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:16_1"><span class="lab"><font color="Green" title="E19">A11</font></span></a>, <a class="txt" href="#E8:16_1"><span class="lab"><font color="Green" title="E13">A5</font></span></a></span>;<br/>
<a NAME="E24:16_1"/>
<font color="Maroon" title="c16">pi2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fin_topo.html#K1" title="FIN_TOPO:func.1">U_FT</a> <font color="Maroon" title="c15">pi</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:16_1"><span class="lab"><font color="Green" title="E19">A11</font></span></a>, <a class="txt" href="#E10:16_1"><span class="lab"><font color="Green" title="E15">A7</font></span></a>, <a class="txt" href="#E17:16_1"><span class="lab"><font color="Green" title="E17">A9</font></span></a>, <a class="ref" href="zfmisc_1.html#T27" title="ZFMISC_1:th.27">ZFMISC_1:27</a></span>;<br/>
<a NAME="E25:16_1"/><span class="kw">then </span>
<font color="Maroon" title="c15">pi</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fin_topo.html#K1" title="FIN_TOPO:func.1">U_FT</a> <font color="Maroon" title="c16">pi2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:16_1"><span class="lab"><font color="Green" title="E20">A12</font></span></a>, <a class="ref" href="fin_topo.html#D13" title="FIN_TOPO:def.13">FIN_TOPO:def 13</a></span>;<br/>
<a NAME="E26:16_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="fintopo5.html#K1" title="FINTOPO5:func.1">Im</a> (<span class="p3">(<span class="default"><a href="fintopo4.html#K1" title="FINTOPO4:func.1">Nbdl1</a> <font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c11">i2</font>)</span>)</span>,<span class="p2">(<span class="default"><a href="fintopo5.html#K1" title="FINTOPO5:func.1">Im</a> (<span class="p3">(<span class="default"><a href="fintopo4.html#K1" title="FINTOPO4:func.1">Nbdl1</a> <font color="Maroon" title="c2">m</font></span>)</span>,<font color="Maroon" title="c12">j2</font>)</span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:16_1"><span class="lab"><font color="Green" title="E12">A4</font></span></a>, <a class="txt" href="#E12:16_1"><span class="lab"><font color="Green" title="E16">A8</font></span></a>, <a class="txt" href="#E20:16_1"><span class="lab"><font color="Green" title="E19">A11</font></span></a>, <a class="txt" href="#E19:16_1"><span class="lab"><font color="Green" title="E18">A10</font></span></a>, <a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E27:16_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fin_topo.html#K1" title="FIN_TOPO:func.1">U_FT</a> <font color="Maroon" title="c4">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:16_1"><span class="lab"><font color="Green" title="E15">A7</font></span></a>, <a class="ref" href="fintopo5.html#D2" target="_self" title="FINTOPO5:def.2">Def2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E2:16"/>
 <a href="fintopo5.html#K3" title="FINTOPO5:func.3">FTSL2</a> (<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">m</font>) is  <a href="fin_topo.html#V1" title="FIN_TOPO:attr.1">symmetric</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="fin_topo.html#D13" title="FIN_TOPO:def.13">FIN_TOPO:def 13</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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