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<b>let </b><font color="Maroon" title="c1">n</font>, <font color="Maroon" title="c2">m</font> be  non <a href="xxreal_0.html#NV4" title="XXREAL_0:NV.4">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> ;<br/>

<a NAME="E1:15"/>
for <font color="Olive" title="b1">x</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#K2" title="FINTOPO5:func.2">FTSL2</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">m</font></span>)</span> holds  <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="b1">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">x</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#K2" title="FINTOPO5:func.2">FTSL2</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">m</font></span>)</span>;<br/>

<b>consider </b><font color="Maroon" title="c4">u</font>, <font color="Maroon" title="c5">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E2:15_1"/><i><font color="Green" title="E9">A1</font></i>: 
( <font color="Maroon" title="c4">u</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> &amp; <font color="Maroon" title="c5">y</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> &amp; <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="c4">u</font>,<font color="Maroon" title="c5">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c6">i</font> = <font color="Maroon" title="c4">u</font>, <font color="Maroon" title="c7">j</font> = <font color="Maroon" title="c5">y</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>  <b>by </b><i><a class="txt" href="fintopo5.html#E2:15_1"><i><font color="Green" title="E9">A1</font></i></a></i>;<br/>
<a NAME="E4:15_1"/><i><font color="Green" title="E10">A2</font></i>: 
 <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> #)
 
<b>by </b><i><a class="ref" href="fintopo4.html#D4" title="FINTOPO4:def.4">FINTOPO4:def 4</a></i>;<br/>
<a NAME="E5:15_1"/><i><font color="Green" title="E11">A3</font></i>: 
 <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> #)
 
<b>by </b><i><a class="ref" href="fintopo4.html#D4" title="FINTOPO4:def.4">FINTOPO4:def 4</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c8">pi</font> = <font color="Maroon" title="c6">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> <b>by </b><i><a class="txt" href="fintopo5.html#E2:15_1"><i><font color="Green" title="E9">A1</font></i></a>, <a class="txt" href="fintopo5.html#E5:15_1"><i><font color="Green" title="E11">A3</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c9">pj</font> = <font color="Maroon" title="c7">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> <b>by </b><i><a class="txt" href="fintopo5.html#E2:15_1"><i><font color="Green" title="E9">A1</font></i></a>, <a class="txt" href="fintopo5.html#E4:15_1"><i><font color="Green" title="E10">A2</font></i></a></i>;<br/>
<a NAME="E8:15_1"/>
 <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">filled</a>
 
<b>by </b><i><a class="ref" href="fintopo4.html#T18" title="FINTOPO4:th.18">FINTOPO4:18</a></i>;<br/>
<a NAME="E9:15_1"/><b>then </b>
<font color="Maroon" title="c6">i</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="c8">pi</font>
 
<b>by </b><i><a class="ref" href="fin_topo.html#D4" title="FIN_TOPO:def.4">FIN_TOPO:def 4</a></i>;<br/>
<a NAME="E10:15_1"/><b>then </b><i><font color="Green" title="E12">A4</font></i>: 
<font color="Maroon" title="c6">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">i</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="txt" href="fintopo5.html#E5:15_1"><i><font color="Green" title="E11">A3</font></i></a>, <a class="ref" href="fin_topo.html#D1" title="FIN_TOPO:def.1">FIN_TOPO:def 1</a></i>;<br/>
<a NAME="E11:15_1"/>
 <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">filled</a>
 
<b>by </b><i><a class="ref" href="fintopo4.html#T18" title="FINTOPO4:th.18">FINTOPO4:18</a></i>;<br/>
<a NAME="E12:15_1"/><b>then </b>
<font color="Maroon" title="c7">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="c9">pj</font>
 
<b>by </b><i><a class="ref" href="fin_topo.html#D4" title="FIN_TOPO:def.4">FIN_TOPO:def 4</a></i>;<br/>
<a NAME="E13:15_1"/><b>then </b>
<font color="Maroon" title="c7">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</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> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c7">j</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="ref" href="fin_topo.html#D1" title="FIN_TOPO:def.1">FIN_TOPO:def 1</a></i>;<br/>
<a NAME="E14:15_1"/><b>then </b>
<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"><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> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">i</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>,<span class="p2">(<span class="default"><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> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c7">j</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a>, <a class="txt" href="fintopo5.html#E10:15_1"><i><font color="Green" title="E12">A4</font></i></a>, <a class="txt" href="fintopo5.html#E2:15_1"><i><font color="Green" title="E9">A1</font></i></a>, <a class="txt" href="fintopo5.html#E4:15_1"><i><font color="Green" title="E10">A2</font></i></a></i>;<br/>
<a NAME="E15:15_1"/><b>then </b>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="fintopo5.html#K1" title="FINTOPO5:func.1">Nbdl2</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="ref" href="fintopo5.html#D2" target="_self" title="FINTOPO5:def.2">Def2</a>, <a class="txt" href="fintopo5.html#E2:15_1"><i><font color="Green" title="E9">A1</font></i></a></i>;<br/>
<b>hence </b><a NAME="E16:15_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="c3">x</font>
 <b>by </b><i><a class="ref" href="fin_topo.html#D1" title="FIN_TOPO:def.1">FIN_TOPO:def 1</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:15"/>
 <a href="fintopo5.html#K2" title="FINTOPO5:func.2">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">filled</a>
 <b>by </b><i><a class="ref" href="fin_topo.html#D4" title="FIN_TOPO:def.4">FIN_TOPO:def 4</a></i>;<br/>


</div>
