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<div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:21"/>
<font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/>

<a NAME="E2:21"/><b>then </b><i><font color="Green">E11</font></i>: 
 <a href="fintopo4.html#K2" target="_self">FTSL1</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="fin_topo.html#G1">FT_Space_Str</a>(# <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="fintopo4.html#K1" target="_self">Nbdl1</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> #)
 
<b>by </b><i><a class="ref" href="fintopo4.html#D4" target="_self">Def4</a></i>;<br/>
<a NAME="E3:21"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="fintopo4.html#K2" target="_self">FTSL1</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> holds <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="fin_topo.html#K1">U_FT</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="fintopo4.html#K2" target="_self">FTSL1</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/>

<a NAME="E1:21_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="fintopo4.html#E2:21"><i><font color="Green">E11</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>3</sub></font> = <font color="Maroon">c<sub>2</sub></font> as   <a href="nat_1.html#NM1">Nat</a> ;<br/>
<a NAME="E3:21_1"/>
the <a href="fin_topo.html#U1">Nbd</a> of <span class="p1">(<span class="default"><a href="fintopo4.html#K2" target="_self">FTSL1</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="enumset1.html#K1">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>,1</span>)</span>,<span class="p2">(<span class="default"><a href="rfunct_3.html#K1">min</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="enumset1.html#K1">}</a></span>
 
<b>by </b><i><a class="txt" href="fintopo4.html#E2:21"><i><font color="Green">E11</font></i></a>, <a class="ref" href="fintopo4.html#D3" target="_self">Def3</a></i>;<br/>
<a NAME="E4:21_1"/><b>then </b>
 <a href="fin_topo.html#K1">U_FT</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="enumset1.html#K1">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><a href="limfunc1.html#K1">max</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>,1</span>)</span>,<span class="p2">(<span class="default"><a href="rfunct_3.html#K1">min</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="enumset1.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="fin_topo.html#D1">FIN_TOPO:def 1</a></i>;<br/>
<b>hence </b><a NAME="E5:21_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="fin_topo.html#K1">U_FT</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="enumset1.html#D1">ENUMSET1:def 1</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:21"/>
 <a href="fintopo4.html#K2" target="_self">FTSL1</a> <font color="Maroon">c<sub>1</sub></font> is <a href="fin_topo.html#V2">filled</a>
 <b>by </b><i><a class="ref" href="fin_topo.html#D3">FIN_TOPO:def 3</a></i>;<br/>


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