<?xml version="1.0"?>
<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="E1:25"/>
<font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/>

<a NAME="E2:25"/><b>then </b><i><font color="Green">E13</font></i>: 
 <a href="fintopo4.html#K4" target="_self">FTSC1</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#K3" target="_self">Nbdc1</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> #)
 
<b>by </b><i><a class="ref" href="fintopo4.html#D6" target="_self">Def6</a></i>;<br/>
<a NAME="E3:25"/>
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#K4" target="_self">FTSC1</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="25_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#K4" target="_self">FTSC1</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/>

<a NAME="E1:25_1"/><i><font color="Green">E14</font></i>: 
<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#E2:25"><i><font color="Green">E13</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="E3:25_1"/><i><font color="Green">E15</font></i>: 
the <a href="fin_topo.html#U1">Nbd</a> of <span class="p1">(<span class="default"><a href="fintopo4.html#K4" target="_self">FTSC1</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#R1">=</a>  <a href="fintopo4.html#K3" target="_self">Nbdc1</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="fintopo4.html#E2:25"><i><font color="Green">E13</font></i></a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>4</sub></font> =  <a href="fintopo4.html#K3" target="_self">Nbdc1</a> <font color="Maroon">c<sub>1</sub></font>;<br/>
<a NAME="E4:25_1"/><i><font color="Green">E16</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="txt" href="fintopo4.html#E1:25_1"><i><font color="Green">E14</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E5:25_1"/><b>then </b><i><font color="Green">E17</font></i>: 
( 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>3</sub></font> or <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 1 )
 
<b>by </b><i><a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/>
<a NAME="E6:25_1"/><i><font color="Green">E18</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>1</sub></font> or <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="txt" href="fintopo4.html#E4:25_1"><i><font color="Green">E16</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/>
<a NAME="E7:25_1"/><i><font color="Green">E19</font></i>: 
( ( 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>1</sub></font> implies <span class="p1">(<span class="default"><a href="fintopo4.html#K3" target="_self">Nbdc1</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</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"><font color="Maroon">c<sub>3</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span><a href="enumset1.html#K1">}</a></span> ) &amp; ( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>1</sub></font> implies <span class="p1">(<span class="default"><a href="fintopo4.html#K3" target="_self">Nbdc1</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</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>,<font color="Maroon">c<sub>1</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span><a href="enumset1.html#K1">}</a></span> ) &amp; ( 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> implies <span class="p1">(<span class="default"><a href="fintopo4.html#K3" target="_self">Nbdc1</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</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"><font color="Maroon">c<sub>3</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>,1</span><a href="enumset1.html#K1">}</a></span> ) &amp; ( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> implies <span class="p1">(<span class="default"><a href="fintopo4.html#K3" target="_self">Nbdc1</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="tarski.html#K1">}</a></span> ) )
 
<b>by </b><i><a class="txt" href="fintopo4.html#E1:25_1"><i><font color="Green">E14</font></i></a>, <a class="ref" href="fintopo4.html#D5" target="_self">Def5</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:25_1_1_1"/>
( ( 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>1</sub></font> ) or ( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>1</sub></font> ) or ( 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> ) or ( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> ) )
 </span><b>by </b><i><a class="txt" href="fintopo4.html#E5:25_1"><i><font color="Green">E17</font></i></a>, <a class="txt" href="fintopo4.html#E6:25_1"><i><font color="Green">E18</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_1_1_1_1"><b>case </b></a><a NAME="E1:25_1_1_1_1"/>
( 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>1</sub></font> )
 ;<br/><div class="add"><a NAME="E2:25_1_1_1_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"><font color="Maroon">c<sub>3</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span><a href="enumset1.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="fintopo4.html#E3:25_1"><i><font color="Green">E15</font></i></a>, <a class="txt" href="fintopo4.html#E7:25_1"><i><font color="Green">E19</font></i></a>, <a class="ref" href="fin_topo.html#D1">FIN_TOPO:def 1</a></i>;<br/><b>hence </b><a NAME="E3:25_1_1_1_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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_1_1_1_2"><b>case </b></a><a NAME="E1:25_1_1_1_2"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>1</sub></font> )
 ;<br/><div class="add"><a NAME="E2:25_1_1_1_2"/><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>,<font color="Maroon">c<sub>1</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span><a href="enumset1.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="fintopo4.html#E3:25_1"><i><font color="Green">E15</font></i></a>, <a class="txt" href="fintopo4.html#E7:25_1"><i><font color="Green">E19</font></i></a>, <a class="ref" href="fin_topo.html#D1">FIN_TOPO:def 1</a></i>;<br/><b>hence </b><a NAME="E3:25_1_1_1_2"/>
<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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_1_1_1_3"><b>case </b></a><a NAME="E1:25_1_1_1_3"/>
( 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> )
 ;<br/><div class="add"><a NAME="E2:25_1_1_1_3"/><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"><font color="Maroon">c<sub>3</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>,1</span><a href="enumset1.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="fintopo4.html#E3:25_1"><i><font color="Green">E15</font></i></a>, <a class="txt" href="fintopo4.html#E7:25_1"><i><font color="Green">E19</font></i></a>, <a class="ref" href="fin_topo.html#D1">FIN_TOPO:def 1</a></i>;<br/><b>hence </b><a NAME="E3:25_1_1_1_3"/>
<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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_1_1_1_4"><b>case </b></a><a NAME="E1:25_1_1_1_4"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> )
 ;<br/><div class="add"><a NAME="E2:25_1_1_1_4"/><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="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="fintopo4.html#E3:25_1"><i><font color="Green">E15</font></i></a>, <a class="txt" href="fintopo4.html#E7:25_1"><i><font color="Green">E19</font></i></a>, <a class="ref" href="fin_topo.html#D1">FIN_TOPO:def 1</a></i>;<br/><b>hence </b><a NAME="E3:25_1_1_1_4"/>
<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="tarski.html#D1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E9:25_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>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:25"/>
 <a href="fintopo4.html#K4" target="_self">FTSC1</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/>


</div>
