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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="xboole_0.html#V1">zero</a> <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] means <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a<sub>2</sub></font>,1</span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<a NAME="E1:17"/><i><font color="Green">E9</font></i>: 
 <a href="fintopo4.html#K2">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">Nbdl1</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> #)
 
<b>by </b><i><a class="ref" href="fintopo4.html#D4">FINTOPO4:def 4</a></i>;<br/>
<a NAME="E2:17"/><i><font color="Green">E10</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="fintopo5.html#K2" target="_self">FTSL2</a> <font color="Maroon">c<sub>1</sub></font>,1</span>)</span> holds <br/>ex <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="fintopo4.html#K2">FTSL1</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>2</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:17_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="fintopo5.html#K2" target="_self">FTSL2</a> <font color="Maroon">c<sub>1</sub></font>,1</span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:17_1"/><i><font color="Green">E11</font></i>: 
( <font color="Maroon">c<sub>3</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> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> 1 &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></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">c<sub>5</sub></font> = <font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>4</sub></font> as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="txt" href="fintopo5.html#E3:17_1"><i><font color="Green">E11</font></i></a></i>;<br/>
<a NAME="E5:17_1"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 
<b>by </b><i><a class="txt" href="fintopo5.html#E3:17_1"><i><font color="Green">E11</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E6:17_1"/><b>then </b><i><font color="Green">E12</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font>]
 
<b>by </b><i><a class="txt" href="fintopo5.html#E3:17_1"><i><font color="Green">E11</font></i></a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/>
<a NAME="E7:17_1"/>
 <a href="fintopo4.html#K2">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">Nbdl1</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> #)
 
<b>by </b><i><a class="ref" href="fintopo4.html#D4">FINTOPO4:def 4</a></i>;<br/>
<b>hence </b><a NAME="E8:17_1"/>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="fintopo4.html#K2">FTSL1</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font>] )
 <b>by </b><i><a class="txt" href="fintopo5.html#E3:17_1"><i><font color="Green">E11</font></i></a>, <a class="txt" href="fintopo5.html#E6:17_1"><i><font color="Green">E12</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:17"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM7">Function</a> of <span class="p1">(<span class="default"><a href="fintopo5.html#K2" target="_self">FTSL2</a> <font color="Maroon">c<sub>1</sub></font>,1</span>)</span>,<span class="p1">(<span class="default"><a href="fintopo4.html#K2">FTSL1</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> st <br/>for <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="fintopo5.html#K2" target="_self">FTSL2</a> <font color="Maroon">c<sub>1</sub></font>,1</span>)</span> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>2</sub></font>]
 
<b>from </b><i><a class="ref" href="funct_2.html#S1">FUNCT_2:sch 1</a>(<a class="txt" href="fintopo5.html#E2:17"><i><font color="Green">E10</font></i></a>);<br/></i>
<b>then consider </b><font color="Maroon">c<sub>2</sub></font> being   <a href="struct_0.html#NM7">Function</a> of <span class="p1">(<span class="default"><a href="fintopo5.html#K2" target="_self">FTSL2</a> <font color="Maroon">c<sub>1</sub></font>,1</span>)</span>,<span class="p1">(<span class="default"><a href="fintopo4.html#K2">FTSL1</a> <font color="Maroon">c<sub>1</sub></font></span>)</span><b> such that </b><br/><a NAME="E5:17"/><i><font color="Green">E11</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="fintopo5.html#K2" target="_self">FTSL2</a> <font color="Maroon">c<sub>1</sub></font>,1</span>)</span> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>]
 ;<br/>
<a NAME="E6:17"/>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>2</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:17_2"/><i><font color="Green">E12</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> )
 ;<br/>

<a NAME="E2:17_2"/><b>then </b><i><font color="Green">E13</font></i>: 
<font color="Maroon">c<sub>3</sub></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"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> 1</span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E3:17_2"/><i><font color="Green">E14</font></i>: 
<font color="Maroon">c<sub>4</sub></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"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> 1</span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="txt" href="fintopo5.html#E1:17_2"><i><font color="Green">E12</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E4:17_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,1</span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="fintopo5.html#E5:17"><i><font color="Green">E11</font></i></a>, <a class="txt" href="fintopo5.html#E2:17_2"><i><font color="Green">E13</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:17_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="fintopo5.html#E1:17_2"><i><font color="Green">E12</font></i></a>, <a class="txt" href="fintopo5.html#E5:17"><i><font color="Green">E11</font></i></a>, <a class="txt" href="fintopo5.html#E3:17_2"><i><font color="Green">E14</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:17"/><b>then </b><i><font color="Green">E12</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is <a href="funct_1.html#V2">one-to-one</a>
 
<b>by </b><i><a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/>
<a NAME="E8:17"/><i><font color="Green">E13</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="fintopo4.html#K2">FTSL1</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>2</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:17_3"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="fintopo4.html#K2">FTSL1</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 ;<br/>

<a NAME="E2:17_3"/><b>then </b><i><font color="Green">E14</font></i>: 
<font color="Maroon">c<sub>3</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="fintopo5.html#E1:17"><i><font color="Green">E9</font></i></a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>4</sub></font> = <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,1</span><a href="tarski.html#K4">]</a></span>;<br/>
<a NAME="E3:17_3"/>
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> 1
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E4:17_3"/><b>then </b><i><font color="Green">E15</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,1</span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="fintopo5.html#K2" target="_self">FTSL2</a> <font color="Maroon">c<sub>1</sub></font>,1</span>)</span>
 
<b>by </b><i><a class="txt" href="fintopo5.html#E2:17_3"><i><font color="Green">E14</font></i></a>, <a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>
<a NAME="E5:17_3"/><b>then </b><i><font color="Green">E16</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,1</span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E6:17_3"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <span class="p3"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,1</span><a href="tarski.html#K4">]</a></span></span>)</span>,1</span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,1</span><a href="tarski.html#K4">]</a></span>
 
<b>by </b><i><a class="txt" href="fintopo5.html#E5:17"><i><font color="Green">E11</font></i></a>, <a class="txt" href="fintopo5.html#E4:17_3"><i><font color="Green">E15</font></i></a></i>;<br/>
<a NAME="E7:17_3"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,1</span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>
<b>hence </b><a NAME="E8:17_3"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="fintopo5.html#E5:17_3"><i><font color="Green">E16</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E9:17"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="fintopo4.html#K2">FTSL1</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="relset_1.html#T12">RELSET_1:12</a></i>;<br/>
<a NAME="E10:17"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="fintopo4.html#K2">FTSL1</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="fintopo5.html#E8:17"><i><font color="Green">E13</font></i></a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>
<a NAME="E11:17"/><b>then </b><i><font color="Green">E14</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is <a href="funct_2.html#V2">onto</a>
 
<b>by </b><i><a class="ref" href="funct_2.html#D3">FUNCT_2:def 3</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>3</sub></font> =  <a href="fintopo5.html#K2" target="_self">FTSL2</a> <font color="Maroon">c<sub>1</sub></font>,1;<br/>
<b>set </b><font color="Maroon">c<sub>4</sub></font> =  <a href="fintopo4.html#K2">FTSL1</a> <font color="Maroon">c<sub>1</sub></font>;<br/>
<a NAME="E12:17"/>
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="fintopo5.html#K2" target="_self">FTSL2</a> <font color="Maroon">c<sub>1</sub></font>,1</span>)</span> holds <font color="Maroon">c<sub>2</sub></font> <a href="relat_1.html#K9">.:</a> <span class="p1">(<span class="default">the <a href="fin_topo.html#U1">Nbd</a> of <span class="p2">(<span class="default"><a href="fintopo5.html#K2" target="_self">FTSL2</a> <font color="Maroon">c<sub>1</sub></font>,1</span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="fin_topo.html#U1">Nbd</a> of <span class="p1">(<span class="default"><a href="fintopo4.html#K2">FTSL1</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</sub></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">c<sub>1</sub></font>,1</span>)</span>;<br/>

<b>consider </b><font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>7</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E2:17_4"/><i><font color="Green">E15</font></i>: 
( <font color="Maroon">c<sub>6</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> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> 1 &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></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="E3:17_4"/><i><font color="Green">E16</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="relat_1.html#K9">.:</a> <span class="p1">(<span class="default">the <a href="fin_topo.html#U1">Nbd</a> of <span class="p2">(<span class="default"><a href="fintopo5.html#K2" target="_self">FTSL2</a> <font color="Maroon">c<sub>1</sub></font>,1</span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> the <a href="fin_topo.html#U1">Nbd</a> of <span class="p1">(<span class="default"><a href="fintopo4.html#K2">FTSL1</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_4_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>8</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:17_4_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="relat_1.html#K9">.:</a> <span class="p1">(<span class="default">the <a href="fin_topo.html#U1">Nbd</a> of <span class="p2">(<span class="default"><a href="fintopo5.html#K2" target="_self">FTSL2</a> <font color="Maroon">c<sub>1</sub></font>,1</span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>9</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:17_4_1"/><i><font color="Green">E17</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>9</sub></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">c<sub>1</sub></font>,1</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<a NAME="E4:17_4_1"/><i><font color="Green">E18</font></i>: 
<span class="p1">(<span class="default"><a href="fintopo5.html#K1" target="_self">Nbdl2</a> <font color="Maroon">c<sub>1</sub></font>,1</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</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">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="fintopo4.html#K1">Nbdl1</a> 1</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="txt" href="fintopo5.html#E2:17_4"><i><font color="Green">E15</font></i></a>, <a class="ref" href="fintopo5.html#D2" target="_self">Def2</a></i>;<br/>
<a NAME="E5:17_4_1"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="fintopo5.html#K2" target="_self">FTSL2</a> <font color="Maroon">c<sub>1</sub></font>,1</span>)</span>
 
<b>by </b><i><a class="txt" href="fintopo5.html#E3:17_4_1"><i><font color="Green">E17</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E6:17_4_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,1</span><a href="tarski.html#K4">]</a></span>
 
<b>by </b><i><a class="txt" href="fintopo5.html#E5:17"><i><font color="Green">E11</font></i></a></i>;<br/>
<a NAME="E7:17_4_1"/><b>then </b><i><font color="Green">E20</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> 1 )
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a>, <a class="txt" href="fintopo5.html#E2:17_4"><i><font color="Green">E15</font></i></a></i>;<br/>
<a NAME="E8:17_4_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>,1</span><a href="tarski.html#K4">]</a></span>
 
<b>by </b><i><a class="txt" href="fintopo5.html#E5:17"><i><font color="Green">E11</font></i></a>, <a class="txt" href="fintopo5.html#E5:17_4_1"><i><font color="Green">E19</font></i></a></i>;<br/>
<b>hence </b><a NAME="E9:17_4_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> the <a href="fin_topo.html#U1">Nbd</a> of <span class="p1">(<span class="default"><a href="fintopo4.html#K2">FTSL1</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="fintopo5.html#E1:17"><i><font color="Green">E9</font></i></a>, <a class="txt" href="fintopo5.html#E3:17_4_1"><i><font color="Green">E17</font></i></a>, <a class="txt" href="fintopo5.html#E7:17_4_1"><i><font color="Green">E20</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a>, <a class="txt" href="fintopo5.html#E4:17_4_1"><i><font color="Green">E18</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:17_4"/>
the <a href="fin_topo.html#U1">Nbd</a> of <span class="p1">(<span class="default"><a href="fintopo4.html#K2">FTSL1</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> <a href="relat_1.html#K9">.:</a> <span class="p1">(<span class="default">the <a href="fin_topo.html#U1">Nbd</a> of <span class="p2">(<span class="default"><a href="fintopo5.html#K2" target="_self">FTSL2</a> <font color="Maroon">c<sub>1</sub></font>,1</span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_4_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>8</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:17_4_2"/><i><font color="Green">E17</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> the <a href="fin_topo.html#U1">Nbd</a> of <span class="p1">(<span class="default"><a href="fintopo4.html#K2">FTSL1</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 ;<br/>

<a NAME="E2:17_4_2"/>
<span class="p1">(<span class="default"><a href="fintopo4.html#K1">Nbdl1</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</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>
 
<b>by </b><i><a class="ref" href="funct_2.html#T7">FUNCT_2:7</a>, <a class="txt" href="fintopo5.html#E1:17"><i><font color="Green">E9</font></i></a></i>;<br/>
<a NAME="E3:17_4_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="fintopo4.html#K1">Nbdl1</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a>, <a class="txt" href="fintopo5.html#E8:17"><i><font color="Green">E13</font></i></a>, <a class="txt" href="fintopo5.html#E1:17"><i><font color="Green">E9</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>9</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:17_4_2"/><i><font color="Green">E18</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a>, <a class="txt" href="fintopo5.html#E1:17_4_2"><i><font color="Green">E17</font></i></a>, <a class="txt" href="fintopo5.html#E1:17"><i><font color="Green">E9</font></i></a></i>;<br/>
<a NAME="E6:17_4_2"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="fintopo4.html#K1">Nbdl1</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="fintopo5.html#E1:17"><i><font color="Green">E9</font></i></a>, <a class="txt" href="fintopo5.html#E1:17_4_2"><i><font color="Green">E17</font></i></a>, <a class="txt" href="fintopo5.html#E5:17_4_2"><i><font color="Green">E18</font></i></a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>11</sub></font> = 1;<br/>
<a NAME="E7:17_4_2"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="fintopo5.html#K2" target="_self">FTSL2</a> <font color="Maroon">c<sub>1</sub></font>,1</span>)</span>
 
<b>by </b><i><a class="txt" href="fintopo5.html#E5:17_4_2"><i><font color="Green">E18</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E8:17_4_2"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>,1</span><a href="tarski.html#K4">]</a></span>
 
<b>by </b><i><a class="txt" href="fintopo5.html#E5:17"><i><font color="Green">E11</font></i></a></i>;<br/>
<a NAME="E9:17_4_2"/><b>then </b><i><font color="Green">E21</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</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> &amp; 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> 1 &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>,1</span><a href="tarski.html#K4">]</a></span> )
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a>, <a class="txt" href="fintopo5.html#E7:17_4_2"><i><font color="Green">E20</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>7</sub></font> as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="txt" href="fintopo5.html#E2:17_4"><i><font color="Green">E15</font></i></a></i>;<br/>
<a NAME="E11:17_4_2"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a>, <a class="txt" href="fintopo5.html#E2:17_4"><i><font color="Green">E15</font></i></a></i>;<br/>
<a NAME="E12:17_4_2"/><b>then </b><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/>
<span class="p1">(<span class="default"><a href="fintopo4.html#K1">Nbdl1</a> 1</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> = 
<span class="p1"><a href="enumset1.html#K1">{</a><span class="default"><font color="Maroon">c<sub>12</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>12</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>12</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,1</span>)</span></span><a href="enumset1.html#K1">}</a></span>
<b>by </b><i><a class="ref" href="fintopo4.html#D3">FINTOPO4:def 3</a>, <a class="txt" href="fintopo5.html#E2:17_4"><i><font color="Green">E15</font></i></a></i>
<br/>.= 
<span class="p1"><a href="enumset1.html#K1">{</a><span class="default">1,<span class="p2">(<span class="default"><a href="limfunc1.html#K1">max</a> 0,1</span>)</span>,<span class="p2">(<span class="default"><a href="rfunct_3.html#K1">min</a> 2,1</span>)</span></span><a href="enumset1.html#K1">}</a></span>
<b>by </b><i><a class="ref" href="nat_2.html#T10">NAT_2:10</a>, <a class="txt" href="fintopo5.html#E12:17_4_2"><i><font color="Green">E22</font></i></a></i>
<br/>.= 
<span class="p1"><a href="enumset1.html#K1">{</a><span class="default">1,1,<span class="p2">(<span class="default"><a href="rfunct_3.html#K1">min</a> 2,1</span>)</span></span><a href="enumset1.html#K1">}</a></span>
<b>by </b><i><a class="ref" href="xxreal_0.html#D9">XXREAL_0:def 9</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K2">{</a><span class="default">1,<span class="p2">(<span class="default"><a href="rfunct_3.html#K1">min</a> 2,1</span>)</span></span><a href="tarski.html#K2">}</a></span>
<b>by </b><i><a class="ref" href="enumset1.html#T70">ENUMSET1:70</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K2">{</a><span class="default">1,1</span><a href="tarski.html#K2">}</a></span>
<b>by </b><i><a class="ref" href="xxreal_0.html#D8">XXREAL_0:def 8</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span>
<b>by </b><i><a class="ref" href="enumset1.html#T69">ENUMSET1:69</a></i>
;<br/>
<a NAME="E14:17_4_2"/><b>then </b><i><font color="Green">E23</font></i>: 
1 <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="fintopo4.html#K1">Nbdl1</a> 1</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/>
<a NAME="E15:17_4_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,1</span><a href="tarski.html#K4">]</a></span>
 
<b>by </b><i><a class="txt" href="fintopo5.html#E5:17"><i><font color="Green">E11</font></i></a></i>;<br/>
<a NAME="E16:17_4_2"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="fintopo4.html#K1">Nbdl1</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="fintopo5.html#E6:17_4_2"><i><font color="Green">E19</font></i></a>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a>, <a class="txt" href="fintopo5.html#E2:17_4"><i><font color="Green">E15</font></i></a></i>;<br/>
<a NAME="E17:17_4_2"/><b>then </b><i><font color="Green">E24</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>,1</span><a href="tarski.html#K4">]</a></span> <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">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="fintopo4.html#K1">Nbdl1</a> 1</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="txt" href="fintopo5.html#E14:17_4_2"><i><font color="Green">E23</font></i></a>, <a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>
<a NAME="E18:17_4_2"/>
<span class="p1">(<span class="default"><a href="fintopo5.html#K1" target="_self">Nbdl2</a> <font color="Maroon">c<sub>1</sub></font>,1</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</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">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="fintopo4.html#K1">Nbdl1</a> 1</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="txt" href="fintopo5.html#E2:17_4"><i><font color="Green">E15</font></i></a>, <a class="ref" href="fintopo5.html#D2" target="_self">Def2</a></i>;<br/>
<b>hence </b><a NAME="E19:17_4_2"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="relat_1.html#K9">.:</a> <span class="p1">(<span class="default">the <a href="fin_topo.html#U1">Nbd</a> of <span class="p2">(<span class="default"><a href="fintopo5.html#K2" target="_self">FTSL2</a> <font color="Maroon">c<sub>1</sub></font>,1</span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a>, <a class="txt" href="fintopo5.html#E5:17_4_2"><i><font color="Green">E18</font></i></a>, <a class="txt" href="fintopo5.html#E17:17_4_2"><i><font color="Green">E24</font></i></a>, <a class="txt" href="fintopo5.html#E9:17_4_2"><i><font color="Green">E21</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:17_4"/>
<font color="Maroon">c<sub>2</sub></font> <a href="relat_1.html#K9">.:</a> <span class="p1">(<span class="default">the <a href="fin_topo.html#U1">Nbd</a> of <span class="p2">(<span class="default"><a href="fintopo5.html#K2" target="_self">FTSL2</a> <font color="Maroon">c<sub>1</sub></font>,1</span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="fin_topo.html#U1">Nbd</a> of <span class="p1">(<span class="default"><a href="fintopo4.html#K2">FTSL1</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="fintopo5.html#E3:17_4"><i><font color="Green">E16</font></i></a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E13:17"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="fintopo5.html#R1" target="_self">is_homeomorphism</a> 
 
<b>by </b><i><a class="ref" href="fintopo5.html#D1" target="_self">Def1</a>, <a class="txt" href="fintopo5.html#E7:17"><i><font color="Green">E12</font></i></a>, <a class="txt" href="fintopo5.html#E11:17"><i><font color="Green">E14</font></i></a></i>;<br/>
<b>hence </b><a NAME="E14:17"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM7">Function</a> of <span class="p1">(<span class="default"><a href="fintopo5.html#K2" target="_self">FTSL2</a> <font color="Maroon">c<sub>1</sub></font>,1</span>)</span>,<span class="p1">(<span class="default"><a href="fintopo4.html#K2">FTSL1</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> st <font color="Olive">b<sub>1</sub></font> <a href="fintopo5.html#R1" target="_self">is_homeomorphism</a> 
 ;<br/>


</div>
