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

<b>let </b><font color="Maroon" title="c1">n</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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">h</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="fintopo5.html#K5" title="FINTOPO5:func.5">FTSS2</a> <font color="Maroon" title="c1">n</font>,1</span>)</span>,<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> st <font color="Olive" title="b1">h</font> is <a href="fintopo5.html#V1" title="FINTOPO5:attr.1">being_homeomorphism</a></span><br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b><span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">$2,1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1;<br/>
<a NAME="E1:24"/><i><font color="Green" title="E10">A1</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/>
<a NAME="E2:24"/><i><font color="Green" title="E11">A2</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="fintopo5.html#K5" title="FINTOPO5:func.5">FTSS2</a> <font color="Maroon" title="c1">n</font>,1</span>)</span> holds <br/> ex <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</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> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="fintopo5.html#K5" title="FINTOPO5:func.5">FTSS2</a> <font color="Maroon" title="c1">n</font>,1</span>)</span> implies  ex <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</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> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">x</font>,<font color="Olive" title="b1">y</font>] ) )</span><br/>

<b>assume </b><a NAME="E1:24_1"/>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="fintopo5.html#K5" title="FINTOPO5:func.5">FTSS2</a> <font color="Maroon" title="c1">n</font>,1</span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</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> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">x</font>,<font color="Olive" title="b1">y</font>] )</span><br/>

<b>then consider </b><font color="Maroon" title="c3">u</font>, <font color="Maroon" title="c4">v</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:24_1"/><i><font color="Green" title="E12">A3</font></i>: 
( <font color="Maroon" title="c3">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="c4">v</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> 1 &amp; <font color="Maroon" title="c2">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="c3">u</font>,<font color="Maroon" title="c4">v</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="c5">nu</font> = <font color="Maroon" title="c3">u</font>, <font color="Maroon" title="c6">nv</font> = <font color="Maroon" title="c4">v</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#E3:24_1"><i><font color="Green" title="E12">A3</font></i></a></i>;<br/>
<a NAME="E5:24_1"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">nv</font> &amp; <font color="Maroon" title="c6">nv</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 )
 
<b>by </b><i><a class="txt" href="fintopo5.html#E3:24_1"><i><font color="Green" title="E12">A3</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E6:24_1"/><b>then </b><i><font color="Green" title="E13">A4</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c5">nu</font>]
 
<b>by </b><i><a class="txt" href="fintopo5.html#E3:24_1"><i><font color="Green" title="E12">A3</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/>
<a NAME="E7:24_1"/>
 <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>hence </b><a NAME="E8:24_1"/>
 ex <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</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> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">x</font>,<font color="Olive" title="b1">y</font>] )
 <b>by </b><i><a class="txt" href="fintopo5.html#E3:24_1"><i><font color="Green" title="E12">A3</font></i></a>, <a class="txt" href="fintopo5.html#E6:24_1"><i><font color="Green" title="E13">A4</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E3:24"/>
 ex <font color="Olive" title="b1">f</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="fintopo5.html#K5" title="FINTOPO5:func.5">FTSS2</a> <font color="Maroon" title="c1">n</font>,1</span>)</span>,<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> st <br/>for <font color="Olive" title="b2">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="fintopo5.html#K5" title="FINTOPO5:func.5">FTSS2</a> <font color="Maroon" title="c1">n</font>,1</span>)</span> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">x</font>,<font color="Olive" title="b1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">x</font>]
 
<b>from </b><i><a class="ref" href="funct_2.html#S1" title="FUNCT_2:sch.1">FUNCT_2:sch 1</a>(<a class="txt" href="fintopo5.html#E2:24"><i><font color="Green" title="E11">A2</font></i></a>);<br/></i>
<b>then consider </b><font color="Maroon" title="c2">f</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="fintopo5.html#K5" title="FINTOPO5:func.5">FTSS2</a> <font color="Maroon" title="c1">n</font>,1</span>)</span>,<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> such that </b><br/><a NAME="E5:24"/><i><font color="Green" title="E12">A5</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="fintopo5.html#K5" title="FINTOPO5:func.5">FTSS2</a> <font color="Maroon" title="c1">n</font>,1</span>)</span> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>]
 ;<br/>
<a NAME="E6:24"/>
for <font color="Olive" title="b1">x1</font>, <font color="Olive" title="b2">x2</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">f</font> &amp; <font color="Olive" title="b2">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">f</font> &amp; <font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">x2</font> holds <br/><font color="Olive" title="b1">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">x2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">x1</font>, <font color="Maroon" title="c4">x2</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">f</font> &amp; <font color="Maroon" title="c4">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">f</font> &amp; <font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x2</font> implies <font color="Maroon" title="c3">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x2</font> )</span><br/>

<b>assume </b><a NAME="E1:24_2"/><i><font color="Green" title="E13">A6</font></i>: 
( <font color="Maroon" title="c3">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">f</font> &amp; <font color="Maroon" title="c4">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">f</font> &amp; <font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x2</font></span><br/>

<a NAME="E2:24_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">x1</font></span>)</span>,1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x1</font>
 
<b>by </b><i><a class="txt" href="fintopo5.html#E5:24"><i><font color="Green" title="E12">A5</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:24_2"/>
<font color="Maroon" title="c3">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x2</font>
 <b>by </b><i><a class="txt" href="fintopo5.html#E5:24"><i><font color="Green" title="E12">A5</font></i></a>, <a class="txt" href="fintopo5.html#E1:24_2"><i><font color="Green" title="E13">A6</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E7:24"/><b>then </b><i><font color="Green" title="E13">A7</font></i>: 
<font color="Maroon" title="c2">f</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 
<b>by </b><i><a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></i>;<br/>
<a NAME="E8:24"/><i><font color="Green" title="E14">A8</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</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> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">f</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</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> or <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">f</font> )</span><br/>

<b>assume </b><a NAME="E1:24_3"/><i><font color="Green" title="E15">A9</font></i>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">f</font></span><br/>

<b>set </b><font color="Maroon" title="c4">z</font> = <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">x</font>,1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>;<br/>
<a NAME="E2:24_3"/>
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 1
 
;<br/>
<a NAME="E3:24_3"/><b>then </b><i><font color="Green" title="E16">A10</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">x</font>,1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="fintopo5.html#K5" title="FINTOPO5:func.5">FTSS2</a> <font color="Maroon" title="c1">n</font>,1</span>)</span>
 
<b>by </b><i><a class="txt" href="fintopo5.html#E1:24"><i><font color="Green" title="E10">A1</font></i></a>, <a class="txt" href="fintopo5.html#E1:24_3"><i><font color="Green" title="E15">A9</font></i></a>, <a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></i>;<br/>
<a NAME="E4:24_3"/><b>then </b><i><font color="Green" title="E17">A11</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">x</font>,1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">f</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E5:24_3"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">x</font>,1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span>)</span>,1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <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="c3">x</font>,1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 
<b>by </b><i><a class="txt" href="fintopo5.html#E5:24"><i><font color="Green" title="E12">A5</font></i></a>, <a class="txt" href="fintopo5.html#E3:24_3"><i><font color="Green" title="E16">A10</font></i></a></i>;<br/>
<a NAME="E6:24_3"/><b>then </b>
<font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">x</font>,1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x</font>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/>
<b>hence </b><a NAME="E7:24_3"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">f</font>
 <b>by </b><i><a class="txt" href="fintopo5.html#E4:24_3"><i><font color="Green" title="E17">A11</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E9:24"/><b>then </b>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</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="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/>
<a NAME="E10:24"/><b>then </b><i><font color="Green" title="E15">A12</font></i>: 
<font color="Maroon" title="c2">f</font> is <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a>
 
<b>by </b><i><a class="ref" href="funct_2.html#D3" title="FUNCT_2:def.3">FUNCT_2:def 3</a></i>;<br/>
<b>set </b><font color="Maroon" title="c3">FT1</font> =  <a href="fintopo5.html#K5" title="FINTOPO5:func.5">FTSS2</a> <font color="Maroon" title="c1">n</font>,1;<br/>
<b>set </b><font color="Maroon" title="c4">FT2</font> =  <a href="fintopo4.html#K2" title="FINTOPO4:func.2">FTSL1</a> <font color="Maroon" title="c1">n</font>;<br/>
<a NAME="E11:24"/>
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#K5" title="FINTOPO5:func.5">FTSS2</a> <font color="Maroon" title="c1">n</font>,1</span>)</span> holds  <font color="Maroon" title="c2">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1">(<span class="default"><a href="fin_topo.html#K1" title="FIN_TOPO:func.1">U_FT</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fintopo5.html#K1" title="FINTOPO5:func.1">Im</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="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">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#K5" title="FINTOPO5:func.5">FTSS2</a> <font color="Maroon" title="c1">n</font>,1</span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1">(<span class="default"><a href="fin_topo.html#K1" title="FIN_TOPO:func.1">U_FT</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fintopo5.html#K1" title="FINTOPO5:func.1">Im</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="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">x</font></span>)</span></span><br/>

<b>consider </b><font color="Maroon" title="c6">u</font>, <font color="Maroon" title="c7">v</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E2:24_4"/><i><font color="Green" title="E16">A13</font></i>: 
( <font color="Maroon" title="c6">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="c7">v</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> 1 &amp; <font color="Maroon" title="c5">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="c6">u</font>,<font color="Maroon" title="c7">v</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/>
<a NAME="E3:24_4"/><i><font color="Green" title="E17">A14</font></i>: 
<font color="Maroon" title="c2">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1">(<span class="default"><a href="fin_topo.html#K1" title="FIN_TOPO:func.1">U_FT</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="fintopo5.html#K1" title="FINTOPO5:func.1">Im</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="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">x</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_4_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1">(<span class="default"><a href="fin_topo.html#K1" title="FIN_TOPO:func.1">U_FT</a> <font color="Maroon" title="c5">x</font></span>)</span> or <font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fintopo5.html#K1" title="FINTOPO5:func.1">Im</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="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">x</font></span>)</span> )</span><br/>

<b>assume </b><a NAME="E1:24_4_1"/>
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1">(<span class="default"><a href="fin_topo.html#K1" title="FIN_TOPO:func.1">U_FT</a> <font color="Maroon" title="c5">x</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fintopo5.html#K1" title="FINTOPO5:func.1">Im</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="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">x</font></span>)</span></span><br/>

<b>then consider </b><font color="Maroon" title="c9">x2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:24_4_1"/><i><font color="Green" title="E18">A15</font></i>: 
( <font color="Maroon" title="c9">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">f</font> &amp; <font color="Maroon" title="c9">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fintopo5.html#K1" title="FINTOPO5:func.1">Im</a> <span class="p1">(<span class="default"><a href="fintopo5.html#K4" title="FINTOPO5:func.4">Nbds2</a> <font color="Maroon" title="c1">n</font>,1</span>)</span>,<font color="Maroon" title="c5">x</font> &amp; <font color="Maroon" title="c8">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x2</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/>
<a NAME="E4:24_4_1"/><i><font color="Green" title="E19">A16</font></i>: 
 <a href="fintopo5.html#K1" title="FINTOPO5:func.1">Im</a> <span class="p1">(<span class="default"><a href="fintopo5.html#K4" title="FINTOPO5:func.4">Nbds2</a> <font color="Maroon" title="c1">n</font>,1</span>)</span>,<font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">u</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></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> 1</span>)</span>,<font color="Maroon" title="c7">v</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</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="c6">u</font></span>)</span>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c7">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="txt" href="fintopo5.html#E2:24_4"><i><font color="Green" title="E16">A13</font></i></a>, <a class="ref" href="fintopo5.html#D4" target="_self" title="FINTOPO5:def.4">Def4</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c10">u2</font>, <font color="Maroon" title="c11">v2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E6:24_4_1"/><i><font color="Green" title="E20">A17</font></i>: 
( <font color="Maroon" title="c10">u2</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="c11">v2</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> 1 &amp; <font color="Maroon" title="c9">x2</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="c10">u2</font>,<font color="Maroon" title="c11">v2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 <b>by </b><i><a class="txt" href="fintopo5.html#E3:24_4_1"><i><font color="Green" title="E18">A15</font></i></a>, <a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></i>;<br/>
<a NAME="E7:24_4_1"/><i><font color="Green" title="E21">A18</font></i>: 
<font color="Maroon" title="c5">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"><span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">x</font></span>)</span>,1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 
<b>by </b><i><a class="txt" href="fintopo5.html#E5:24"><i><font color="Green" title="E12">A5</font></i></a></i>;<br/>
<a NAME="E8:24_4_1"/>
<font color="Maroon" title="c9">x2</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"><span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x2</font></span>)</span>,1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 
<b>by </b><i><a class="txt" href="fintopo5.html#E5:24"><i><font color="Green" title="E12">A5</font></i></a>, <a class="txt" href="fintopo5.html#E3:24_4_1"><i><font color="Green" title="E18">A15</font></i></a></i>;<br/>
<a NAME="E9:24_4_1"/><b>then </b><i><font color="Green" title="E22">A19</font></i>: 
( <font color="Maroon" title="c10">u2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x2</font> &amp; <font color="Maroon" title="c11">v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 
<b>by </b><i><a class="txt" href="fintopo5.html#E6:24_4_1"><i><font color="Green" title="E20">A17</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_4_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_4_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:24_4_1_1_1"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">u2</font>,<font color="Maroon" title="c11">v2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <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"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">u</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></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> 1</span>)</span>,<font color="Maroon" title="c7">v</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> or <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">u2</font>,<font color="Maroon" title="c11">v2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <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="c6">u</font></span>)</span>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c7">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> )
 </span><b>by </b><i><a class="txt" href="fintopo5.html#E3:24_4_1"><i><font color="Green" title="E18">A15</font></i></a>, <a class="txt" href="fintopo5.html#E4:24_4_1"><i><font color="Green" title="E19">A16</font></i></a>, <a class="txt" href="fintopo5.html#E6:24_4_1"><i><font color="Green" title="E20">A17</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_4_1_1_1_1"><b>suppose </b></a><a NAME="E1:24_4_1_1_1_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">u2</font>,<font color="Maroon" title="c11">v2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <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"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">u</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></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> 1</span>)</span>,<font color="Maroon" title="c7">v</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c10">u2</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="c6">u</font></span><br/><div class="add"><a NAME="E2:24_4_1_1_1_1"/><b>then </b><i><font color="Green" title="E23">A20</font></i>: 
<font color="Maroon" title="c10">u2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">u</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c12">pu</font> = <font color="Maroon" title="c6">u</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#E1:24"><i><font color="Green" title="E10">A1</font></i></a>, <a class="txt" href="fintopo5.html#E2:24_4"><i><font color="Green" title="E16">A13</font></i></a></i>;<br/><a NAME="E4:24_4_1_1_1_1"/>
 <a href="fintopo4.html#K2" title="FINTOPO4:func.2">FTSL1</a> <font color="Maroon" title="c1">n</font> is <a href="orders_2.html#V3" title="ORDERS_2:attr.3">filled</a>
 <b>by </b><i><a class="ref" href="fintopo4.html#T18" title="FINTOPO4:th.18">FINTOPO4:18</a></i>;<br/><a NAME="E5:24_4_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c6">u</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="c12">pu</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/><b>hence </b><a NAME="E6:24_4_1_1_1_1"/>
<font color="Maroon" title="c10">u2</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="c6">u</font>
 <b>by </b><i><a class="txt" href="fintopo5.html#E1:24"><i><font color="Green" title="E10">A1</font></i></a>, <a class="txt" href="fintopo5.html#E2:24_4_1_1_1_1"><i><font color="Green" title="E23">A20</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_4_1_1_1_2"><b>suppose </b></a><a NAME="E1:24_4_1_1_1_2"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">u2</font>,<font color="Maroon" title="c11">v2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <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="c6">u</font></span>)</span>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c7">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c10">u2</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="c6">u</font></span><br/><div class="add"><b>hence </b><a NAME="E2:24_4_1_1_1_2"/>
<font color="Maroon" title="c10">u2</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="c6">u</font>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E11:24_4_1"/>
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fintopo5.html#K1" title="FINTOPO5:func.1">Im</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="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">x</font></span>)</span>
 <b>by </b><i><a class="txt" href="fintopo5.html#E1:24"><i><font color="Green" title="E10">A1</font></i></a>, <a class="txt" href="fintopo5.html#E2:24_4"><i><font color="Green" title="E16">A13</font></i></a>, <a class="txt" href="fintopo5.html#E3:24_4_1"><i><font color="Green" title="E18">A15</font></i></a>, <a class="txt" href="fintopo5.html#E7:24_4_1"><i><font color="Green" title="E21">A18</font></i></a>, <a class="txt" href="fintopo5.html#E9:24_4_1"><i><font color="Green" title="E22">A19</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E4:24_4"/>
 <a href="fintopo5.html#K1" title="FINTOPO5:func.1">Im</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="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1">(<span class="default"><a href="fin_topo.html#K1" title="FIN_TOPO:func.1">U_FT</a> <font color="Maroon" title="c5">x</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_4_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fintopo5.html#K1" title="FINTOPO5:func.1">Im</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="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">x</font></span>)</span> or <font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1">(<span class="default"><a href="fin_topo.html#K1" title="FIN_TOPO:func.1">U_FT</a> <font color="Maroon" title="c5">x</font></span>)</span> )</span><br/>

<b>assume </b><a NAME="E1:24_4_2"/><i><font color="Green" title="E18">A21</font></i>: 
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fintopo5.html#K1" title="FINTOPO5:func.1">Im</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="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">x</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1">(<span class="default"><a href="fin_topo.html#K1" title="FIN_TOPO:func.1">U_FT</a> <font color="Maroon" title="c5">x</font></span>)</span></span><br/>

<a NAME="E2:24_4_2"/>
 <a href="fintopo5.html#K1" title="FINTOPO5:func.1">Im</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>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">f</font>
 
<b>by </b><i><a class="txt" href="fintopo5.html#E1:24"><i><font color="Green" title="E10">A1</font></i></a>, <a class="txt" href="fintopo5.html#E8:24"><i><font color="Green" title="E14">A8</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c9">x3</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:24_4_2"/><i><font color="Green" title="E19">A22</font></i>: 
( <font color="Maroon" title="c9">x3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">f</font> &amp; <font color="Maroon" title="c8">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x3</font> )
 <b>by </b><i><a class="txt" href="fintopo5.html#E1:24"><i><font color="Green" title="E10">A1</font></i></a>, <a class="txt" href="fintopo5.html#E1:24_4_2"><i><font color="Green" title="E18">A21</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>set </b><font color="Maroon" title="c10">u2</font> = <font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x3</font>;<br/>
<b>set </b><font color="Maroon" title="c11">v2</font> = 1;<br/>
<a NAME="E5:24_4_2"/><i><font color="Green" title="E20">A23</font></i>: 
<font color="Maroon" title="c9">x3</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"><span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x3</font></span>)</span>,1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 
<b>by </b><i><a class="txt" href="fintopo5.html#E5:24"><i><font color="Green" title="E12">A5</font></i></a>, <a class="txt" href="fintopo5.html#E4:24_4_2"><i><font color="Green" title="E19">A22</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c12">nv</font> = <font color="Maroon" title="c7">v</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#E2:24_4"><i><font color="Green" title="E16">A13</font></i></a></i>;<br/>
<a NAME="E7:24_4_2"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">nv</font> &amp; <font color="Maroon" title="c12">nv</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 )
 
<b>by </b><i><a class="txt" href="fintopo5.html#E2:24_4"><i><font color="Green" title="E16">A13</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E8:24_4_2"/><b>then </b><i><font color="Green" title="E21">A24</font></i>: 
<font color="Maroon" title="c12">nv</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/>
<i><font color="Green" title="E22">A25</font></i>:  <a href="fintopo5.html#K1" title="FINTOPO5:func.1">Im</a> <span class="p1">(<span class="default"><a href="fintopo4.html#K1" title="FINTOPO4:func.1">Nbdl1</a> 1</span>)</span>,<font color="Maroon" title="c7">v</font> = 
<span class="p1"><a href="enumset1.html#K1" title="ENUMSET1:func.1">{</a><span class="default"><font color="Maroon" title="c12">nv</font>,<span class="p2">(<span class="default"><a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">nv</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,1</span>)</span>,<span class="p2">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">nv</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,1</span>)</span></span><a href="enumset1.html#K1" title="ENUMSET1:func.1">}</a></span>
<b>by </b><i><a class="txt" href="fintopo5.html#E2:24_4"><i><font color="Green" title="E16">A13</font></i></a>, <a class="ref" href="fintopo4.html#D3" title="FINTOPO4:def.3">FINTOPO4:def 3</a></i>
<br/>.= 
<span class="p1"><a href="enumset1.html#K1" title="ENUMSET1:func.1">{</a><span class="default">1,<span class="p2">(<span class="default"><a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span>)</span>,<span class="p2">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> 2,1</span>)</span></span><a href="enumset1.html#K1" title="ENUMSET1:func.1">}</a></span>
<b>by </b><i><a class="txt" href="fintopo5.html#E8:24_4_2"><i><font color="Green" title="E21">A24</font></i></a>, <a class="ref" href="nat_2.html#T10" title="NAT_2:th.10">NAT_2:10</a></i>
<br/>.= 
<span class="p1"><a href="enumset1.html#K1" title="ENUMSET1:func.1">{</a><span class="default">1,1,<span class="p2">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> 2,1</span>)</span></span><a href="enumset1.html#K1" title="ENUMSET1:func.1">}</a></span>
<b>by </b><i><a class="ref" href="xxreal_0.html#D9" title="XXREAL_0:def.9">XXREAL_0:def 9</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default">1,<span class="p2">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> 2,1</span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
<b>by </b><i><a class="ref" href="enumset1.html#T70" title="ENUMSET1:th.70">ENUMSET1:70</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default">1,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
<b>by </b><i><a class="ref" href="xxreal_0.html#D8" title="XXREAL_0:def.8">XXREAL_0:def 8</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
<b>by </b><i><a class="ref" href="enumset1.html#T69" title="ENUMSET1:th.69">ENUMSET1:69</a></i>
;<br/>
<a NAME="E10:24_4_2"/><b>then </b><i><font color="Green" title="E23">A26</font></i>: 
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fintopo5.html#K1" title="FINTOPO5:func.1">Im</a> <span class="p1">(<span class="default"><a href="fintopo4.html#K1" title="FINTOPO4:func.1">Nbdl1</a> 1</span>)</span>,<font color="Maroon" title="c7">v</font>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T37" title="ZFMISC_1:th.37">ZFMISC_1:37</a></i>;<br/>
<a NAME="E11:24_4_2"/>
<font color="Maroon" title="c5">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"><span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">x</font></span>)</span>,1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 
<b>by </b><i><a class="txt" href="fintopo5.html#E5:24"><i><font color="Green" title="E12">A5</font></i></a></i>;<br/>
<a NAME="E12:24_4_2"/><b>then </b>
<font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fintopo5.html#K1" title="FINTOPO5:func.1">Im</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="c6">u</font>
 
<b>by </b><i><a class="txt" href="fintopo5.html#E1:24"><i><font color="Green" title="E10">A1</font></i></a>, <a class="txt" href="fintopo5.html#E2:24_4"><i><font color="Green" title="E16">A13</font></i></a>, <a class="txt" href="fintopo5.html#E1:24_4_2"><i><font color="Green" title="E18">A21</font></i></a>, <a class="txt" href="fintopo5.html#E4:24_4_2"><i><font color="Green" title="E19">A22</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/>
<a NAME="E13:24_4_2"/><b>then </b><i><font color="Green" title="E24">A27</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x3</font></span>)</span>,1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <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="c6">u</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> 1</span>)</span>,<font color="Maroon" title="c7">v</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="txt" href="fintopo5.html#E10:24_4_2"><i><font color="Green" title="E23">A26</font></i></a>, <a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></i>;<br/>
<b>set </b><font color="Maroon" title="c13">X</font> =  <a href="fintopo5.html#K1" title="FINTOPO5:func.1">Im</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="c6">u</font>;<br/>
<b>set </b><font color="Maroon" title="c14">Y</font> =  <a href="fintopo5.html#K1" title="FINTOPO5:func.1">Im</a> <span class="p1">(<span class="default"><a href="fintopo4.html#K1" title="FINTOPO4:func.1">Nbdl1</a> 1</span>)</span>,<font color="Maroon" title="c7">v</font>;<br/>
<a NAME="E14:24_4_2"/><i><font color="Green" title="E25">A28</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x3</font></span>)</span>,1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <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="c6">u</font></span>)</span>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c7">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">u</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></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> 1</span>)</span>,<font color="Maroon" title="c7">v</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="txt" href="fintopo5.html#E8:24_4_2"><i><font color="Green" title="E21">A24</font></i></a>, <a class="txt" href="fintopo5.html#E9:24_4_2"><i><font color="Green" title="E22">A25</font></i></a>, <a class="txt" href="fintopo5.html#E13:24_4_2"><i><font color="Green" title="E24">A27</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E15:24_4_2"/>
 <a href="fintopo5.html#K1" title="FINTOPO5:func.1">Im</a> <span class="p1">(<span class="default"><a href="fintopo5.html#K4" title="FINTOPO5:func.4">Nbds2</a> <font color="Maroon" title="c1">n</font>,1</span>)</span>,<font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">u</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></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> 1</span>)</span>,<font color="Maroon" title="c7">v</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</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="c6">u</font></span>)</span>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c7">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="txt" href="fintopo5.html#E2:24_4"><i><font color="Green" title="E16">A13</font></i></a>, <a class="ref" href="fintopo5.html#D4" target="_self" title="FINTOPO5:def.4">Def4</a></i>;<br/>
<b>hence </b><a NAME="E16:24_4_2"/>
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1">(<span class="default"><a href="fin_topo.html#K1" title="FIN_TOPO:func.1">U_FT</a> <font color="Maroon" title="c5">x</font></span>)</span>
 <b>by </b><i><a class="txt" href="fintopo5.html#E4:24_4_2"><i><font color="Green" title="E19">A22</font></i></a>, <a class="txt" href="fintopo5.html#E5:24_4_2"><i><font color="Green" title="E20">A23</font></i></a>, <a class="txt" href="fintopo5.html#E14:24_4_2"><i><font color="Green" title="E25">A28</font></i></a>, <a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:24_4"/>
<font color="Maroon" title="c2">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1">(<span class="default"><a href="fin_topo.html#K1" title="FIN_TOPO:func.1">U_FT</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fintopo5.html#K1" title="FINTOPO5:func.1">Im</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="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">x</font></span>)</span>
 <b>by </b><i><a class="txt" href="fintopo5.html#E3:24_4"><i><font color="Green" title="E17">A14</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E12:24"/><b>then </b>
<font color="Maroon" title="c2">f</font> is <a href="fintopo5.html#V1" title="FINTOPO5:attr.1">being_homeomorphism</a>
 
<b>by </b><i><a class="txt" href="fintopo5.html#E7:24"><i><font color="Green" title="E13">A7</font></i></a>, <a class="txt" href="fintopo5.html#E10:24"><i><font color="Green" title="E15">A12</font></i></a>, <a class="ref" href="fintopo5.html#D1" target="_self" title="FINTOPO5:def.1">Def1</a></i>;<br/>
<b>hence </b><a NAME="E13:24"/>
 ex <font color="Olive" title="b1">h</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="fintopo5.html#K5" title="FINTOPO5:func.5">FTSS2</a> <font color="Maroon" title="c1">n</font>,1</span>)</span>,<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> st <font color="Olive" title="b1">h</font> is <a href="fintopo5.html#V1" title="FINTOPO5:attr.1">being_homeomorphism</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
