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

<b>let </b><font color="Maroon">T</font> be  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a>, <font color="Maroon">S</font> be  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a>;<br/>

<b>set </b><font color="Maroon">F</font> =  <a href="t_0topsp.html#K4" target="_self">T_0-canonical_map</a> <font color="Maroon">T</font>;<br/>
<b>set </b><font color="Maroon">G</font> =  <a href="t_0topsp.html#K4" target="_self">T_0-canonical_map</a> <font color="Maroon">S</font>;<br/>
<b>set </b><font color="Maroon">TR</font> =  <a href="t_0topsp.html#K3" target="_self">T_0-reflex</a> <font color="Maroon">T</font>;<br/>
<b>set </b><font color="Maroon">SR</font> =  <a href="t_0topsp.html#K3" target="_self">T_0-reflex</a> <font color="Maroon">S</font>;<br/>
<b>given </b><font color="Maroon">h</font> being   <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><a href="t_0topsp.html#K3" target="_self">T_0-reflex</a> <font color="Maroon">S</font></span>)</span>,<span class="p1">(<span class="default"><a href="t_0topsp.html#K3" target="_self">T_0-reflex</a> <font color="Maroon">T</font></span>)</span><b> such that </b><a NAME="E1:22"/><i><font color="Green">E16</font></i>: 
<font color="Maroon">h</font> <a href="tops_2.html#NR1" target="_self">is_homeomorphism</a> 
 <b>and </b><br/><a NAME="E2:22"/><i><font color="Green">E19</font></i>: 
 <a href="t_0topsp.html#K4" target="_self">T_0-canonical_map</a> <font color="Maroon">T</font>,<font color="Maroon">h</font> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><a href="t_0topsp.html#K4" target="_self">T_0-canonical_map</a> <font color="Maroon">S</font></span>)</span> <a href="rfinseq.html#R1">are_fiberwise_equipotent</a> 
 ;<br/>

<b>consider </b><font color="Maroon">f</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E4:22"/><i><font color="Green">E22</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="t_0topsp.html#K4" target="_self">T_0-canonical_map</a> <font color="Maroon">T</font></span>)</span> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="t_0topsp.html#K4" target="_self">T_0-canonical_map</a> <font color="Maroon">S</font></span>)</span></span>)</span> &amp; <font color="Maroon">f</font> is <a href="funct_1.html#V2">one-to-one</a> &amp;  <a href="t_0topsp.html#K4" target="_self">T_0-canonical_map</a> <font color="Maroon">T</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="t_0topsp.html#K4" target="_self">T_0-canonical_map</a> <font color="Maroon">S</font></span>)</span></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">f</font> )
 <b>by </b><i>, <a class="ref" href="rfinseq.html#T3">RFINSEQ:3</a></i>;<br/>
<a NAME="E5:22"/><i><font color="Green">E30</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> )
 
<b>by </b><i><a class="txt" href="t_0topsp.html#E2:22"><i><font color="Green">E19</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">f</font> = <font color="Maroon">f</font> as   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">T</font>,<font color="Maroon">S</font> <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T11">RELSET_1:11</a></i>;<br/>
<b>take </b>
<font color="Maroon">f</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="t_0topsp.html#D1" target="_self">T_0TOPSP:def 1</a><br/>

<b>thus </b><a NAME="E7:22"/><i><font color="Green">E37</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">S</font> )
 <b>by </b><i/>; <i><font color="Red">:: according to </font></i><a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a><br/>

<b>thus </b><a NAME="E8:22"/>
<font color="Maroon">f</font> is <a href="funct_1.html#V2">one-to-one</a>
 <b>by </b><i><a class="txt" href="t_0topsp.html#E2:22"><i><font color="Green">E19</font></i></a></i>;<br/>

<a NAME="E9:22"/><i><font color="Green">E39</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><a href="t_0topsp.html#K3" target="_self">T_0-reflex</a> <font color="Maroon">S</font></span>)</span> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">h</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><a href="t_0topsp.html#K3" target="_self">T_0-reflex</a> <font color="Maroon">T</font></span>)</span> &amp; <font color="Maroon">h</font> is <a href="funct_1.html#V2">one-to-one</a> &amp; <font color="Maroon">h</font> is <a href="pre_topc.html#V5">continuous</a> &amp; <font color="Maroon">h</font> <a href="tops_2.html#NK3" target="_self">"</a>  is <a href="pre_topc.html#V5">continuous</a> )
 
<b>by </b><i>, <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>;<br/>
<a NAME="E10:22"/>
for <font color="Olive">A</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">S</font>  st <font color="Olive">A</font> is <a href="pre_topc.html#V3">open</a> holds <br/><font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Olive">A</font> is <a href="pre_topc.html#V3">open</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">A</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">S</font>;<br/>

<b>assume </b><a NAME="E1:22_1"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">A</font> is <a href="pre_topc.html#V3">open</a>
 ;<br/>

<a NAME="E2:22_1"/>
 <a href="t_0topsp.html#K4" target="_self">T_0-canonical_map</a> <font color="Maroon">S</font> is <a href="t_0topsp.html#V1" target="_self">open</a>
 
<b>by </b><i/>;<br/>
<a NAME="E3:22_1"/><b>then </b><i><font color="Green">E44</font></i>: 
<span class="p1">(<span class="default"><a href="t_0topsp.html#K4" target="_self">T_0-canonical_map</a> <font color="Maroon">S</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">A</font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="ref" href="t_0topsp.html#T2" target="_self">Th2</a>, </i>;<br/>
<a NAME="E4:22_1"/>
<font color="Maroon">h</font> <a href="tops_2.html#NK3" target="_self">"</a>  is <a href="pre_topc.html#V5">continuous</a>
 
<b>by </b><i>, <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>;<br/>
<a NAME="E5:22_1"/><b>then </b><i><font color="Green">E67</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="relset_1.html#K11">"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="t_0topsp.html#K4" target="_self">T_0-canonical_map</a> <font color="Maroon">S</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">A</font></span>)</span> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i>, <a class="ref" href="tops_2.html#T55">TOPS_2:55</a></i>;<br/>
<a NAME="E6:22_1"/>
<font color="Maroon">h</font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="t_0topsp.html#K4" target="_self">T_0-canonical_map</a> <font color="Maroon">S</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="relat_1.html#K10">"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="t_0topsp.html#K4" target="_self">T_0-canonical_map</a> <font color="Maroon">S</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">A</font></span>)</span>
 
<b>by </b><i><a class="txt" href="t_0topsp.html#E4:22"><i><font color="Green">E22</font></i></a>, <a class="ref" href="funct_1.html#T154">FUNCT_1:154</a></i>;<br/>
<a NAME="E7:22_1"/><b>then </b>
<font color="Maroon">h</font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="t_0topsp.html#K4" target="_self">T_0-canonical_map</a> <font color="Maroon">S</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">A</font></span>)</span> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="txt" href="t_0topsp.html#E4:22"><i><font color="Green">E22</font></i></a>, , <a class="ref" href="tops_2.html#D4">TOPS_2:def 4</a></i>;<br/>
<a NAME="E8:22_1"/><b>then </b><i><font color="Green">E68</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="t_0topsp.html#K4" target="_self">T_0-canonical_map</a> <font color="Maroon">S</font></span>)</span></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">A</font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="ref" href="relat_1.html#T159">RELAT_1:159</a></i>;<br/>
<b>set </b><font color="Maroon">g</font> = <font color="Maroon">h</font> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><a href="t_0topsp.html#K4" target="_self">T_0-canonical_map</a> <font color="Maroon">S</font></span>)</span>;<br/>
<b>set </b><font color="Maroon">B</font> = <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="t_0topsp.html#K4" target="_self">T_0-canonical_map</a> <font color="Maroon">S</font></span>)</span></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">A</font>;<br/>
<a NAME="E9:22_1"/><i><font color="Green">E70</font></i>: 
<span class="p1">(<span class="default"><a href="t_0topsp.html#K4" target="_self">T_0-canonical_map</a> <font color="Maroon">T</font></span>)</span> <a href="relset_1.html#K11">"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="partfun1.html#K1">*</a> <span class="p3">(<span class="default"><a href="t_0topsp.html#K4" target="_self">T_0-canonical_map</a> <font color="Maroon">S</font></span>)</span></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">A</font></span>)</span> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i>, <a class="ref" href="tops_2.html#T55">TOPS_2:55</a></i>;<br/>
<a NAME="E10:22_1"/><i><font color="Green">E71</font></i>: 
for <font color="Olive">x1</font>, <font color="Olive">x2</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font>  st <font color="Olive">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="t_0topsp.html#K4" target="_self">T_0-canonical_map</a> <font color="Maroon">S</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">x1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="t_0topsp.html#K4" target="_self">T_0-canonical_map</a> <font color="Maroon">S</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">x2</font> holds <br/><font color="Olive">x2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x1</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font>, <font color="Maroon">x2</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:22_1_1"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>and </b><br/><a NAME="E2:22_1_1"/><i><font color="Green">E73</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="t_0topsp.html#K4" target="_self">T_0-canonical_map</a> <font color="Maroon">S</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="t_0topsp.html#K4" target="_self">T_0-canonical_map</a> <font color="Maroon">S</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x2</font>
 ;<br/>

<a NAME="E3:22_1_1"/>
<font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="t_0topsp.html#K4" target="_self">T_0-canonical_map</a> <font color="Maroon">S</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x1</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="t_0topsp.html#K4" target="_self">T_0-canonical_map</a> <font color="Maroon">S</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x2</font>
 
<b>by </b><i><a class="txt" href="t_0topsp.html#E5:22"><i><font color="Green">E30</font></i></a>, <a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>;<br/>
<a NAME="E4:22_1_1"/><b>then </b>
<font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="t_0topsp.html#K4" target="_self">T_0-canonical_map</a> <font color="Maroon">S</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x1</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="t_0topsp.html#K4" target="_self">T_0-canonical_map</a> <font color="Maroon">S</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>;<br/>
<a NAME="E5:22_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="t_0topsp.html#K4" target="_self">T_0-canonical_map</a> <font color="Maroon">S</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="t_0topsp.html#K4" target="_self">T_0-canonical_map</a> <font color="Maroon">S</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x2</font>
 
<b>by </b><i><a class="txt" href="t_0topsp.html#E4:22"><i><font color="Green">E22</font></i></a>, <a class="ref" href="funct_2.html#T25">FUNCT_2:25</a></i>;<br/>
<b>hence </b><a NAME="E6:22_1_1"/>
<font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="t_0topsp.html#T2" target="_self">Th2</a>, , </i>;<br/>


</div><b>end;</b></div>
<a NAME="E11:22_1"/>
<span class="p1">(<span class="default"><a href="t_0topsp.html#K4" target="_self">T_0-canonical_map</a> <font color="Maroon">T</font></span>)</span> <a href="relset_1.html#K11">"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="partfun1.html#K1">*</a> <span class="p3">(<span class="default"><a href="t_0topsp.html#K4" target="_self">T_0-canonical_map</a> <font color="Maroon">S</font></span>)</span></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="partfun1.html#K1">*</a> <span class="p3">(<span class="default"><a href="t_0topsp.html#K4" target="_self">T_0-canonical_map</a> <font color="Maroon">S</font></span>)</span></span>)</span> <a href="relset_1.html#K11">"</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="partfun1.html#K1">*</a> <span class="p4">(<span class="default"><a href="t_0topsp.html#K4" target="_self">T_0-canonical_map</a> <font color="Maroon">S</font></span>)</span></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">A</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="t_0topsp.html#E2:22"><i><font color="Green">E19</font></i></a>, <a class="ref" href="relat_1.html#T181">RELAT_1:181</a></i>;<br/>
<b>hence </b><a NAME="E12:22_1"/>
<font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">A</font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i>, , </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E11:22"/>
<font color="Maroon">f</font> is <a href="pre_topc.html#V5">continuous</a>
 <b>by </b><i><a class="ref" href="tops_2.html#T55">TOPS_2:55</a></i>;<br/>

<a NAME="E12:22"/>
for <font color="Olive">A</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>  st <font color="Olive">A</font> is <a href="pre_topc.html#V3">open</a> holds <br/><span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="relset_1.html#K11">"</a> <font color="Olive">A</font> is <a href="pre_topc.html#V3">open</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">A</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>;<br/>

<b>assume </b><a NAME="E1:22_2"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">A</font> is <a href="pre_topc.html#V3">open</a>
 ;<br/>

<b>set </b><font color="Maroon">g</font> = <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><a href="t_0topsp.html#K4" target="_self">T_0-canonical_map</a> <font color="Maroon">T</font></span>)</span>;<br/>
<a NAME="E2:22_2"/><i><font color="Green">E75</font></i>: 
for <font color="Olive">x1</font>, <font color="Olive">x2</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font>  st <font color="Olive">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="t_0topsp.html#K4" target="_self">T_0-canonical_map</a> <font color="Maroon">T</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">x1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="t_0topsp.html#K4" target="_self">T_0-canonical_map</a> <font color="Maroon">T</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">x2</font> holds <br/><font color="Olive">x2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x1</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font>, <font color="Maroon">x2</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:22_2_1"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>and </b><br/><a NAME="E2:22_2_1"/><i><font color="Green">E77</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="t_0topsp.html#K4" target="_self">T_0-canonical_map</a> <font color="Maroon">T</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="t_0topsp.html#K4" target="_self">T_0-canonical_map</a> <font color="Maroon">T</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x2</font>
 ;<br/>

<a NAME="E3:22_2_1"/><i><font color="Green">E78</font></i>: 
<font color="Maroon">h</font> <a href="tops_2.html#NK3" target="_self">"</a>  is <a href="funct_1.html#V2">one-to-one</a>
 
<b>by </b><i><a class="txt" href="t_0topsp.html#E4:22"><i><font color="Green">E22</font></i></a>, <a class="ref" href="tops_2.html#T63">TOPS_2:63</a></i>;<br/>
<a NAME="E4:22_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="t_0topsp.html#K4" target="_self">T_0-canonical_map</a> <font color="Maroon">T</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x1</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="t_0topsp.html#K4" target="_self">T_0-canonical_map</a> <font color="Maroon">T</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x2</font>
 
<b>by </b><i>, <a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>;<br/>
<a NAME="E5:22_2_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="t_0topsp.html#K4" target="_self">T_0-canonical_map</a> <font color="Maroon">T</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x1</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="t_0topsp.html#K4" target="_self">T_0-canonical_map</a> <font color="Maroon">T</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>;<br/>
<a NAME="E6:22_2_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="t_0topsp.html#K4" target="_self">T_0-canonical_map</a> <font color="Maroon">T</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="t_0topsp.html#K4" target="_self">T_0-canonical_map</a> <font color="Maroon">T</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x2</font>
 
<b>by </b><i>, <a class="ref" href="funct_2.html#T25">FUNCT_2:25</a></i>;<br/>
<b>hence </b><a NAME="E7:22_2_1"/>
<font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="t_0topsp.html#T2" target="_self">Th2</a>, , </i>;<br/>


</div><b>end;</b></div>
<b>set </b><font color="Maroon">B</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="t_0topsp.html#K4" target="_self">T_0-canonical_map</a> <font color="Maroon">T</font></span>)</span></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">A</font>;<br/>
<a NAME="E3:22_2"/><i><font color="Green">E79</font></i>: 
<span class="p1">(<span class="default"><a href="t_0topsp.html#K4" target="_self">T_0-canonical_map</a> <font color="Maroon">S</font></span>)</span> <a href="relset_1.html#K11">"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="partfun1.html#K1">*</a> <span class="p3">(<span class="default"><a href="t_0topsp.html#K4" target="_self">T_0-canonical_map</a> <font color="Maroon">T</font></span>)</span></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">A</font></span>)</span> is <a href="pre_topc.html#V3">open</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_2_2"><b>proof </b></a><div class="add">

<a NAME="E1:22_2_2"/>
 <a href="t_0topsp.html#K4" target="_self">T_0-canonical_map</a> <font color="Maroon">T</font> is <a href="t_0topsp.html#V1" target="_self">open</a>
 
<b>by </b><i/>;<br/>
<a NAME="E2:22_2_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="t_0topsp.html#K4" target="_self">T_0-canonical_map</a> <font color="Maroon">T</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">A</font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="ref" href="t_0topsp.html#T2" target="_self">Th2</a>, </i>;<br/>
<a NAME="E3:22_2_2"/><b>then </b><i><font color="Green">E80</font></i>: 
<font color="Maroon">h</font> <a href="relset_1.html#K11">"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="t_0topsp.html#K4" target="_self">T_0-canonical_map</a> <font color="Maroon">T</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">A</font></span>)</span> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="txt" href="t_0topsp.html#E4:22"><i><font color="Green">E22</font></i></a>, <a class="ref" href="tops_2.html#T55">TOPS_2:55</a></i>;<br/>
<a NAME="E4:22_2_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="t_0topsp.html#K4" target="_self">T_0-canonical_map</a> <font color="Maroon">T</font></span>)</span></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="t_0topsp.html#K4" target="_self">T_0-canonical_map</a> <font color="Maroon">T</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">A</font></span>)</span>
 
<b>by </b><i><a class="ref" href="relat_1.html#T159">RELAT_1:159</a></i>;<br/>
<a NAME="E5:22_2_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="t_0topsp.html#K4" target="_self">T_0-canonical_map</a> <font color="Maroon">S</font></span>)</span> <a href="relset_1.html#K11">"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="partfun1.html#K1">*</a> <span class="p3">(<span class="default"><a href="t_0topsp.html#K4" target="_self">T_0-canonical_map</a> <font color="Maroon">T</font></span>)</span></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="t_0topsp.html#K4" target="_self">T_0-canonical_map</a> <font color="Maroon">S</font></span>)</span> <a href="relset_1.html#K11">"</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="relset_1.html#K11">"</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="t_0topsp.html#K4" target="_self">T_0-canonical_map</a> <font color="Maroon">T</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">A</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="t_0topsp.html#E4:22"><i><font color="Green">E22</font></i></a>, <a class="ref" href="tops_2.html#T68">TOPS_2:68</a></i>;<br/>
<b>hence </b><a NAME="E6:22_2_2"/>
<span class="p1">(<span class="default"><a href="t_0topsp.html#K4" target="_self">T_0-canonical_map</a> <font color="Maroon">S</font></span>)</span> <a href="relset_1.html#K11">"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="partfun1.html#K1">*</a> <span class="p3">(<span class="default"><a href="t_0topsp.html#K4" target="_self">T_0-canonical_map</a> <font color="Maroon">T</font></span>)</span></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">A</font></span>)</span> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i>, <a class="ref" href="tops_2.html#T55">TOPS_2:55</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:22_2"/>
 <a href="t_0topsp.html#K4" target="_self">T_0-canonical_map</a> <font color="Maroon">T</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">h</font> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="t_0topsp.html#K4" target="_self">T_0-canonical_map</a> <font color="Maroon">S</font></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font></span>)</span>
 
<b>by </b><i><a class="txt" href="t_0topsp.html#E2:22"><i><font color="Green">E19</font></i></a>, <a class="ref" href="relat_1.html#T55">RELAT_1:55</a></i>;<br/>
<a NAME="E5:22_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><a href="t_0topsp.html#K4" target="_self">T_0-canonical_map</a> <font color="Maroon">T</font></span>)</span> <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">h</font></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="t_0topsp.html#K4" target="_self">T_0-canonical_map</a> <font color="Maroon">S</font></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font></span>)</span>
 
<b>by </b><i><a class="ref" href="relat_1.html#T55">RELAT_1:55</a></i>;<br/>
<a NAME="E6:22_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><a href="t_0topsp.html#K4" target="_self">T_0-canonical_map</a> <font color="Maroon">T</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6">id</a> <span class="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">h</font></span>)</span></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="t_0topsp.html#K4" target="_self">T_0-canonical_map</a> <font color="Maroon">S</font></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font></span>)</span>
 
<b>by </b><i><a class="txt" href="t_0topsp.html#E4:22"><i><font color="Green">E22</font></i></a>, <a class="ref" href="tops_2.html#T65">TOPS_2:65</a></i>;<br/>
<a NAME="E7:22_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><a href="t_0topsp.html#K4" target="_self">T_0-canonical_map</a> <font color="Maroon">T</font></span>)</span> <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="t_0topsp.html#K3" target="_self">T_0-reflex</a> <font color="Maroon">S</font></span>)</span></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="t_0topsp.html#K4" target="_self">T_0-canonical_map</a> <font color="Maroon">S</font></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E8:22_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="t_0topsp.html#K4" target="_self">T_0-canonical_map</a> <font color="Maroon">T</font></span>)</span></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="t_0topsp.html#K4" target="_self">T_0-canonical_map</a> <font color="Maroon">S</font></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#T23">FUNCT_2:23</a></i>;<br/>
<a NAME="E9:22_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="t_0topsp.html#K4" target="_self">T_0-canonical_map</a> <font color="Maroon">T</font></span>)</span></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><a href="t_0topsp.html#K4" target="_self">T_0-canonical_map</a> <font color="Maroon">S</font></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="relat_1.html#T55">RELAT_1:55</a></i>;<br/>
<a NAME="E10:22_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="t_0topsp.html#K4" target="_self">T_0-canonical_map</a> <font color="Maroon">T</font></span>)</span></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><a href="t_0topsp.html#K4" target="_self">T_0-canonical_map</a> <font color="Maroon">S</font></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span>)</span>
 
<b>by </b><i><a class="txt" href="t_0topsp.html#E2:22"><i><font color="Green">E19</font></i></a>, , <a class="ref" href="tops_2.html#T65">TOPS_2:65</a></i>;<br/>
<a NAME="E11:22_2"/><b>then </b>
 <a href="t_0topsp.html#K4" target="_self">T_0-canonical_map</a> <font color="Maroon">S</font> <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="t_0topsp.html#K4" target="_self">T_0-canonical_map</a> <font color="Maroon">T</font></span>)</span></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#T23">FUNCT_2:23</a></i>;<br/>
<a NAME="E12:22_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="t_0topsp.html#K4" target="_self">T_0-canonical_map</a> <font color="Maroon">S</font></span>)</span> <a href="relset_1.html#K11">"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="partfun1.html#K1">*</a> <span class="p3">(<span class="default"><a href="t_0topsp.html#K4" target="_self">T_0-canonical_map</a> <font color="Maroon">T</font></span>)</span></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="relset_1.html#K11">"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="partfun1.html#K1">*</a> <span class="p3">(<span class="default"><a href="t_0topsp.html#K4" target="_self">T_0-canonical_map</a> <font color="Maroon">T</font></span>)</span></span>)</span> <a href="relset_1.html#K11">"</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">h</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="partfun1.html#K1">*</a> <span class="p4">(<span class="default"><a href="t_0topsp.html#K4" target="_self">T_0-canonical_map</a> <font color="Maroon">T</font></span>)</span></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">A</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="relat_1.html#T181">RELAT_1:181</a></i>;<br/>
<b>hence </b><a NAME="E13:22_2"/>
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="relset_1.html#K11">"</a> <font color="Maroon">A</font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i>, , </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E13:22"/>
<font color="Maroon">f</font> <a href="tops_2.html#NK3" target="_self">"</a>  is <a href="pre_topc.html#V5">continuous</a>
 <b>by </b><i><a class="ref" href="tops_2.html#T55">TOPS_2:55</a></i>;<br/>


</div>
