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

<b>thus </b><a NAME="E1:12_1_1"/>
( <font color="Maroon">X</font> <a href="waybel18.html#R1">is_Retract_of</a> <font color="Maroon">Y</font> implies  ex <font color="Olive">c</font> being <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">X</font>,<font color="Maroon">Y</font> ex <font color="Olive">r</font> being <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">Y</font>,<font color="Maroon">X</font> st <font color="Olive">r</font> <a href="partfun1.html#K1">*</a> <font color="Olive">c</font> <a href="funct_2.html#R2">=</a>  <a href="grcat_1.html#K7">id</a> <font color="Maroon">X</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1_1_1"><b>proof </b></a><div class="add">

<b>given </b><font color="Maroon">f</font> being   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">Y</font>,<font color="Maroon">Y</font><b> such that </b><a NAME="E1:12_1_1_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">f</font> is <a href="pre_topc.html#V5">continuous</a>
 <b>and </b><br/><a NAME="E2:12_1_1_1"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">f</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">f</font>
 <b>and </b><br/><a NAME="E3:12_1_1_1"/><i><font color="Green">E44</font></i>: 
 <a href="waybel18.html#K7">Image</a> <font color="Maroon">f</font>,<font color="Maroon">X</font> <a href="borsuk_3.html#R2">are_homeomorphic</a> 
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel18.html#D8">WAYBEL18:def 8</a><br/>

<b>consider </b><font color="Maroon">h</font> being   <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><a href="waybel18.html#K7">Image</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">X</font><b> such that </b><br/><a NAME="E5:12_1_1_1"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">h</font> <a href="tops_2.html#NR1" target="_self">is_homeomorphism</a> 
 <b>by </b><i>, <a class="ref" href="t_0topsp.html#D1">T_0TOPSP:def 1</a></i>;<br/>
<a NAME="E6:12_1_1_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/>
<b>then reconsider </b><font color="Maroon">c</font> = <span class="p1">(<span class="default"><a href="tmap_1.html#K4">incl</a> <span class="p2">(<span class="default"><a href="waybel18.html#K7">Image</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> as  <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">X</font>,<font color="Maroon">Y</font> <b>by </b><i><a class="ref" href="tops_2.html#T58">TOPS_2:58</a></i>;<br/>
<a NAME="E8:12_1_1_1"/>
( <font color="Maroon">h</font> is <a href="pre_topc.html#V5">continuous</a> &amp;  <a href="waybel18.html#K8">corestr</a> <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#D5">TOPS_2:def 5</a>, <a class="ref" href="waybel18.html#T11">WAYBEL18:11</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">r</font> = <font color="Maroon">h</font> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><a href="waybel18.html#K8">corestr</a> <font color="Maroon">f</font></span>)</span> as  <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">Y</font>,<font color="Maroon">X</font> <b>by </b><i><a class="ref" href="tops_2.html#T58">TOPS_2:58</a></i>;<br/>
<b>take </b>
<font color="Maroon">c</font>
;<br/>

<b>take </b>
<font color="Maroon">r</font>
;<br/>

<a NAME="E10:12_1_1_1"/><i><font color="Green">E54</font></i>: 
(  <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> <font color="Maroon">X</font> &amp; <font color="Maroon">h</font> is <a href="funct_1.html#V2">one-to-one</a> )
 
<b>by </b><i>, <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>;<br/>
<b>thus </b><font color="Maroon">r</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c</font> = 
<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="waybel18.html#K8">corestr</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="tmap_1.html#K4">incl</a> <span class="p4">(<span class="default"><a href="waybel18.html#K7">Image</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="relat_1.html#T55">RELAT_1:55</a></i>
<br/>.= 
<font color="Maroon">h</font> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel18.html#K8">corestr</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p3">(<span class="default"><a href="tmap_1.html#K4">incl</a> <span class="p4">(<span class="default"><a href="waybel18.html#K7">Image</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">h</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/>.= 
<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="grcat_1.html#K7">id</a> <span class="p3">(<span class="default"><a href="waybel18.html#K7">Image</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="yellow14.html#T36">YELLOW14:36</a></i>
<br/>.= 
<font color="Maroon">h</font> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span>
<b>by </b><i><a class="ref" href="grcat_1.html#T12">GRCAT_1:12</a></i>
<br/>.= 
 <a href="grcat_1.html#K7">id</a> <font color="Maroon">X</font>
<b>by </b><i>, <a class="ref" href="tops_2.html#T65">TOPS_2:65</a></i>
;<br/>


</div><b>end;</b></div>

<b>given </b><font color="Maroon">c</font> being  <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">X</font>,<font color="Maroon">Y</font>, <font color="Maroon">r</font> being  <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">Y</font>,<font color="Maroon">X</font><b> such that </b><a NAME="E2:12_1_1"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">r</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c</font> <a href="funct_2.html#R2">=</a>  <a href="grcat_1.html#K7">id</a> <font color="Maroon">X</font>
 ;<br/>

<b>take </b><font color="Maroon">f</font> = <font color="Maroon">c</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">r</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel18.html#D8">WAYBEL18:def 8</a><br/>

<b>thus </b><a NAME="E3:12_1_1"/>
<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#T58">TOPS_2:58</a></i>;<br/>

<b>thus </b><font color="Maroon">f</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font> = 
<font color="Maroon">c</font> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">r</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="relat_1.html#T55">RELAT_1:55</a></i>
<br/>.= 
<font color="Maroon">c</font> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="grcat_1.html#K7">id</a> <font color="Maroon">X</font></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">r</font></span>)</span>
<b>by </b><i>, <a class="ref" href="relat_1.html#T55">RELAT_1:55</a></i>
<br/>.= 
<font color="Maroon">f</font>
<b>by </b><i><a class="ref" href="grcat_1.html#T12">GRCAT_1:12</a></i>
;<br/>

<a NAME="E5:12_1_1"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">r</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c</font> <a href="funct_2.html#R2">=</a>  <a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>
 
<b>by </b><i/>;<br/>
<i><font color="Green">E65</font></i>:  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c</font> = 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>
<br/>.= 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">r</font>
<b>by </b><i>, <a class="ref" href="funct_2.html#T24">FUNCT_2:24</a></i>
;<br/>
<a NAME="E7:12_1_1"/><i><font color="Green">E68</font></i>: 
 <a href="waybel18.html#K7">Image</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="waybel18.html#K7">Image</a> <font color="Maroon">c</font>
 
<b>by </b><i>, <a class="ref" href="relat_1.html#T47">RELAT_1:47</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">cf</font> =  <a href="waybel18.html#K8">corestr</a> <font color="Maroon">c</font> as   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">X</font>,<span class="p1">(<span class="default"><a href="waybel18.html#K7">Image</a> <font color="Maroon">f</font></span>)</span> ;<br/>
<b>take </b><font color="Maroon">h</font> = <font color="Maroon">cf</font> <a href="tops_2.html#NK3" target="_self">"</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="t_0topsp.html#D1">T_0TOPSP:def 1</a><br/>

<b>thus </b><a NAME="E9:12_1_1"/>
 <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="waybel18.html#K7">Image</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>; <i><font color="Red">:: according to </font></i><a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a><br/>

<a NAME="E10:12_1_1"/><i><font color="Green">E69</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="waybel18.html#K8">corestr</a> <font color="Maroon">c</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><a href="waybel18.html#K7">Image</a> <font color="Maroon">c</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D3">FUNCT_2:def 3</a></i>;<br/>
<a NAME="E11:12_1_1"/>
<font color="Maroon">c</font> is <a href="funct_1.html#V2">one-to-one</a>
 
<b>by </b><i>, <a class="ref" href="funct_2.html#T29">FUNCT_2:29</a></i>;<br/>
<a NAME="E12:12_1_1"/><b>then </b><i><font color="Green">E71</font></i>: 
 <a href="waybel18.html#K8">corestr</a> <font color="Maroon">c</font> is <a href="funct_1.html#V2">one-to-one</a>
 
;<br/>
<b>hence </b><a NAME="E13:12_1_1"/>
 <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> <font color="Maroon">X</font>
 <b>by </b><i>, , <a class="ref" href="tops_2.html#T62">TOPS_2:62</a></i>;<br/>

<a NAME="E14:12_1_1"/>
<span class="p1">(<span class="default"><a href="waybel18.html#K8">corestr</a> <font color="Maroon">c</font></span>)</span> <a href="funct_1.html#K2">"</a>  is <a href="funct_1.html#V2">one-to-one</a>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#T62">FUNCT_1:62</a></i>;<br/>
<b>hence </b><a NAME="E15:12_1_1"/>
<font color="Maroon">h</font> is <a href="funct_1.html#V2">one-to-one</a>
 <b>by </b><i>, , , <a class="ref" href="tops_2.html#D4">TOPS_2:def 4</a></i>;<br/>

<a NAME="E16:12_1_1"/><i><font color="Green">E72</font></i>: 
 <a href="waybel18.html#K8">corestr</a> <font color="Maroon">c</font> is <a href="pre_topc.html#V5">continuous</a>
 
<b>by </b><i><a class="ref" href="waybel18.html#T11">WAYBEL18:11</a></i>;<br/>
<a NAME="E17:12_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">cf</font></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">cf</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="grcat_1.html#K7">id</a> <font color="Maroon">X</font></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">cf</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E18:12_1_1"/><b>then </b>
<font color="Maroon">r</font> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">cf</font> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">cf</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="grcat_1.html#K7">id</a> <font color="Maroon">X</font></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">cf</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="relat_1.html#T55">RELAT_1:55</a></i>;<br/>
<a NAME="E19:12_1_1"/><b>then </b>
<font color="Maroon">r</font> <a href="partfun1.html#K1">*</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#K5">rng</a> <font color="Maroon">cf</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="grcat_1.html#K7">id</a> <font color="Maroon">X</font></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">cf</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span>
 
<b>by </b><i>, , , <a class="ref" href="tops_2.html#T65">TOPS_2:65</a></i>;<br/>
<a NAME="E20:12_1_1"/><b>then </b>
<font color="Maroon">r</font> <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 <span class="p2">(<span class="default"><a href="waybel18.html#K7">Image</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="grcat_1.html#K7">id</a> <font color="Maroon">X</font></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">cf</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E21:12_1_1"/><b>then </b>
<font color="Maroon">r</font> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><a href="grcat_1.html#K7">id</a> <span class="p2">(<span class="default"><a href="waybel18.html#K7">Image</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="grcat_1.html#K7">id</a> <font color="Maroon">X</font></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">cf</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span>
 
;<br/>
<a NAME="E22:12_1_1"/><b>then </b><i><font color="Green">E73</font></i>: 
<font color="Maroon">r</font> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><a href="grcat_1.html#K7">id</a> <span class="p2">(<span class="default"><a href="waybel18.html#K7">Image</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">cf</font> <a href="tops_2.html#NK3" target="_self">"</a> 
 
<b>by </b><i><a class="ref" href="grcat_1.html#T12">GRCAT_1:12</a></i>;<br/>
<a NAME="E23:12_1_1"/>
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel18.html#K7">Image</a> <font color="Maroon">c</font></span>)</span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font>
 
<b>by </b><i><a class="ref" href="borsuk_1.html#T35">BORSUK_1:35</a></i>;<br/>
<a NAME="E24:12_1_1"/><b>then </b>
 <a href="grcat_1.html#K7">id</a> <span class="p1">(<span class="default"><a href="waybel18.html#K7">Image</a> <font color="Maroon">c</font></span>)</span> is   <a href="funct_2.html#NM1">Function</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel18.html#K7">Image</a> <font color="Maroon">c</font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T9">FUNCT_2:9</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">q</font> = <font color="Maroon">r</font> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><a href="grcat_1.html#K7">id</a> <span class="p2">(<span class="default"><a href="waybel18.html#K7">Image</a> <font color="Maroon">c</font></span>)</span></span>)</span> as   <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><a href="waybel18.html#K7">Image</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">X</font> <b>by </b><i>, <a class="ref" href="funct_2.html#T19">FUNCT_2:19</a></i>;<br/>
<a NAME="E26:12_1_1"/>
for <font color="Olive">P</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>  st <font color="Olive">P</font> is <a href="pre_topc.html#V3">open</a> holds <br/><font color="Maroon">q</font> <a href="relset_1.html#K11">"</a> <font color="Olive">P</font> is <a href="pre_topc.html#V3">open</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1_1_4"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:12_1_1_4"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">P</font> is <a href="pre_topc.html#V3">open</a>
 ;<br/>

<a NAME="E2:12_1_1_4"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">q</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">P</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="grcat_1.html#K7">id</a> <span class="p2">(<span class="default"><a href="waybel18.html#K7">Image</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="relset_1.html#K11">"</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">P</font></span>)</span>
 
<b>by </b><i><a class="ref" href="relat_1.html#T181">RELAT_1:181</a></i>;<br/>
<a NAME="E3:12_1_1_4"/>
<font color="Maroon">r</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">P</font> 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="E4:12_1_1_4"/><b>then </b><i><font color="Green">E76</font></i>: 
<font color="Maroon">r</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">P</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">Y</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/>
<a NAME="E5:12_1_1_4"/><i><font color="Green">E90</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="grcat_1.html#K7">id</a> <span class="p2">(<span class="default"><a href="waybel18.html#K7">Image</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><a href="waybel18.html#K7">Image</a> <font color="Maroon">c</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E6:12_1_1_4"/>
<span class="p1">(<span class="default"><a href="grcat_1.html#K7">id</a> <span class="p2">(<span class="default"><a href="waybel18.html#K7">Image</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="relset_1.html#K11">"</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">P</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">P</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2">(<span class="default"><a href="waybel18.html#K7">Image</a> <font color="Maroon">c</font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1_1_4_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:12_1_1_4_1"/>
<span class="p1">(<span class="default"><a href="grcat_1.html#K7">id</a> <span class="p2">(<span class="default"><a href="waybel18.html#K7">Image</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="relset_1.html#K11">"</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">P</font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">P</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2">(<span class="default"><a href="waybel18.html#K7">Image</a> <font color="Maroon">c</font></span>)</span></span>)</span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1_1_4_1_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:12_1_1_4_1_1"/><i><font color="Green">E91</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="grcat_1.html#K7">id</a> <span class="p2">(<span class="default"><a href="waybel18.html#K7">Image</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="relset_1.html#K11">"</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">P</font></span>)</span>
 ;<br/>

<a NAME="E2:12_1_1_4_1_1"/><b>then </b><i><font color="Green">E92</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="grcat_1.html#K7">id</a> <span class="p2">(<span class="default"><a href="waybel18.html#K7">Image</a> <font color="Maroon">c</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/>
<a NAME="E3:12_1_1_4_1_1"/>
<span class="p1">(<span class="default"><a href="grcat_1.html#K7">id</a> <span class="p2">(<span class="default"><a href="waybel18.html#K7">Image</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">r</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">P</font>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/>
<a NAME="E4:12_1_1_4_1_1"/><b>then </b>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">r</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">P</font>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#T35">FUNCT_1:35</a></i>;<br/>
<b>hence </b><a NAME="E5:12_1_1_4_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">P</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2">(<span class="default"><a href="waybel18.html#K7">Image</a> <font color="Maroon">c</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>


</div><b>end;</b></div>

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

<b>assume </b><a NAME="E2:12_1_1_4_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">P</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2">(<span class="default"><a href="waybel18.html#K7">Image</a> <font color="Maroon">c</font></span>)</span></span>)</span>
 ;<br/>

<a NAME="E3:12_1_1_4_1"/><b>then </b><i><font color="Green">E93</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">r</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">P</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><a href="waybel18.html#K7">Image</a> <font color="Maroon">c</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E4:12_1_1_4_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="grcat_1.html#K7">id</a> <span class="p2">(<span class="default"><a href="waybel18.html#K7">Image</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">r</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">P</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#T35">FUNCT_1:35</a></i>;<br/>
<b>hence </b><a NAME="E5:12_1_1_4_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="grcat_1.html#K7">id</a> <span class="p2">(<span class="default"><a href="waybel18.html#K7">Image</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="relset_1.html#K11">"</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">P</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:12_1_1_4"/>
<font color="Maroon">q</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">P</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><a href="waybel18.html#K7">Image</a> <font color="Maroon">f</font></span>)</span>
 <b>by </b><i>, , , <a class="ref" href="pre_topc.html#D9">PRE_TOPC:def 9</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E27:12_1_1"/>
<font color="Maroon">h</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/>

<b>thus </b><a NAME="E28:12_1_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#T64">TOPS_2:64</a></i>;<br/>


</div>
