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<div class="add">

<b>let </b><font color="Maroon">Y</font> be  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a>;<br/><b>let </b><font color="Maroon">W</font> be  non <a href="struct_0.html#V3">empty</a>  <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">Y</font>;<br/>



<b>set </b><font color="Maroon">ci</font> =  <a href="waybel18.html#K8">corestr</a> <span class="p1">(<span class="default"><a href="tmap_1.html#K4">incl</a> <font color="Maroon">W</font></span>)</span>;<br/>
<b>thus </b><a NAME="E1:53"/>
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="waybel18.html#K8">corestr</a> <span class="p2">(<span class="default"><a href="tmap_1.html#K4">incl</a> <font color="Maroon">W</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">W</font>
 <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/>

<b>thus </b><a NAME="E2:53"/><i><font color="Green">E38</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="waybel18.html#K8">corestr</a> <span class="p2">(<span class="default"><a href="tmap_1.html#K4">incl</a> <font color="Maroon">W</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> <span class="p2">(<span class="default"><a href="tmap_1.html#K4">incl</a> <font color="Maroon">W</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="funct_2.html#D3">FUNCT_2:def 3</a></i>;<br/>

<i><font color="Green">E39</font></i>:  <a href="waybel18.html#K8">corestr</a> <span class="p1">(<span class="default"><a href="tmap_1.html#K4">incl</a> <font color="Maroon">W</font></span>)</span> = 
 <a href="tmap_1.html#K4">incl</a> <font color="Maroon">W</font>
<b>by </b><i><a class="ref" href="waybel18.html#D7">WAYBEL18:def 7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="grcat_1.html#K7">id</a> <font color="Maroon">Y</font></span>)</span> <a href="tmap_1.html#K2">|</a> <font color="Maroon">W</font>
<b>by </b><i><a class="ref" href="tmap_1.html#D6">TMAP_1:def 6</a></i>
<br/>.= 
<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">Y</font></span>)</span> <a href="partfun1.html#K2">|</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W</font>
<b>by </b><i><a class="ref" href="tmap_1.html#D3">TMAP_1:def 3</a></i>
;<br/>
<b>hence </b><a NAME="E4:53"/><i><font color="Green">E40</font></i>: 
 <a href="waybel18.html#K8">corestr</a> <span class="p1">(<span class="default"><a href="tmap_1.html#K4">incl</a> <font color="Maroon">W</font></span>)</span> is <a href="funct_1.html#V2">one-to-one</a>
 <b>by </b><i><a class="ref" href="funct_1.html#T84">FUNCT_1:84</a></i>;<br/>

<b>thus </b><a NAME="E5:53"/>
 <a href="waybel18.html#K8">corestr</a> <span class="p1">(<span class="default"><a href="tmap_1.html#K4">incl</a> <font color="Maroon">W</font></span>)</span> 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="E6:53"/>
for <font color="Olive">P</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">W</font>  st <font color="Olive">P</font> is <a href="pre_topc.html#V3">open</a> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel18.html#K8">corestr</a> <span class="p3">(<span class="default"><a href="tmap_1.html#K4">incl</a> <font color="Maroon">W</font></span>)</span></span>)</span> <a href="tops_2.html#K2">/"</a> </span>)</span> <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="53_2"><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">W</font>;<br/>

<b>assume </b><a NAME="E1:53_2"/>
<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">W</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a><br/>

<b>then consider </b><font color="Maroon">Q</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font><b> such that </b><br/><a NAME="E3:53_2"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">Q</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">Y</font> &amp; <font color="Maroon">P</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Q</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">W</font></span>)</span> )
 <b>by </b><i><a class="ref" href="pre_topc.html#D9">PRE_TOPC:def 9</a></i>;<br/>
<a NAME="E4:53_2"/><i><font color="Green">E44</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W</font> is  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</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="E5:53_2"/><b>then </b><i><font color="Green">E45</font></i>: 
<font color="Maroon">P</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<i><font color="Green">E71</font></i>:  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">W</font> = 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font></span>)</span> <a href="partfun1.html#K2">|</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W</font></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="grcat_1.html#K7">id</a> <font color="Maroon">Y</font></span>)</span> <a href="tmap_1.html#K2">|</a> <font color="Maroon">W</font></span>)</span>
<b>by </b><i><a class="ref" href="tmap_1.html#D3">TMAP_1:def 3</a></i>
<br/>.= 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="tmap_1.html#K4">incl</a> <font color="Maroon">W</font></span>)</span>
<b>by </b><i><a class="ref" href="tmap_1.html#D6">TMAP_1:def 6</a></i>
<br/>.= 
 <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><a href="waybel18.html#K7">Image</a> <span class="p2">(<span class="default"><a href="tmap_1.html#K4">incl</a> <font color="Maroon">W</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="waybel18.html#T10">WAYBEL18:10</a></i>
;<br/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel18.html#K8">corestr</a> <span class="p3">(<span class="default"><a href="tmap_1.html#K4">incl</a> <font color="Maroon">W</font></span>)</span></span>)</span> <a href="tops_2.html#K2">/"</a> </span>)</span> <a href="relset_1.html#K11">"</a> <font color="Maroon">P</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font></span>)</span> <a href="partfun1.html#K2">|</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">P</font>
<b>by </b><i>, , , <a class="ref" href="tops_2.html#T67">TOPS_2:67</a></i>
<br/>.= 
<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">Y</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">P</font>
<b>by </b><i><a class="ref" href="tmap_1.html#T4">TMAP_1:4</a></i>
<br/>.= 
<font color="Maroon">P</font>
<b>by </b><i>, <a class="ref" href="funct_1.html#T162">FUNCT_1:162</a></i>
;<br/>
<b>hence </b><a NAME="E8:53_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel18.html#K8">corestr</a> <span class="p3">(<span class="default"><a href="tmap_1.html#K4">incl</a> <font color="Maroon">W</font></span>)</span></span>)</span> <a href="tops_2.html#K2">/"</a> </span>)</span> <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> <span class="p2">(<span class="default"><a href="tmap_1.html#K4">incl</a> <font color="Maroon">W</font></span>)</span></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="E7:53"/>
<span class="p1">(<span class="default"><a href="waybel18.html#K8">corestr</a> <span class="p2">(<span class="default"><a href="tmap_1.html#K4">incl</a> <font color="Maroon">W</font></span>)</span></span>)</span> <a href="tops_2.html#K2">/"</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>
