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

<b>let </b><font color="Maroon">Y</font> be   <a href="t_0topsp.html#NM1">T_0-TopSpace</a>;<br/>

<b>assume </b><a NAME="E1:47"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">S<sub>4</sub></font>[<font color="Maroon">Y</font>]
 ;<br/>

<b>let </b><font color="Maroon">X</font> be  <a href="waybel11.html#V4">Scott</a>  <a href="yellow_9.html#M1">TopAugmentation</a> of  <a href="yellow_1.html#K2">InclPoset</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">Y</font>;<br/>

<b>reconsider </b><font color="Maroon">f</font> =  <a href="grcat_1.html#K7">id</a> <font color="Maroon">X</font> 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">X</font> ;<br/>
<a NAME="E3:47"/><i><font color="Green">E47</font></i>: 
 <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">X</font> #) <a href="hidden.html#R1">=</a>  <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">Y</font></span>)</span>,the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">Y</font></span>)</span> #)
 
<b>by </b><i><a class="ref" href="yellow_9.html#D4">YELLOW_9:def 4</a></i>;<br/>
<a NAME="E4:47"/>
 <a href="waybel26.html#K5">*graph</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">W</font>,<font color="Olive">y</font></span><a href="tarski.html#K4">]</a></span> where <font color="Olive">W</font> is  <a href="pre_topc.html#V3">open</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font>, <font color="Olive">y</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Y</font> : <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Olive">W</font> </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:47_1"/>
 <a href="waybel26.html#K5">*graph</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">W</font>,<font color="Olive">y</font></span><a href="tarski.html#K4">]</a></span> where <font color="Olive">W</font> is  <a href="pre_topc.html#V3">open</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font>, <font color="Olive">y</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Y</font> : <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Olive">W</font> </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="47_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">e</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="E1:47_1_1"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a>  <a href="waybel26.html#K5">*graph</a> <font color="Maroon">f</font>
 ;<br/>

<b>then consider </b><font color="Maroon">a</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">b</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:47_1_1"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="ref" href="relat_1.html#D1">RELAT_1:def 1</a></i>;<br/>
<a NAME="E4:47_1_1"/><i><font color="Green">E53</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> )
 
<b>by </b><i>, , <a class="ref" href="waybel26.html#T39">WAYBEL26:39</a></i>;<br/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> = 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">Y</font></span>)</span>
<b>by </b><i>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>
<br/>.= 
the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">Y</font>
<b>by </b><i><a class="ref" href="yellow_1.html#T1">YELLOW_1:1</a></i>
;<br/>
<b>then reconsider </b><font color="Maroon">a</font> = <font color="Maroon">a</font> as  <a href="pre_topc.html#V3">open</a> <a href="struct_0.html#NM2">Subset</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="E7:47_1_1"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#T35">FUNCT_1:35</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">b</font> = <font color="Maroon">b</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Y</font> <b>by </b><i/>;<br/>
<a NAME="E9:47_1_1"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a</font>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#T35">FUNCT_1:35</a></i>;<br/>
<b>hence </b><a NAME="E10:47_1_1"/>
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">W</font>,<font color="Olive">y</font></span><a href="tarski.html#K4">]</a></span> where <font color="Olive">W</font> is  <a href="pre_topc.html#V3">open</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font>, <font color="Olive">y</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Y</font> : <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Olive">W</font> </span>}</span> 
 <b>by </b><i/>;<br/>


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

<b>let </b><font color="Maroon">e</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="E2:47_1"/>
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">W</font>,<font color="Olive">y</font></span><a href="tarski.html#K4">]</a></span> where <font color="Olive">W</font> is  <a href="pre_topc.html#V3">open</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font>, <font color="Olive">y</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Y</font> : <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Olive">W</font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">W</font> being  <a href="pre_topc.html#V3">open</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font>, <font color="Maroon">y</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Y</font><b> such that </b><br/><a NAME="E4:47_1"/><i><font color="Green">E54</font></i>: 
( <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">W</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font> )
 ;<br/>
<a NAME="E5:47_1"/>
<font color="Maroon">W</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="E6:47_1"/><b>then </b><i><font color="Green">E72</font></i>: 
<font color="Maroon">W</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">Y</font></span>)</span>
 
<b>by </b><i><a class="ref" href="yellow_1.html#T1">YELLOW_1:1</a></i>;<br/>
<a NAME="E7:47_1"/><b>then </b><i><font color="Green">E74</font></i>: 
<font color="Maroon">W</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 
<b>by </b><i>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E8:47_1"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">W</font> <a href="hidden.html#R1">=</a> <font color="Maroon">W</font>
 
<b>by </b><i>, , <a class="ref" href="funct_1.html#T35">FUNCT_1:35</a></i>;<br/>
<b>hence </b><a NAME="E9:47_1"/>
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a>  <a href="waybel26.html#K5">*graph</a> <font color="Maroon">f</font>
 <b>by </b><i>, , <a class="ref" href="waybel26.html#T39">WAYBEL26:39</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:47"/>
<span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">W</font>,<font color="Olive">y</font></span><a href="tarski.html#K4">]</a></span> where <font color="Olive">W</font> is  <a href="pre_topc.html#V3">open</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font>, <font color="Olive">y</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Y</font> : <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Olive">W</font> </span>}</span>  is  <a href="pre_topc.html#V3">open</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">X</font>,<font color="Maroon">Y</font></span><a href="borsuk_1.html#K5">:]</a></span>
 <b>by </b><i/>;<br/>


</div>
