<?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="E1:49"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">S<sub>6</sub></font>[<font color="Maroon">Y</font>]
 ;<br/>

<b>let </b><font color="Maroon">X</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">T</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>let </b><font color="Maroon">f</font> be  <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">T</font>;<br/>





<a NAME="E2:49"/><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">T</font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">T</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="E3:49"/>
( the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">X</font> is    <a href="cantor_1.html#M1">Basis</a> of <font color="Maroon">X</font> &amp; the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">Y</font> is    <a href="cantor_1.html#M1">Basis</a> of <font color="Maroon">Y</font> )
 
<b>by </b><i><a class="ref" href="cantor_1.html#T2">CANTOR_1:2</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">B</font> = <span class="p1">{<span class="default"> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Olive">a</font>,<font color="Olive">b</font></span><a href="borsuk_1.html#K6">:]</a></span> where <font color="Olive">a</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>, <font color="Olive">b</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font> : ( <font color="Olive">a</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">X</font> &amp; <font color="Olive">b</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">Y</font> ) </span>}</span>  as    <a href="cantor_1.html#M1">Basis</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><a class="ref" href="yellow_9.html#T40">YELLOW_9:40</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">s</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:49_1"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">s</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">s1</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">s2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:49_1"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">s1</font>,<font color="Maroon">s2</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="E4:49_1"/><i><font color="Green">E53</font></i>: 
( <font color="Maroon">s1</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">s2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s1</font> )
 <b>by </b><i>, , <a class="ref" href="waybel26.html#T39">WAYBEL26:39</a></i>;<br/><a NAME="E5:49_1"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E6:49_1"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s1</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>
 ;<br/><a NAME="E7:49_1"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s1</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="yellow_1.html#T1">YELLOW_1:1</a></i>;<br/><a NAME="E8:49_1"/><b>then </b>
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i>, , <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><b>hence </b><a NAME="E9:49_1"/>
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</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><a class="ref" href="borsuk_1.html#D5">BORSUK_1:def 5</a></i>;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">A</font> =  <a href="waybel26.html#K5">*graph</a> <font color="Maroon">f</font> as   <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><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">p</font> be   <a href="pre_topc.html#NM2">Point</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>;<br/><b>assume </b><a NAME="E1:49_2"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 ;<br/><b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:49_2"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</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="E4:49_2"/><i><font color="Green">E74</font></i>: 
( <font color="Maroon">x</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">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> )
 <b>by </b><i>, , <a class="ref" href="waybel26.html#T39">WAYBEL26:39</a></i>;<br/><b>then reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font> ;<br/><a NAME="E6:49_2"/>
<font color="Maroon">f</font> <a href="yellow_6.html#K3">.</a> <font color="Maroon">x</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/>;<br/><a NAME="E7:49_2"/><b>then </b><i><font color="Green">E75</font></i>: 
<font color="Maroon">f</font> <a href="yellow_6.html#K3">.</a> <font color="Maroon">x</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="yellow_1.html#T1">YELLOW_1:1</a></i>;<br/><b>then reconsider </b><font color="Maroon">y</font> = <font color="Maroon">y</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Y</font> <b>by </b><i/>;<br/><b>reconsider </b><font color="Maroon">W</font> = <font color="Maroon">f</font> <a href="yellow_6.html#K3">.</a> <font color="Maroon">x</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="E10:49_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="tops_1.html#K1">Int</a> <font color="Maroon">W</font>
 <b>by </b><i>, <a class="ref" href="tops_1.html#T55">TOPS_1:55</a></i>;<br/><b>then reconsider </b><font color="Maroon">W</font> = <font color="Maroon">W</font> as  <a href="pre_topc.html#V3">open</a>  <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Maroon">y</font> <b>by </b><i><a class="ref" href="connsp_2.html#D1">CONNSP_2:def 1</a></i>;<br/><b>consider </b><font color="Maroon">H</font> being  <a href="pre_topc.html#V3">open</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E13:49_2"/><i><font color="Green">E76</font></i>: 
( <font color="Maroon">W</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H</font> &amp;  <a href="setfam_1.html#K1">meet</a> <font color="Maroon">H</font> is    <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Maroon">y</font> )
 <b>by </b><i/>;<br/><b>reconsider </b><font color="Maroon">mH</font> =  <a href="setfam_1.html#K1">meet</a> <font color="Maroon">H</font> as    <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Maroon">y</font> <b>by </b><i/>;<br/><a NAME="E15:49_2"/><i><font color="Green">E77</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="tops_1.html#K1">Int</a> <font color="Maroon">mH</font> &amp;  <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">mH</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="tops_1.html#K1">Int</a> <font color="Maroon">mH</font> )
 <b>by </b><i><a class="ref" href="connsp_2.html#D1">CONNSP_2:def 1</a>, <a class="ref" href="tops_1.html#T45">TOPS_1:45</a></i>;<br/><b>then reconsider </b><font color="Maroon">V</font> =  <a href="tops_1.html#K1">Int</a> <font color="Maroon">mH</font> as  <a href="pre_topc.html#V3">open</a>  <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Maroon">y</font> <b>by </b><i><a class="ref" href="connsp_2.html#D1">CONNSP_2:def 1</a></i>;<br/><b>reconsider </b><font color="Maroon">fH</font> = <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">H</font> as  <a href="pre_topc.html#V3">open</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> <b>by </b><i><a class="ref" href="tops_2.html#T55">TOPS_2:55</a></i>;<br/><b>reconsider </b><font color="Maroon">a</font> = <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">fH</font>,<font color="Maroon">V</font></span><a href="borsuk_1.html#K6">:]</a></span> as   <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> ;<br/><b>take </b><font color="Maroon">a</font> = <font color="Maroon">a</font>;<br/><a NAME="E19:49_2"/>
( <font color="Maroon">V</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">fH</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">X</font> )
 <b>by </b><i><a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/><b>hence </b><a NAME="E20:49_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 ;<br/><a NAME="E21:49_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">fH</font>
 <b>by </b><i>, , <a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/><b>hence </b><a NAME="E22:49_2"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a</font>
 <b>by </b><i>, , <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><b>thus </b><a NAME="E23:49_2"/>
<font color="Maroon">a</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">s</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="E1:49_2_1"/><i><font color="Green">E78</font></i>: 
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a</font>
 ;<br/>

<b>then consider </b><font color="Maroon">s1</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">s2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:49_2_1"/><i><font color="Green">E79</font></i>: 
<font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">s1</font>,<font color="Maroon">s2</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="E4:49_2_1"/><i><font color="Green">E80</font></i>: 
( <font color="Maroon">s1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">fH</font> &amp; <font color="Maroon">s2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font> )
 
<b>by </b><i>, , <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E5:49_2_1"/><b>then </b><i><font color="Green">E81</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/>
<a NAME="E6:49_2_1"/>
<font color="Maroon">V</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">mH</font>
 
<b>by </b><i><a class="ref" href="tops_1.html#T44">TOPS_1:44</a></i>;<br/>
<a NAME="E7:49_2_1"/><b>then </b><i><font color="Green">E82</font></i>: 
<font color="Maroon">s2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s1</font>
 
<b>by </b><i>, <a class="ref" href="waybel29.html#T1" target="_self">Th1</a>, <a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/>
<a NAME="E8:49_2_1"/>
<font color="Maroon">s1</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_1.html#D13">FUNCT_1:def 13</a></i>;<br/>
<b>hence </b><a NAME="E9:49_2_1"/>
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>by </b><i>, , <a class="ref" href="waybel26.html#T39">WAYBEL26:39</a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E8:49"/>
 <a href="waybel26.html#K5">*graph</a> <font color="Maroon">f</font> 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><a class="ref" href="yellow_9.html#T31">YELLOW_9:31</a></i>;<br/>


</div>
