<?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/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:53_1"/>
 <a href="yellow_1.html#K2">InclPoset</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">Y</font> is <a href="waybel_3.html#V3">continuous</a>
 ;<br/><a NAME="E2:53_1"/><b>then </b>
<font color="Maroon">S<sub>6</sub></font>[<font color="Maroon">Y</font>]
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:53_1"/>
for <font color="Olive">X</font> being non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a><br/> for <font color="Olive">f</font> being <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM6">Function</a> of <font color="Olive">X</font>,<span class="p1">(<span class="default"><a href="waybel29.html#K1" target="_self">Sigma</a> <span class="p2">(<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></span>)</span> holds   <a href="waybel26.html#K5">*graph</a> <font color="Olive">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="Olive">X</font>,<font color="Maroon">Y</font></span><a href="borsuk_1.html#K5">:]</a></span>
 <b>by </b><i/>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:53"/><i><font color="Green">E46</font></i>: 
for <font color="Olive">X</font> being non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a><br/> for <font color="Olive">f</font> being <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM6">Function</a> of <font color="Olive">X</font>,<span class="p1">(<span class="default"><a href="waybel29.html#K1" target="_self">Sigma</a> <span class="p2">(<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></span>)</span> holds   <a href="waybel26.html#K5">*graph</a> <font color="Olive">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="Olive">X</font>,<font color="Maroon">Y</font></span><a href="borsuk_1.html#K5">:]</a></span>
 ;<br/>

<a NAME="E3:53"/>
<font color="Maroon">S<sub>4</sub></font>[<font color="Maroon">Y</font>]
 
<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">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="E1:53_2"/><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="yellow_1.html#K2">InclPoset</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">Y</font> &amp;  <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="waybel29.html#K1" target="_self">Sigma</a> <span class="p2">(<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></span>)</span>,the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="waybel29.html#K1" target="_self">Sigma</a> <span class="p2">(<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></span>)</span> #) <a href="hidden.html#R1">=</a>  <a href="yellow_1.html#K2">InclPoset</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">Y</font> &amp; the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font> <a href="hidden.html#R1">=</a> the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><a href="waybel29.html#K1" target="_self">Sigma</a> <span class="p2">(<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></span>)</span> )
 
<b>by </b><i>, <a class="ref" href="yellow_9.html#D4">YELLOW_9:def 4</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">g</font> = <font color="Maroon">f</font> as   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">X</font>,<span class="p1">(<span class="default"><a href="waybel29.html#K1" target="_self">Sigma</a> <span class="p2">(<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></span>)</span> ;<br/>
<a NAME="E3:53_2"/>
(  <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">X</font> #) <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">X</font> #) &amp;  <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font> #) <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel29.html#K1" target="_self">Sigma</a> <span class="p2">(<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></span>)</span>,the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><a href="waybel29.html#K1" target="_self">Sigma</a> <span class="p2">(<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></span>)</span> #) )
 
<b>by </b><i/>;<br/>
<a NAME="E4:53_2"/><b>then </b>
( <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">X</font>,<font color="Maroon">T</font></span><a href="borsuk_1.html#K5">:]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">X</font>,<span class="p2">(<span class="default"><a href="waybel29.html#K1" target="_self">Sigma</a> <span class="p3">(<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></span>)</span></span><a href="borsuk_1.html#K5">:]</a></span> &amp; <font color="Maroon">g</font> is <a href="pre_topc.html#V5">continuous</a> )
 
<b>by </b><i>, <a class="ref" href="yellow12.html#T36">YELLOW12:36</a></i>;<br/>
<b>hence </b><a NAME="E5:53_2"/>
 <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/>;<br/>


</div><b>end;</b></div>
<a NAME="E4:53"/><b>then </b>
<font color="Maroon">S<sub>5</sub></font>[<font color="Maroon">Y</font>]
 
<b>by </b><i/>;<br/>
<a NAME="E5:53"/><b>then </b>
<font color="Maroon">S<sub>6</sub></font>[<font color="Maroon">Y</font>]
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E6:53"/>
 <a href="yellow_1.html#K2">InclPoset</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">Y</font> is <a href="waybel_3.html#V3">continuous</a>
 <b>by </b><i/>;<br/>


</div>
