<?xml version="1.0"?>
<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>, <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">f</font> be  <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">X</font></span>)</span>,<span class="p1">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">Y</font></span>)</span>;<br/><b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">X</font></span>)</span>, <font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">X</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_1.html#D2">WAYBEL_1:def 2</a><br/>





<b>assume </b><a NAME="E1:40"/>
<font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">y</font>
 ;<br/>

<b>then consider </b><font color="Maroon">A</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E3:40"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">y</font></span><a href="domain_1.html#K6">}</a></span>
 <b>and </b><br/><a NAME="E4:40"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A</font>
 <b>by </b><i/>;<br/>
<a NAME="E5:40"/><i><font color="Green">E44</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">Y</font></span>)</span>
 
<b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">Z</font> = <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="yellow_6.html#K3">.</a> <font color="Maroon">y</font></span>)</span></span><a href="domain_1.html#K6">}</a></span> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font> <b>by </b><i/>;<br/>
<a NAME="E7:40"/>
for <font color="Olive">G</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font>  st <font color="Olive">G</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">f</font> <a href="yellow_6.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">G</font> holds <br/><font color="Maroon">Z</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Olive">G</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1"><b>proof </b></a><div class="add">

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

<b>assume </b><b>that </b><br/><a NAME="E1:40_1"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">G</font> is <a href="pre_topc.html#V3">open</a>
 <b>and </b><br/><a NAME="E2:40_1"/><i><font color="Green">E54</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> <font color="Maroon">G</font>
 ;<br/>

<a NAME="E3:40_1"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">X</font></span>)</span>
 
<b>by </b><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>,<font color="Maroon">Y</font> <b>by </b><i/>;<br/>
<a NAME="E5:40_1"/>
(  <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 <span class="p1">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">X</font></span>)</span>,the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">X</font></span>)</span> #) &amp;  <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font>,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">Y</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="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">Y</font></span>)</span>,the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">Y</font></span>)</span> #) )
 
<b>by </b><i/>;<br/>
<a NAME="E6:40_1"/><b>then </b>
<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/>
<a NAME="E7:40_1"/><b>then </b><i><font color="Green">E55</font></i>: 
<font color="Maroon">g</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">G</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="E8:40_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">g</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">G</font>
 
<b>by </b><i>, <a class="ref" href="funct_2.html#T46">FUNCT_2:46</a></i>;<br/>
<a NAME="E9:40_1"/><b>then </b>
<font color="Maroon">A</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">g</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">G</font>
 
<b>by </b><i>, , <a class="ref" href="pre_topc.html#D13">PRE_TOPC:def 13</a></i>;<br/>
<b>then consider </b><font color="Maroon">m</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E11:40_1"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">G</font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#T4">XBOOLE_0:4</a></i>;<br/>
<a NAME="E12:40_1"/><i><font color="Green">E65</font></i>: 
( <font color="Maroon">m</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">m</font> <a href="hidden.html#R2">in</a> <font color="Maroon">g</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">G</font> )
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E13:40_1"/><b>then </b>
<font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 
<b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E14:40_1"/><b>then </b>
( <font color="Maroon">f</font> <a href="yellow_6.html#K3">.</a> <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> &amp; <font color="Maroon">f</font> <a href="yellow_6.html#K3">.</a> <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> )
 
<b>by </b><i>, <a class="ref" href="funct_2.html#T46">FUNCT_2:46</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E15:40_1"/><b>then </b>
<font color="Maroon">Z</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">G</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pre_topc.html#K1">{}</a> <font color="Maroon">Y</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>hence </b><a NAME="E16:40_1"/>
<font color="Maroon">Z</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">G</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:40"/><b>then </b>
<font color="Maroon">f</font> <a href="yellow_6.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">Z</font>
 
<b>by </b><i>, <a class="ref" href="pre_topc.html#D13">PRE_TOPC:def 13</a></i>;<br/>
<b>hence </b><a NAME="E9:40"/>
<font color="Maroon">f</font> <a href="yellow_6.html#K3">.</a> <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">f</font> <a href="yellow_6.html#K3">.</a> <font color="Maroon">y</font>
 <b>by </b><i/>;<br/>


</div>
