<?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="struct_0.html#NM6">Function</a> of <font color="Maroon">X</font>,<font color="Maroon">Y</font>;<br/>



<b>assume </b><a NAME="E1:26"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">f</font> is <a href="pre_topc.html#V5">continuous</a>
 ;<br/>

<a NAME="E2:26"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">f</font> is   <a href="funct_2.html#NM1">Function</a> of  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>, <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T3">FUNCT_2:3</a></i>;<br/>
<a NAME="E3:26"/>
for <font color="Olive">R</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="waybel18.html#K7" target="_self">Image</a> <font color="Maroon">f</font></span>)</span>  st <font color="Olive">R</font> is <a href="pre_topc.html#V3">open</a> holds <br/><span class="p1">(<span class="default"><a href="waybel18.html#K8" target="_self">corestr</a> <font color="Maroon">f</font></span>)</span> <a href="relset_1.html#K11">"</a> <font color="Olive">R</font> is <a href="pre_topc.html#V3">open</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">R</font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="waybel18.html#K7" target="_self">Image</a> <font color="Maroon">f</font></span>)</span>;<br/>

<b>assume </b><a NAME="E1:26_1"/>
<font color="Maroon">R</font> is <a href="pre_topc.html#V3">open</a>
 ;<br/>

<a NAME="E2:26_1"/><b>then </b>
<font color="Maroon">R</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" target="_self">Image</a> <font color="Maroon">f</font></span>)</span>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<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="E4:26_1"/><i><font color="Green">E31</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>
 <b>and </b><br/><a NAME="E5:26_1"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">R</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> <span class="p2">(<span class="default"><a href="waybel18.html#K7" target="_self">Image</a> <font color="Maroon">f</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="pre_topc.html#D9">PRE_TOPC:def 9</a></i>;<br/>
<b>reconsider </b><font color="Maroon">Q</font> = <font color="Maroon">Q</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font> ;<br/>
<a NAME="E7:26_1"/>
<font color="Maroon">Q</font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i>, <a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/>
<a NAME="E8:26_1"/><b>then </b><i><font color="Green">E33</font></i>: 
<font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">Q</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="E9:26_1"/>
 <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><a href="waybel18.html#K7" target="_self">Image</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font>
 
<b>by </b><i/>;<br/>
<b>then </b><i><font color="Green">E35</font></i>: <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2">(<span class="default"><a href="waybel18.html#K7" target="_self">Image</a> <font color="Maroon">f</font></span>)</span></span>)</span> = 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
<b>by </b><i>, <a class="ref" href="funct_2.html#T48">FUNCT_2:48</a></i>
<br/>.= 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>
;<br/>
<a NAME="E11:26_1"/>
the <a href="struct_0.html#U1">carrier</a> of <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">X</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D1">PRE_TOPC:def 1</a></i>;<br/>
<a NAME="E12:26_1"/><b>then </b><i><font color="Green">E36</font></i>: 
<font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2">(<span class="default"><a href="waybel18.html#K7" target="_self">Image</a> <font color="Maroon">f</font></span>)</span></span>)</span> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i>, <a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/>
<a NAME="E13:26_1"/>
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">Q</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p3">(<span class="default"><a href="waybel18.html#K7" target="_self">Image</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p1">(<span class="default"><font color="Maroon">Q</font> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p3">(<span class="default"><a href="waybel18.html#K7" target="_self">Image</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_1.html#T137">FUNCT_1:137</a></i>;<br/>
<a NAME="E14:26_1"/><b>then </b>
<font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p1">(<span class="default"><font color="Maroon">Q</font> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p3">(<span class="default"><a href="waybel18.html#K7" target="_self">Image</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i>, , <a class="ref" href="tops_1.html#T38">TOPS_1:38</a></i>;<br/>
<b>hence </b><a NAME="E15:26_1"/>
<span class="p1">(<span class="default"><a href="waybel18.html#K8" target="_self">corestr</a> <font color="Maroon">f</font></span>)</span> <a href="relset_1.html#K11">"</a> <font color="Maroon">R</font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:26"/>
 <a href="waybel18.html#K8" target="_self">corestr</a> <font color="Maroon">f</font> 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>
