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

<b>let </b><font color="Maroon" title="c1">X</font>, <font color="Maroon" title="c2">Y</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>;<br/><b>let </b><font color="Maroon" title="c3">f</font> be   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font>;<br/>



<b>assume </b><a NAME="E1:26"/><i><font color="Green" title="E14">A1</font></i>: 
<font color="Maroon" title="c3">f</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 ;<br/>

<a NAME="E2:26"/><i><font color="Green" title="E15">A2</font></i>: 
<font color="Maroon" title="c3">f</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">f</font>, <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c3">f</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T3" title="FUNCT_2:th.3">FUNCT_2:3</a></i>;<br/>
<a NAME="E3:26"/><i><font color="Green" title="E16">A3</font></i>: 
(  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c2">Y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp;  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="waybel18.html#K7" title="WAYBEL18:func.7">Image</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 
;<br/>
<a NAME="E4:26"/>
for <font color="Olive" title="b1">R</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="waybel18.html#K7" title="WAYBEL18:func.7">Image</a> <font color="Maroon" title="c3">f</font></span>)</span>  st <font color="Olive" title="b1">R</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> holds <br/><span class="p1">(<span class="default"><a href="waybel18.html#K8" title="WAYBEL18:func.8">corestr</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Olive" title="b1">R</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">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" title="c4">R</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="waybel18.html#K7" title="WAYBEL18:func.7">Image</a> <font color="Maroon" title="c3">f</font></span>)</span>;<br/>

<b>assume </b><a NAME="E1:26_1"/>
<font color="Maroon" title="c4">R</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 ;<br/>

<a NAME="E2:26_1"/><b>then </b>
<font color="Maroon" title="c4">R</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <span class="p1">(<span class="default"><a href="waybel18.html#K7" title="WAYBEL18:func.7">Image</a> <font color="Maroon" title="c3">f</font></span>)</span>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c5">Q</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">Y</font><b> such that </b><br/><a NAME="E4:26_1"/><i><font color="Green" title="E17">A4</font></i>: 
<font color="Maroon" title="c5">Q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c2">Y</font>
 <b>and </b><br/><a NAME="E5:26_1"/><i><font color="Green" title="E18">A5</font></i>: 
<font color="Maroon" title="c4">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">Q</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p2">(<span class="default"><a href="waybel18.html#K7" title="WAYBEL18:func.7">Image</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="pre_topc.html#D9" title="PRE_TOPC:def.9">PRE_TOPC:def 9</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c6">Q</font> = <font color="Maroon" title="c5">Q</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">Y</font> ;<br/>
<a NAME="E7:26_1"/>
<font color="Maroon" title="c6">Q</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 
<b>by </b><i><a class="txt" href="waybel18.html#E4:26_1"><i><font color="Green" title="E17">A4</font></i></a>, <a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></i>;<br/>
<a NAME="E8:26_1"/><b>then </b><i><font color="Green" title="E19">A6</font></i>: 
<font color="Maroon" title="c3">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c6">Q</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 
<b>by </b><i><a class="txt" href="waybel18.html#E1:26"><i><font color="Green" title="E14">A1</font></i></a>, <a class="txt" href="waybel18.html#E3:26"><i><font color="Green" title="E16">A3</font></i></a>, <a class="ref" href="tops_2.html#T55" title="TOPS_2:th.55">TOPS_2:55</a></i>;<br/>
<a NAME="E9:26_1"/>
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="waybel18.html#K7" title="WAYBEL18:func.7">Image</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c3">f</font>
 
<b>by </b><i><a class="ref" href="waybel18.html#T10" target="_self" title="WAYBEL18:th.10">Th10</a></i>;<br/>
<b>then </b><i><font color="Green" title="E20">A7</font></i>: <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p2">(<span class="default"><a href="waybel18.html#K7" title="WAYBEL18:func.7">Image</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span> = 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">f</font>
<b>by </b><i><a class="txt" href="waybel18.html#E2:26"><i><font color="Green" title="E15">A2</font></i></a>, <a class="ref" href="funct_2.html#T48" title="FUNCT_2:th.48">FUNCT_2:48</a></i>
<br/>.= 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>
;<br/>
<a NAME="E11:26_1"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">X</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D1" title="PRE_TOPC:def.1">PRE_TOPC:def 1</a></i>;<br/>
<a NAME="E12:26_1"/><b>then </b><i><font color="Green" title="E21">A8</font></i>: 
<font color="Maroon" title="c3">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p2">(<span class="default"><a href="waybel18.html#K7" title="WAYBEL18:func.7">Image</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 
<b>by </b><i><a class="txt" href="waybel18.html#E10:26_1"><i><font color="Green" title="E20">A7</font></i></a>, <a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></i>;<br/>
<a NAME="E13:26_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c6">Q</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p2">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p3">(<span class="default"><a href="waybel18.html#K7" title="WAYBEL18:func.7">Image</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">Q</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p3">(<span class="default"><a href="waybel18.html#K7" title="WAYBEL18:func.7">Image</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_1.html#T137" title="FUNCT_1:th.137">FUNCT_1:137</a></i>;<br/>
<b>hence </b><a NAME="E14:26_1"/>
<span class="p1">(<span class="default"><a href="waybel18.html#K8" title="WAYBEL18:func.8">corestr</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c4">R</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>by </b><i><a class="txt" href="waybel18.html#E5:26_1"><i><font color="Green" title="E18">A5</font></i></a>, <a class="txt" href="waybel18.html#E8:26_1"><i><font color="Green" title="E19">A6</font></i></a>, <a class="txt" href="waybel18.html#E12:26_1"><i><font color="Green" title="E21">A8</font></i></a>, <a class="ref" href="tops_1.html#T38" title="TOPS_1:th.38">TOPS_1:38</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:26"/>
 <a href="waybel18.html#K8" title="WAYBEL18:func.8">corestr</a> <font color="Maroon" title="c3">f</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 <b>by </b><i><a class="txt" href="waybel18.html#E3:26"><i><font color="Green" title="E16">A3</font></i></a>, <a class="ref" href="tops_2.html#T55" title="TOPS_2:th.55">TOPS_2:55</a></i>;<br/>


</div>
