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

<b>let </b><font color="Maroon" title="c1">X</font> be  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>;<br/><b>let </b><font color="Maroon" title="c2">X0</font> be  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a>  <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c1">X</font>;<br/><b>let </b><font color="Maroon" title="c3">x0</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">X0</font>;<br/>





<b>reconsider </b><font color="Maroon" title="c4">A</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">X0</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> <b>by </b><i><a class="ref" href="tsep_1.html#T1" title="TSEP_1:th.1">TSEP_1:1</a></i>;<br/>
<a NAME="E2:122"/><i><font color="Green" title="E93">A1</font></i>: 
<span class="p1">(<span class="default"><a href="tmap_1.html#K7" title="TMAP_1:func.7">modid</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c4">A</font></span>)</span> <a href="tmap_1.html#K2" title="TMAP_1:func.2">|</a> <font color="Maroon" title="c2">X0</font> <a href="tmap_1.html#R1" title="TMAP_1:pred.1">is_continuous_at</a> <font color="Maroon" title="c3">x0</font>
 
<b>by </b><i><a class="ref" href="tmap_1.html#T110" target="_self" title="TMAP_1:th.110">Th110</a></i>;<br/>
<a NAME="E3:122"/>
<font color="Maroon" title="c1">X</font> <a href="tmap_1.html#K6" title="TMAP_1:func.6">modified_with_respect_to</a> <font color="Maroon" title="c4">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font> <a href="tmap_1.html#K8" title="TMAP_1:func.8">modified_with_respect_to</a> <font color="Maroon" title="c2">X0</font>
 
<b>by </b><i><a class="ref" href="tmap_1.html#D10" target="_self" title="TMAP_1:def.10">Def10</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c5">f</font> = <span class="p1">(<span class="default"><a href="tmap_1.html#K7" title="TMAP_1:func.7">modid</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c4">A</font></span>)</span> <a href="tmap_1.html#K2" title="TMAP_1:func.2">|</a> <font color="Maroon" title="c2">X0</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c2">X0</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="tmap_1.html#K8" title="TMAP_1:func.8">modified_with_respect_to</a> <font color="Maroon" title="c2">X0</font></span>)</span> ;<br/>
<a NAME="E5:122"/><i><font color="Green" title="E94">A2</font></i>: 
<font color="Maroon" title="c5">f</font> <a href="tmap_1.html#R1" title="TMAP_1:pred.1">is_continuous_at</a> <font color="Maroon" title="c3">x0</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="122_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="122_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">W</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="tmap_1.html#K8" title="TMAP_1:func.8">modified_with_respect_to</a> <font color="Maroon" title="c2">X0</font></span>)</span>;<br/><b>assume </b><a NAME="E1:122_1_1"/><i><font color="Green" title="E95">A3</font></i>: 
( <font color="Maroon" title="c6">W</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">W</font> )
 ;<br/><a NAME="E2:122_1_1"/><b>then </b>
<font color="Maroon" title="c6">W</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"><font color="Maroon" title="c1">X</font> <a href="tmap_1.html#K8" title="TMAP_1:func.8">modified_with_respect_to</a> <font color="Maroon" title="c2">X0</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/><a NAME="E3:122_1_1"/><b>then </b><i><font color="Green" title="E96">A4</font></i>: 
<font color="Maroon" title="c6">W</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font> <a href="tmap_1.html#K5" title="TMAP_1:func.5">-extension_of_the_topology_of</a> <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="ref" href="tmap_1.html#T114" target="_self" title="TMAP_1:th.114">Th114</a></i>;<br/><a NAME="E4:122_1_1"/><b>then </b><i><font color="Green" title="E97">A5</font></i>: 
<font color="Maroon" title="c6">W</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"><font color="Maroon" title="c1">X</font> <a href="tmap_1.html#K6" title="TMAP_1:func.6">modified_with_respect_to</a> <font color="Maroon" title="c4">A</font></span>)</span>
 ;<br/><b>reconsider </b><font color="Maroon" title="c7">W0</font> = <font color="Maroon" title="c6">W</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="tmap_1.html#K6" title="TMAP_1:func.6">modified_with_respect_to</a> <font color="Maroon" title="c4">A</font></span>)</span> <b>by </b><i><a class="txt" href="tmap_1.html#E3:122_1_1"><i><font color="Green" title="E96">A4</font></i></a></i>;<br/><a NAME="E6:122_1_1"/><i><font color="Green" title="E98">A6</font></i>: 
<font color="Maroon" title="c7">W0</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>by </b><i><a class="txt" href="tmap_1.html#E4:122_1_1"><i><font color="Green" title="E97">A5</font></i></a>, <a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></i>;<br/><b>thus </b><a NAME="E7:122_1_1"/>
 ex <font color="Olive" title="b1">V</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">X0</font> st <br/>( <font color="Olive" title="b1">V</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c3">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">V</font> &amp; <font color="Maroon" title="c5">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Olive" title="b1">V</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">W</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="122_1_1_1"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon" title="c8">V</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">X0</font><b> such that </b><br/><a NAME="E2:122_1_1_1"/><i><font color="Green" title="E99">A7</font></i>: 
( <font color="Maroon" title="c8">V</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c3">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">V</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="tmap_1.html#K7" title="TMAP_1:func.7">modid</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c4">A</font></span>)</span> <a href="tmap_1.html#K2" title="TMAP_1:func.2">|</a> <font color="Maroon" title="c2">X0</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c8">V</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">W0</font> )
 <b>by </b><i><a class="txt" href="tmap_1.html#E2:122"><i><font color="Green" title="E93">A1</font></i></a>, <a class="txt" href="tmap_1.html#E1:122_1_1"><i><font color="Green" title="E95">A3</font></i></a>, <a class="txt" href="tmap_1.html#E6:122_1_1"><i><font color="Green" title="E98">A6</font></i></a>, <a class="ref" href="tmap_1.html#T48" target="_self" title="TMAP_1:th.48">Th48</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c8">V</font>
;<br/>

<b>thus </b><a NAME="E3:122_1_1_1"/>
( <font color="Maroon" title="c8">V</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c3">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">V</font> &amp; <font color="Maroon" title="c5">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c8">V</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">W</font> )
 <b>by </b><i><a class="txt" href="tmap_1.html#E2:122_1_1_1"><i><font color="Green" title="E99">A7</font></i></a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E2:122_1"/>
<font color="Maroon" title="c5">f</font> <a href="tmap_1.html#R1" title="TMAP_1:pred.1">is_continuous_at</a> <font color="Maroon" title="c3">x0</font>
 <b>by </b><i><a class="ref" href="tmap_1.html#T48" target="_self" title="TMAP_1:th.48">Th48</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:122"/>
<span class="p1">(<span class="default"><a href="tmap_1.html#K7" title="TMAP_1:func.7">modid</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c4">A</font></span>)</span> <a href="tmap_1.html#K2" title="TMAP_1:func.2">|</a> <font color="Maroon" title="c2">X0</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <span class="p1">(<span class="default"><a href="tmap_1.html#K9" title="TMAP_1:func.9">modid</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">X0</font></span>)</span> <a href="tmap_1.html#K2" title="TMAP_1:func.2">|</a> <font color="Maroon" title="c2">X0</font>
 
<b>by </b><i><a class="ref" href="tmap_1.html#D11" target="_self" title="TMAP_1:def.11">Def11</a></i>;<br/>
<b>hence </b><a NAME="E7:122"/>
<span class="p1">(<span class="default"><a href="tmap_1.html#K9" title="TMAP_1:func.9">modid</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">X0</font></span>)</span> <a href="tmap_1.html#K2" title="TMAP_1:func.2">|</a> <font color="Maroon" title="c2">X0</font> <a href="tmap_1.html#R1" title="TMAP_1:pred.1">is_continuous_at</a> <font color="Maroon" title="c3">x0</font>
 <b>by </b><i><a class="txt" href="tmap_1.html#E5:122"><i><font color="Green" title="E94">A2</font></i></a></i>;<br/>


</div>
