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<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">X0</font> be  non <a href="struct_0.html#V3">empty</a>  <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X</font>, <font color="Maroon">X1</font> be  non <a href="struct_0.html#V3">empty</a>  <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X</font>;<br/>



<b>assume </b><a NAME="E1:125"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">X0</font> <a href="tsep_1.html#R1">misses</a> <font color="Maroon">X1</font>
 ;<br/>

<b>let </b><font color="Maroon">x1</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X1</font>;<br/>

<b>reconsider </b><font color="Maroon">A</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X0</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> <b>by </b><i><a class="ref" href="tsep_1.html#T1">TSEP_1:1</a></i>;<br/>
<a NAME="E3:125"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X1</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">A</font>
 
<b>by </b><i>, <a class="ref" href="tsep_1.html#D3">TSEP_1:def 3</a></i>;<br/>
<a NAME="E4:125"/><b>then </b><i><font color="Green">E29</font></i>: 
<span class="p1">(<span class="default"><a href="tmap_1.html#K7" target="_self">modid</a> <font color="Maroon">X</font>,<font color="Maroon">A</font></span>)</span> <a href="tmap_1.html#K2" target="_self">|</a> <font color="Maroon">X1</font> <a href="tmap_1.html#R1" target="_self">is_continuous_at</a> <font color="Maroon">x1</font>
 
<b>by </b><i/>;<br/>
<a NAME="E5:125"/>
<font color="Maroon">X</font> <a href="tmap_1.html#K6" target="_self">modified_with_respect_to</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font> <a href="tmap_1.html#K8" target="_self">modified_with_respect_to</a> <font color="Maroon">X0</font>
 
<b>by </b><i/>;<br/>
<b>then reconsider </b><font color="Maroon">f</font> = <span class="p1">(<span class="default"><a href="tmap_1.html#K7" target="_self">modid</a> <font color="Maroon">X</font>,<font color="Maroon">A</font></span>)</span> <a href="tmap_1.html#K2" target="_self">|</a> <font color="Maroon">X1</font> as   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">X1</font>,<span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="tmap_1.html#K8" target="_self">modified_with_respect_to</a> <font color="Maroon">X0</font></span>)</span> ;<br/>
<a NAME="E7:125"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">f</font> <a href="tmap_1.html#R1" target="_self">is_continuous_at</a> <font color="Maroon">x1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="125_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="125_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">W</font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="tmap_1.html#K8" target="_self">modified_with_respect_to</a> <font color="Maroon">X0</font></span>)</span>;<br/><b>assume </b><a NAME="E1:125_1_1"/><i><font color="Green">E36</font></i>: 
( <font color="Maroon">W</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font> )
 ;<br/><a NAME="E2:125_1_1"/><b>then </b>
<font color="Maroon">W</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="tmap_1.html#K8" target="_self">modified_with_respect_to</a> <font color="Maroon">X0</font></span>)</span>
 <b>by </b><i><a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/><a NAME="E3:125_1_1"/><b>then </b><i><font color="Green">E38</font></i>: 
<font color="Maroon">W</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="tmap_1.html#K5" target="_self">-extension_of_the_topology_of</a> <font color="Maroon">X</font>
 <b>by </b><i/>;<br/><a NAME="E4:125_1_1"/><b>then </b><i><font color="Green">E39</font></i>: 
<font color="Maroon">W</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="tmap_1.html#K6" target="_self">modified_with_respect_to</a> <font color="Maroon">A</font></span>)</span>
 ;<br/><b>reconsider </b><font color="Maroon">W0</font> = <font color="Maroon">W</font> as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="tmap_1.html#K6" target="_self">modified_with_respect_to</a> <font color="Maroon">A</font></span>)</span> <b>by </b><i/>;<br/><a NAME="E6:125_1_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">W0</font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i><a class="ref" href="tmap_1.html#T20" target="_self">Th20</a>, <a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/><b>thus </b><a NAME="E7:125_1_1"/>
 ex <font color="Olive">V</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X1</font> st <br/>( <font color="Olive">V</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Olive">V</font> &amp; <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Olive">V</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">W</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="125_1_1_1"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon">V</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X1</font><b> such that </b><br/><a NAME="E2:125_1_1_1"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">V</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="tmap_1.html#K7" target="_self">modid</a> <font color="Maroon">X</font>,<font color="Maroon">A</font></span>)</span> <a href="tmap_1.html#K2" target="_self">|</a> <font color="Maroon">X1</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">V</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">W0</font> )
 <b>by </b><i>, , <a class="ref" href="tmap_1.html#T21" target="_self">Th21</a>, <a class="txt" href="tmap_1.html#E3:125_1_1"><i><font color="Green">E38</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon">V</font>
;<br/>

<b>thus </b><a NAME="E3:125_1_1_1"/>
( <font color="Maroon">V</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font> &amp; <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">V</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">W</font> )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E2:125_1"/>
<font color="Maroon">f</font> <a href="tmap_1.html#R1" target="_self">is_continuous_at</a> <font color="Maroon">x1</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E8:125"/>
<span class="p1">(<span class="default"><a href="tmap_1.html#K7" target="_self">modid</a> <font color="Maroon">X</font>,<font color="Maroon">A</font></span>)</span> <a href="tmap_1.html#K2" target="_self">|</a> <font color="Maroon">X1</font> <a href="funct_2.html#R1">=</a> <span class="p1">(<span class="default"><a href="tmap_1.html#K9" target="_self">modid</a> <font color="Maroon">X</font>,<font color="Maroon">X0</font></span>)</span> <a href="tmap_1.html#K2" target="_self">|</a> <font color="Maroon">X1</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E9:125"/>
<span class="p1">(<span class="default"><a href="tmap_1.html#K9" target="_self">modid</a> <font color="Maroon">X</font>,<font color="Maroon">X0</font></span>)</span> <a href="tmap_1.html#K2" target="_self">|</a> <font color="Maroon">X1</font> <a href="tmap_1.html#R1" target="_self">is_continuous_at</a> <font color="Maroon">x1</font>
 <b>by </b><i/>;<br/>


</div>
