<?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>;<br/><b>let </b><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">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>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>;<br/><b>let </b><font color="Maroon">g</font> be   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">X0</font>,<font color="Maroon">Y</font>;<br/>









<b>assume </b><a NAME="E1:88"/><i><font color="Green">E23</font></i>: 
 <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X1</font>,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">X1</font> #) <a href="hidden.html#R1">=</a> <font color="Maroon">X0</font>
 ;<br/>

<b>let </b><font color="Maroon">x0</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X0</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>assume </b><a NAME="E2:88"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">x0</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x1</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">Y1</font> =  <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X1</font>,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">X1</font> #) as   <a href="pre_topc.html#NM1">TopSpace</a> <b>by </b><i/>;<br/>
<a NAME="E4:88"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">Y1</font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X0</font>
 
<b>by </b><i><a class="txt" href="tmap_1.html#E12"><i><font color="Green">Lemma69</font></i></a>, <a class="ref" href="tsep_1.html#T2">TSEP_1:2</a></i>;<br/>
<a NAME="E5:88"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X1</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X0</font>
 
<b>by </b><i><a class="txt" href="tmap_1.html#E12"><i><font color="Green">Lemma69</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon">A</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X1</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X0</font> ;<br/>
<a NAME="E7:88"/>
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">X0</font>
 
<b>by </b><i><a class="txt" href="tmap_1.html#E12"><i><font color="Green">Lemma69</font></i></a></i>;<br/>
<a NAME="E8:88"/><b>then </b>
<font color="Maroon">A</font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="ref" href="tops_1.html#T53">TOPS_1:53</a></i>;<br/>
<a NAME="E9:88"/><b>then </b><i><font color="Green">E36</font></i>: 
<font color="Maroon">X1</font> is  <a href="tsep_1.html#V1">open</a>  <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X0</font>
 
<b>by </b><i>, , <a class="ref" href="tsep_1.html#T16">TSEP_1:16</a></i>;<br/>
<b>assume </b><a NAME="E10:88"/>
<font color="Maroon">g</font> <a href="tmap_1.html#K3" 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>
 ;<br/>

<b>hence </b><a NAME="E11:88"/>
<font color="Maroon">g</font> <a href="tmap_1.html#R1" target="_self">is_continuous_at</a> <font color="Maroon">x0</font>
 <b>by </b><i>, <a class="ref" href="tmap_1.html#T11" target="_self">Th11</a>, </i>;<br/>


</div>
