<?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">f</font> be  <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">X</font>,<font color="Maroon">Y</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/>







<a NAME="E1:69"/>
for <font color="Olive">x0</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X0</font> holds  <font color="Maroon">f</font> <a href="tmap_1.html#K2" target="_self">|</a> <font color="Maroon">X0</font> <a href="tmap_1.html#R1" target="_self">is_continuous_at</a> <font color="Olive">x0</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x0</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X0</font>;<br/>

<a NAME="E1:69_1"/>
( the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X0</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> &amp; <font color="Maroon">x0</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X0</font> )
 
<b>by </b><i><a class="ref" href="borsuk_1.html#T35">BORSUK_1:35</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x0</font> as   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font> ;<br/>
<a NAME="E3:69_1"/>
<font color="Maroon">f</font> <a href="tmap_1.html#R1" target="_self">is_continuous_at</a> <font color="Maroon">x</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E4:69_1"/>
<font color="Maroon">f</font> <a href="tmap_1.html#K2" target="_self">|</a> <font color="Maroon">X0</font> <a href="tmap_1.html#R1" target="_self">is_continuous_at</a> <font color="Maroon">x0</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:69"/>
<font color="Maroon">f</font> <a href="tmap_1.html#K2" target="_self">|</a> <font color="Maroon">X0</font> is  <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">X0</font>,<font color="Maroon">Y</font>
 <b>by </b><i/>;<br/>


</div>
