<?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="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> <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>let </b><font color="Maroon" title="c4">X0</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c1">X</font>;<br/>







<a NAME="E1:65"/>
for <font color="Olive" title="b1">x0</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c4">X0</font> holds  <font color="Maroon" title="c3">f</font> <a href="tmap_1.html#K2" title="TMAP_1:func.2">|</a> <font color="Maroon" title="c4">X0</font> <a href="tmap_1.html#R1" title="TMAP_1:pred.1">is_continuous_at</a> <font color="Olive" title="b1">x0</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="65_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">x0</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c4">X0</font>;<br/>

<a NAME="E1:65_1"/>
( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">X0</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font> &amp; <font color="Maroon" title="c5">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">X0</font> )
 
<b>by </b><i><a class="ref" href="borsuk_1.html#T35" title="BORSUK_1:th.35">BORSUK_1:35</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c6">x</font> = <font color="Maroon" title="c5">x0</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font> ;<br/>
<a NAME="E3:65_1"/>
<font color="Maroon" title="c3">f</font> <a href="tmap_1.html#R1" title="TMAP_1:pred.1">is_continuous_at</a> <font color="Maroon" title="c6">x</font>
 
<b>by </b><i><a class="ref" href="tmap_1.html#T49" target="_self" title="TMAP_1:th.49">Th49</a></i>;<br/>
<b>hence </b><a NAME="E4:65_1"/>
<font color="Maroon" title="c3">f</font> <a href="tmap_1.html#K2" title="TMAP_1:func.2">|</a> <font color="Maroon" title="c4">X0</font> <a href="tmap_1.html#R1" title="TMAP_1:pred.1">is_continuous_at</a> <font color="Maroon" title="c5">x0</font>
 <b>by </b><i><a class="ref" href="tmap_1.html#T64" target="_self" title="TMAP_1:th.64">Th64</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:65"/>
<font color="Maroon" title="c3">f</font> <a href="tmap_1.html#K2" title="TMAP_1:func.2">|</a> <font color="Maroon" title="c4">X0</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c4">X0</font>,<font color="Maroon" title="c2">Y</font>
 <b>by </b><i><a class="ref" href="tmap_1.html#T49" target="_self" title="TMAP_1:th.49">Th49</a></i>;<br/>


</div>
