<?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>, <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">Y0</font> be  non <a href="struct_0.html#V3">empty</a>  <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">Y</font>;<br/><b>let </b><font color="Maroon">f</font> be   <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">g</font> be   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">X</font>,<font color="Maroon">Y0</font>;<br/>







<b>assume </b><b>that </b><br/><a NAME="E1:11"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">f</font> <a href="funct_2.html#R1">=</a> <font color="Maroon">g</font>
 <b>and </b><br/><a NAME="E2:11"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">f</font> is <a href="pre_topc.html#V5">continuous</a>
 ;<br/>

<b>let </b><font color="Maroon">W</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="borsuk_1.html#D3">BORSUK_1:def 3</a><br/><b>let </b><font color="Maroon">G</font> be    <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">W</font>;<br/>



<b>consider </b><font color="Maroon">V</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y0</font><b> such that </b><br/><a NAME="E4:11"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">V</font> is <a href="pre_topc.html#V3">open</a>
 <b>and </b><br/><a NAME="E5:11"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">V</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">G</font>
 <b>and </b><br/><a NAME="E6:11"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">W</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font>
 <b>by </b><i><a class="ref" href="connsp_2.html#T8">CONNSP_2:8</a></i>;<br/>
<b>consider </b><font color="Maroon">C</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font><b> such that </b><br/><a NAME="E8:11"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">C</font> is <a href="pre_topc.html#V3">open</a>
 <b>and </b><br/><a NAME="E9:11"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">C</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">Y0</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">V</font>
 <b>by </b><i>, <a class="ref" href="tops_2.html#T32">TOPS_2:32</a></i>;<br/>
<a NAME="E10:11"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">W</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">Y0</font>
 
;<br/>
<a NAME="E11:11"/>
 <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">Y0</font> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">Y</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D9">PRE_TOPC:def 9</a></i>;<br/>
<a NAME="E12:11"/><b>then </b>
<font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">W</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">Y</font>
 
<b>by </b><i/>;<br/>
<b>then reconsider </b><font color="Maroon">p</font> = <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">W</font> as   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">Y</font> ;<br/>
<a NAME="E14:11"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 
<b>by </b><i>, , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E15:11"/><b>then </b>
<font color="Maroon">C</font> is    <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Maroon">p</font>
 
<b>by </b><i>, <a class="ref" href="connsp_2.html#T5">CONNSP_2:5</a></i>;<br/>
<b>then consider </b><font color="Maroon">H</font> being    <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Maroon">W</font><b> such that </b><br/><a NAME="E17:11"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">H</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">C</font>
 <b>by </b><i>, , <a class="ref" href="borsuk_1.html#D3">BORSUK_1:def 3</a></i>;<br/>
<b>take </b>
<font color="Maroon">H</font>
;<br/>

<a NAME="E18:11"/>
<font color="Maroon">g</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">H</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">V</font>
 
<b>by </b><i>, , , <a class="ref" href="xboole_1.html#T19">XBOOLE_1:19</a></i>;<br/>
<b>hence </b><a NAME="E19:11"/>
<font color="Maroon">g</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">H</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">G</font>
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>


</div>
