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<b>let </b><font color="Maroon">T</font> be  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#V2">TopSpace-like</a>  <a href="topgrp_1.html#L1" target="_self">TopGrStr</a> ;<br/>

<b>assume </b><a NAME="E1:77"/><i><font color="Green">E30</font></i>: 
for <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font><br/> for <font color="Olive">W</font> being   <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Olive">a</font> <a href="group_1.html#K1">*</a> <font color="Olive">b</font>  ex <font color="Olive">A</font> being   <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Olive">a</font> ex <font color="Olive">B</font> being   <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Olive">b</font> st <font color="Olive">A</font> <a href="group_2.html#K2">*</a> <font color="Olive">B</font> <a href="tarski.html#R1">c=</a> <font color="Olive">W</font>
 ;<br/>

<b>let </b><font color="Maroon">f</font> be   <a href="struct_0.html#NM6">Function</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">T</font>,<font color="Maroon">T</font></span><a href="borsuk_1.html#K5">:]</a></span>,<font color="Maroon">T</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="topgrp_1.html#D8" target="_self">TOPGRP_1:def 8</a><br/>

<b>assume </b><a NAME="E2:77"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">f</font> <a href="funct_2.html#R1">=</a> the <a href="group_1.html#U1">mult</a> of <font color="Maroon">T</font>
 ;<br/>

<a NAME="E3:77"/>
for <font color="Olive">W</font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">T</font>,<font color="Maroon">T</font></span><a href="borsuk_1.html#K5">:]</a></span><br/> for <font color="Olive">G</font> being   <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">W</font>  ex <font color="Olive">H</font> being   <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Olive">W</font> st <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Olive">H</font> <a href="tarski.html#R1">c=</a> <font color="Olive">G</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="77_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">W</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">T</font>,<font color="Maroon">T</font></span><a href="borsuk_1.html#K5">:]</a></span>;<br/><b>let </b><font color="Maroon">G</font> be    <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">W</font>;<br/>



<b>consider </b><font color="Maroon">a</font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font>, <font color="Maroon">b</font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E2:77_1"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">W</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="borsuk_1.html#K7">]</a></span>
 <b>by </b><i><a class="ref" href="borsuk_1.html#T50">BORSUK_1:50</a></i>;<br/>
<a NAME="E3:77_1"/>
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">W</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <font color="Maroon">b</font>
 
<b>by </b><i><a class="txt" href="topgrp_1.html#E2:77"><i><font color="Green">E40</font></i></a>, </i>;<br/>
<b>then consider </b><font color="Maroon">A</font> being    <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Maroon">a</font>, <font color="Maroon">B</font> being    <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Maroon">b</font><b> such that </b><br/><a NAME="E5:77_1"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">A</font> <a href="group_2.html#K2">*</a> <font color="Maroon">B</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">G</font>
 <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">H</font> = <span class="p1"><a href="borsuk_1.html#K8">[:</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">B</font></span><a href="borsuk_1.html#K8">:]</a></span> as    <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Maroon">W</font> <b>by </b><i><a class="txt" href="topgrp_1.html#E2:77"><i><font color="Green">E40</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon">H</font>
;<br/>

<b>thus </b><a NAME="E7:77_1"/>
<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">G</font>
 <b>by </b><i>, <a class="ref" href="topgrp_1.html#T5" target="_self">Th5</a>, </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:77"/>
<font color="Maroon">f</font> is <a href="pre_topc.html#V5">continuous</a>
 <b>by </b><i><a class="ref" href="borsuk_1.html#D3">BORSUK_1:def 3</a></i>;<br/>


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