<?xml version="1.0"?>
<div class="add">

<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#V4" target="_self">BinContinuous</a>  <a href="topgrp_1.html#L1" target="_self">TopGrStr</a> ;<br/><b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font>, <font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font>;<br/><b>let </b><font color="Maroon">W</font> be    <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <font color="Maroon">b</font>;<br/>





<a NAME="E1:76"/>
the <a href="struct_0.html#U1">carrier</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> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="borsuk_1.html#D5">BORSUK_1:def 5</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">f</font> = the <a href="group_1.html#U1">mult</a> of <font color="Maroon">T</font> as   <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> ;<br/>
<a NAME="E3:76"/>
<font color="Maroon">f</font> is <a href="pre_topc.html#V5">continuous</a>
 
<b>by </b><i/>;<br/>
<b>then consider </b><font color="Maroon">H</font> being    <a href="connsp_2.html#M1">a_neighborhood</a> of <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> such that </b><br/><a NAME="E5:76"/><i><font color="Green">E30</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">W</font>
 <b>by </b><i><a class="ref" href="borsuk_1.html#D3">BORSUK_1:def 3</a></i>;<br/>
<a NAME="E6:76"/><i><font color="Green">E40</font></i>: 
<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> <a href="hidden.html#R2">in</a>  <a href="tops_1.html#K1">Int</a> <font color="Maroon">H</font>
 
<b>by </b><i><a class="ref" href="connsp_2.html#D1">CONNSP_2:def 1</a></i>;<br/>
<b>consider </b><font color="Maroon">F</font> being   <a href="struct_0.html#NM3">Subset-Family</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><b> such that </b><br/><a NAME="E8:76"/><i><font color="Green">E47</font></i>: 
 <a href="tops_1.html#K1">Int</a> <font color="Maroon">H</font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">F</font>
 <b>and </b><br/><a NAME="E9:76"/><i><font color="Green">E49</font></i>: 
for <font color="Olive">e</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> holds <br/> ex <font color="Olive">X1</font>, <font color="Olive">Y1</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> st <br/>( <font color="Olive">e</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Olive">X1</font>,<font color="Olive">Y1</font></span><a href="borsuk_1.html#K6">:]</a></span> &amp; <font color="Olive">X1</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Olive">Y1</font> is <a href="pre_topc.html#V3">open</a> )
 <b>by </b><i><a class="ref" href="borsuk_1.html#T45">BORSUK_1:45</a></i>;<br/>
<b>consider </b><font color="Maroon">M</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E11:76"/><i><font color="Green">E50</font></i>: 
( <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> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> &amp; <font color="Maroon">M</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> )
 <b>by </b><i>, , <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">A</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>, <font color="Maroon">B</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E13:76"/><i><font color="Green">E52</font></i>: 
( <font color="Maroon">M</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">B</font></span><a href="borsuk_1.html#K6">:]</a></span> &amp; <font color="Maroon">A</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">B</font> is <a href="pre_topc.html#V3">open</a> )
 <b>by </b><i><a class="txt" href="topgrp_1.html#E6:76"><i><font color="Green">E40</font></i></a>, <a class="ref" href="topgrp_1.html#T5" target="_self">Th5</a></i>;<br/>
<a NAME="E14:76"/>
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> )
 
<b>by </b><i><a class="ref" href="topgrp_1.html#T5" target="_self">Th5</a>, , <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E15:76"/><b>then </b><i><font color="Green">E53</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="tops_1.html#K1">Int</a> <font color="Maroon">A</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="tops_1.html#K1">Int</a> <font color="Maroon">B</font> )
 
<b>by </b><i>, <a class="ref" href="tops_1.html#T55">TOPS_1:55</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">A</font> = <font color="Maroon">A</font> as  <a href="pre_topc.html#V3">open</a>  <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Maroon">a</font> <b>by </b><i>, <a class="ref" href="connsp_2.html#D1">CONNSP_2:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">B</font> = <font color="Maroon">B</font> as  <a href="pre_topc.html#V3">open</a>  <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Maroon">b</font> <b>by </b><i>, , <a class="ref" href="connsp_2.html#D1">CONNSP_2:def 1</a></i>;<br/>
<a NAME="E18:76"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">A</font> <a href="group_2.html#K2">*</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">g</font> <a href="group_1.html#K1">*</a> <font color="Olive">h</font></span>)</span> where <font color="Olive">g</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font>, <font color="Olive">h</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font> : ( <font color="Olive">g</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Olive">h</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="group_2.html#D2">GROUP_2:def 2</a></i>;<br/>
<b>take </b>
<font color="Maroon">A</font>
;<br/>

<b>take </b>
<font color="Maroon">B</font>
;<br/>

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E19:76"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="group_2.html#K2">*</a> <font color="Maroon">B</font>
 ;<br/>

<b>then consider </b><font color="Maroon">g</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font>, <font color="Maroon">h</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E21:76"/><i><font color="Green">E55</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="group_1.html#K1">*</a> <font color="Maroon">h</font> &amp; <font color="Maroon">g</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">h</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> )
 <b>by </b><i/>;<br/>
<a NAME="E22:76"/>
<span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">g</font>,<font color="Maroon">h</font></span><a href="borsuk_1.html#K7">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font>
 
<b>by </b><i>, , <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E23:76"/><b>then </b><i><font color="Green">E84</font></i>: 
<span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">g</font>,<font color="Maroon">h</font></span><a href="borsuk_1.html#K7">]</a></span> <a href="hidden.html#R2">in</a>  <a href="tops_1.html#K1">Int</a> <font color="Maroon">H</font>
 
<b>by </b><i>, <a class="ref" href="topgrp_1.html#T5" target="_self">Th5</a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<a NAME="E24:76"/><i><font color="Green">E85</font></i>: 
 <a href="tops_1.html#K1">Int</a> <font color="Maroon">H</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">H</font>
 
<b>by </b><i><a class="ref" href="tops_1.html#T44">TOPS_1:44</a></i>;<br/>
<a NAME="E25:76"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">H</font>
 
<b>by </b><i>, <a class="ref" href="topgrp_1.html#T7" target="_self">Th7</a>, <a class="txt" href="topgrp_1.html#E8:76"><i><font color="Green">E47</font></i></a>, <a class="ref" href="funct_2.html#T43">FUNCT_2:43</a></i>;<br/>
<b>hence </b><a NAME="E26:76"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font>
 <b>by </b><i><a class="ref" href="topgrp_1.html#T4" target="_self">Th4</a></i>;<br/>


</div>
