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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="topgrp_1.html#NM2">TopGroup</a>;<br/>

<b>assume </b><a NAME="E1:76"/><i><font color="Green">E39</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><br/>for <font color="Olive">b<sub>3</sub></font> being   <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Olive">b<sub>1</sub></font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="group_1.html#K3">"</a> </span>)</span>ex <font color="Olive">b<sub>4</sub></font> being   <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Olive">b<sub>1</sub></font>ex <font color="Olive">b<sub>5</sub></font> being   <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Olive">b<sub>2</sub></font> st <font color="Olive">b<sub>4</sub></font> <a href="group_2.html#K2">*</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>5</sub></font> <a href="group_2.html#K1">"</a> </span>)</span> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>3</sub></font>
 ;<br/>

<a NAME="E2:76"/><i><font color="Green">E40</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><br/>for <font color="Olive">b<sub>2</sub></font> being   <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Olive">b<sub>1</sub></font> <a href="group_1.html#K3">"</a> ex <font color="Olive">b<sub>3</sub></font> being   <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Olive">b<sub>1</sub></font> st <font color="Olive">b<sub>3</sub></font> <a href="group_2.html#K1">"</a>  <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>2</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Maroon">c<sub>2</sub></font> <a href="group_1.html#K3">"</a> ;<br/>



<a NAME="E1:76_1"/>
<font color="Maroon">c<sub>3</sub></font> is    <a href="connsp_2.html#M1">a_neighborhood</a> of <span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="group_1.html#K3">"</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="group_1.html#D5">GROUP_1:def 5</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>4</sub></font> being    <a href="connsp_2.html#M1">a_neighborhood</a> of  <a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>5</sub></font> being    <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E3:76_1"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="group_2.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="group_2.html#K1">"</a> </span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="topgrp_1.html#E1:76"><i><font color="Green">E39</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>5</sub></font>
;<br/>

<b>let </b><font color="Maroon">c<sub>6</sub></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="E4:76_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> <a href="group_2.html#K1">"</a> 
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E6:76_1"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="group_1.html#K3">"</a>  &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="ref" href="group_2.html#T5">GROUP_2:5</a></i>;<br/>
<a NAME="E7:76_1"/>
 <a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="ref" href="connsp_2.html#T6">CONNSP_2:6</a></i>;<br/>
<a NAME="E8:76_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> <a href="group_2.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="group_2.html#K1">"</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="topgrp_1.html#E4:76_1"><i><font color="Green">E42</font></i></a>, <a class="txt" href="topgrp_1.html#E6:76_1"><i><font color="Green">E43</font></i></a>, <a class="ref" href="group_2.html#T12">GROUP_2:12</a></i>;<br/>
<a NAME="E9:76_1"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> <a href="group_1.html#K3">"</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> <a href="group_2.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="group_2.html#K1">"</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="group_1.html#D5">GROUP_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E10:76_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="topgrp_1.html#E3:76_1"><i><font color="Green">E41</font></i></a>, <a class="txt" href="topgrp_1.html#E6:76_1"><i><font color="Green">E43</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:76"/>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><br/>for <font color="Olive">b<sub>3</sub></font> being   <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Olive">b<sub>1</sub></font> <a href="group_1.html#K1">*</a> <font color="Olive">b<sub>2</sub></font>ex <font color="Olive">b<sub>4</sub></font> being   <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Olive">b<sub>1</sub></font>ex <font color="Olive">b<sub>5</sub></font> being   <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Olive">b<sub>2</sub></font> st <font color="Olive">b<sub>4</sub></font> <a href="group_2.html#K2">*</a> <font color="Olive">b<sub>5</sub></font> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>3</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Maroon">c<sub>2</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>3</sub></font>;<br/>



<a NAME="E1:76_2"/>
<font color="Maroon">c<sub>4</sub></font> is    <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Maroon">c<sub>2</sub></font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_1.html#K3">"</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="group_1.html#T19">GROUP_1:19</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>5</sub></font> being    <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>6</sub></font> being    <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K3">"</a> <b> such that </b><br/><a NAME="E3:76_2"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="group_2.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="group_2.html#K1">"</a> </span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="topgrp_1.html#E1:76"><i><font color="Green">E39</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>7</sub></font> being    <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Maroon">c<sub>3</sub></font><b> such that </b><br/><a NAME="E5:76_2"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="group_2.html#K1">"</a>  <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="topgrp_1.html#E2:76"><i><font color="Green">E40</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>5</sub></font>
;<br/>

<b>take </b>
<font color="Maroon">c<sub>7</sub></font>
;<br/>

<b>let </b><font color="Maroon">c<sub>8</sub></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="E6:76_2"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> <a href="group_2.html#K2">*</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>9</sub></font>, <font color="Maroon">c<sub>10</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E8:76_2"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> )
 <b>by </b><i><a class="ref" href="group_2.html#T12">GROUP_2:12</a></i>;<br/>
<a NAME="E9:76_2"/>
<font color="Maroon">c<sub>10</sub></font> <a href="group_1.html#K3">"</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> <a href="group_2.html#K1">"</a> 
 
<b>by </b><i><a class="txt" href="topgrp_1.html#E8:76_2"><i><font color="Green">E43</font></i></a>, <a class="ref" href="group_2.html#T5">GROUP_2:5</a></i>;<br/>
<a NAME="E10:76_2"/><b>then </b>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> <a href="group_2.html#K1">"</a> 
 
<b>by </b><i><a class="txt" href="topgrp_1.html#E5:76_2"><i><font color="Green">E42</font></i></a>, <a class="ref" href="topgrp_1.html#T7" target="_self">Th7</a></i>;<br/>
<a NAME="E11:76_2"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> <a href="group_2.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="group_2.html#K1">"</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="topgrp_1.html#E8:76_2"><i><font color="Green">E43</font></i></a>, <a class="ref" href="group_2.html#T12">GROUP_2:12</a></i>;<br/>
<b>hence </b><a NAME="E12:76_2"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="topgrp_1.html#E3:76_2"><i><font color="Green">E41</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:76"/>
<font color="Maroon">c<sub>1</sub></font> is   <a href="topgrp_1.html#NM3">TopologicalGroup</a>
 <b>by </b><i><a class="txt" href="topgrp_1.html#E2:76"><i><font color="Green">E40</font></i></a>, <a class="ref" href="topgrp_1.html#T38" target="_self">Th38</a>, <a class="ref" href="topgrp_1.html#T40" target="_self">Th40</a></i>;<br/>


</div>
