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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="topgrp_1.html#NM3">TopologicalGroup</a>;<br/><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> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K3">"</a> </span>)</span>;<br/>





<b>consider </b><font color="Maroon">c<sub>5</sub></font> being  <a href="pre_topc.html#V3">open</a>  <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="pre_topc.html#V3">open</a>  <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="E2:75"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="group_2.html#K2">*</a> <font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="topgrp_1.html#T37" target="_self">Th37</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>7</sub></font> being  <a href="pre_topc.html#V3">open</a>  <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="E4:75"/><i><font color="Green">E40</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="ref" href="topgrp_1.html#T39" target="_self">Th39</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/>

<a NAME="E5:75"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="group_2.html#K2">*</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="group_1.html#K1">*</a> <font color="Olive">b<sub>2</sub></font></span>)</span> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>, B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="group_2.html#D2">GROUP_2:def 2</a></i>;<br/>
<a NAME="E6:75"/><i><font color="Green">E42</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>7</sub></font> <a href="group_2.html#K1">"</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="group_1.html#K1">*</a> <font color="Olive">b<sub>2</sub></font></span>)</span> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>, B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> <a href="group_2.html#K1">"</a>  ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="group_2.html#D2">GROUP_2:def 2</a></i>;<br/>
<a NAME="E7:75"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="group_2.html#K1">"</a>  <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="group_1.html#K3">"</a> </span>)</span> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> </span>}</span> 
 
<b>by </b><i><a class="ref" href="group_2.html#D1">GROUP_2:def 1</a></i>;<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="E8:75"/>
<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>7</sub></font> <a href="group_2.html#K1">"</a> </span>)</span>
 ;<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="E10:75"/><i><font color="Green">E44</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> <a href="group_2.html#K1">"</a>  )
 <b>by </b><i><a class="txt" href="topgrp_1.html#E6:75"><i><font color="Green">E42</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>11</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="E12:75"/><i><font color="Green">E45</font></i>: 
( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> <a href="group_1.html#K3">"</a>  &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> )
 <b>by </b><i><a class="txt" href="topgrp_1.html#E7:75"><i><font color="Green">E43</font></i></a>, <a class="txt" href="topgrp_1.html#E10:75"><i><font color="Green">E44</font></i></a></i>;<br/>
<a NAME="E13:75"/>
<font color="Maroon">c<sub>9</sub></font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="group_1.html#K3">"</a> </span>)</span> <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>6</sub></font>
 
<b>by </b><i><a class="txt" href="topgrp_1.html#E4:75"><i><font color="Green">E40</font></i></a>, <a class="txt" href="topgrp_1.html#E5:75"><i><font color="Green">E41</font></i></a>, <a class="txt" href="topgrp_1.html#E10:75"><i><font color="Green">E44</font></i></a>, <a class="txt" href="topgrp_1.html#E12:75"><i><font color="Green">E45</font></i></a></i>;<br/>
<b>hence </b><a NAME="E14:75"/>
<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#E2:75"><i><font color="Green">E39</font></i></a>, <a class="txt" href="topgrp_1.html#E10:75"><i><font color="Green">E44</font></i></a>, <a class="txt" href="topgrp_1.html#E12:75"><i><font color="Green">E45</font></i></a></i>;<br/>


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