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

<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> be    <a href="yellow_8.html#M1">Basis</a> of  <a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be  <a href="tops_1.html#V1">dense</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>





<b>set </b><font color="Maroon">c<sub>4</sub></font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="group_2.html#K4">*</a> <font color="Olive">b<sub>2</sub></font></span>)</span> where B is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>, B is   <a href="pre_topc.html#NM2">Point</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>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> ) </span>}</span> ;<br/>
<a NAME="E1:104"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="group_2.html#K4">*</a> <font color="Olive">b<sub>2</sub></font></span>)</span> where B is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>, B is   <a href="pre_topc.html#NM2">Point</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>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> ) </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="pcomps_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="104_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</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="E1:104_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="group_2.html#K4">*</a> <font color="Olive">b<sub>2</sub></font></span>)</span> where B is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>, B is   <a href="pre_topc.html#NM2">Point</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>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>6</sub></font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>7</sub></font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:104_1"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="group_2.html#K4">*</a> <font color="Maroon">c<sub>7</sub></font>
 <b>and </b><br/><a NAME="E4:104_1"/>
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> )
 ;<br/>
<b>thus </b><a NAME="E5:104_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="pcomps_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="topgrp_1.html#E3:104_1"><i><font color="Green">E55</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:104"/><b>then </b><i><font color="Green">E55</font></i>: 
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="group_2.html#K4">*</a> <font color="Olive">b<sub>2</sub></font></span>)</span> where B is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>, B is   <a href="pre_topc.html#NM2">Point</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>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> ) </span>}</span>  is   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">c<sub>1</sub></font>
 
;<br/>
<a NAME="E3:104"/><i><font color="Green">E56</font></i>: 
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="group_2.html#K4">*</a> <font color="Olive">b<sub>2</sub></font></span>)</span> where B is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>, B is   <a href="pre_topc.html#NM2">Point</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>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> ) </span>}</span>  <a href="tarski.html#R1">c=</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="104_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</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="E1:104_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="group_2.html#K4">*</a> <font color="Olive">b<sub>2</sub></font></span>)</span> where B is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>, B is   <a href="pre_topc.html#NM2">Point</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>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>6</sub></font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>7</sub></font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:104_2"/><i><font color="Green">E57</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="group_2.html#K4">*</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> )
 ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>6</sub></font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<a NAME="E5:104_2"/>
<font color="Maroon">c<sub>8</sub></font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="txt" href="topgrp_1.html#E3:104_2"><i><font color="Green">E57</font></i></a>, <a class="ref" href="yellow_8.html#T21">YELLOW_8:21</a></i>;<br/>
<a NAME="E6:104_2"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font> <a href="group_2.html#K4">*</a> <font color="Maroon">c<sub>7</sub></font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="ref" href="topgrp_1.html#T49" target="_self">Th49</a></i>;<br/>
<b>hence </b><a NAME="E7:104_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="topgrp_1.html#E3:104_2"><i><font color="Green">E57</font></i></a>, <a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:104"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> is <a href="pre_topc.html#V3">open</a> holds <br/>for <font color="Olive">b<sub>2</sub></font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> holds <br/>ex <font color="Olive">b<sub>3</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>4</sub></font> <a href="group_2.html#K4">*</a> <font color="Olive">b<sub>5</sub></font></span>)</span> where B is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>, B is   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font> : ( <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Olive">b<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> ) </span>}</span>  &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>3</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>1</sub></font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="104_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>assume </b><a NAME="E1:104_3"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">c<sub>5</sub></font> is <a href="pre_topc.html#V3">open</a>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>assume </b><a NAME="E2:104_3"/><i><font color="Green">E58</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>
 ;<br/>

<a NAME="E3:104_3"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="group_2.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="group_1.html#K3">"</a> </span>)</span> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="txt" href="topgrp_1.html#E1:104_3"><i><font color="Green">E57</font></i></a>, <a class="ref" href="topgrp_1.html#T49" target="_self">Th49</a></i>;<br/>
<i><font color="Green">E60</font></i>: <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"><span class="p2">(<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#K3">"</a> </span>)</span> = 
<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"><a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#T16">GROUP_1:16</a></i>
<br/>.= 
 <a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="ref" href="group_1.html#D5">GROUP_1:def 5</a></i>
;<br/>
<a NAME="E5:104_3"/>
 <a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="group_1.html#K3">"</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="group_1.html#D6">GROUP_1:def 6</a></i>;<br/>
<a NAME="E6:104_3"/><b>then </b>
 <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>5</sub></font> <a href="group_2.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="group_1.html#K3">"</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="topgrp_1.html#E2:104_3"><i><font color="Green">E58</font></i></a>, <a class="ref" href="group_2.html#T34">GROUP_2:34</a></i>;<br/>
<a NAME="E7:104_3"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="group_2.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="group_1.html#K3">"</a> </span>)</span> 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"><span class="p2">(<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#K3">"</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="topgrp_1.html#E3:104_3"><i><font color="Green">E59</font></i></a>, <a class="txt" href="topgrp_1.html#E4:104_3"><i><font color="Green">E60</font></i></a>, <a class="ref" href="connsp_2.html#T5">CONNSP_2:5</a></i>;<br/>
<b>then 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  <a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E9:104_3"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">c<sub>7</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="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font> <a href="group_2.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="group_1.html#K3">"</a> </span>)</span>
 <b>by </b><i><a class="ref" href="topgrp_1.html#T55" target="_self">Th55</a></i>;<br/>
<a NAME="E10:104_3"/>
 <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>7</sub></font>
 
<b>by </b><i><a class="ref" href="connsp_2.html#T6">CONNSP_2:6</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>8</sub></font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E12:104_3"/><i><font color="Green">E62</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>7</sub></font> )
 <b>by </b><i><a class="ref" href="yellow_8.html#T22">YELLOW_8:22</a></i>;<br/>
<a NAME="E13:104_3"/>
( <font color="Maroon">c<sub>8</sub></font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 
<b>by </b><i><a class="txt" href="topgrp_1.html#E12:104_3"><i><font color="Green">E62</font></i></a>, <a class="ref" href="yellow_8.html#T21">YELLOW_8:21</a></i>;<br/>
<a NAME="E14:104_3"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="group_2.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_2.html#K1">"</a> </span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="ref" href="tops_1.html#T80">TOPS_1:80</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>9</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E16:104_3"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="group_2.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_2.html#K1">"</a> </span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="ref" href="xboole_0.html#T4">XBOOLE_0:4</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>9</sub></font> as   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="topgrp_1.html#E16:104_3"><i><font color="Green">E63</font></i></a></i>;<br/>
<b>take </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>8</sub></font> <a href="group_2.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>;<br/>

<a NAME="E18:104_3"/><i><font color="Green">E64</font></i>: 
( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> &amp; <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#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_2.html#K1">"</a> </span>)</span> )
 
<b>by </b><i><a class="txt" href="topgrp_1.html#E16:104_3"><i><font color="Green">E63</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>12</sub></font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E20:104_3"/><i><font color="Green">E65</font></i>: 
( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> <a href="group_2.html#K1">"</a>  )
 <b>by </b><i><a class="ref" href="group_2.html#T33">GROUP_2:33</a></i>;<br/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>6</sub></font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#D5">GROUP_1:def 5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="group_1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="group_1.html#K3">"</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#D6">GROUP_1:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="group_1.html#K3">"</a> </span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#D4">GROUP_1:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="group_1.html#K3">"</a> </span>)</span>
<b>by </b><i><a class="txt" href="topgrp_1.html#E20:104_3"><i><font color="Green">E65</font></i></a>, <a class="ref" href="group_1.html#D4">GROUP_1:def 4</a></i>
<br/>.= 
<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>12</sub></font> <a href="group_1.html#K3">"</a> </span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#D6">GROUP_1:def 6</a></i>
<br/>.= 
<font color="Maroon">c<sub>12</sub></font> <a href="group_1.html#K3">"</a> 
<b>by </b><i><a class="ref" href="group_1.html#D5">GROUP_1:def 5</a></i>
;<br/>
<a NAME="E22:104_3"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_1.html#K1">*</a> <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#E20:104_3"><i><font color="Green">E65</font></i></a>, <a class="ref" href="topgrp_1.html#T7" target="_self">Th7</a></i>;<br/>
<b>hence </b><a NAME="E23:104_3"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="group_2.html#K4">*</a> <font color="Olive">b<sub>2</sub></font></span>)</span> where B is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>, B is   <a href="pre_topc.html#NM2">Point</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>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="topgrp_1.html#E12:104_3"><i><font color="Green">E62</font></i></a></i>;<br/>

<a NAME="E24:104_3"/>
<font color="Maroon">c<sub>10</sub></font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="group_1.html#D6">GROUP_1:def 6</a></i>;<br/>
<a NAME="E25:104_3"/><b>then </b><i><font color="Green">E66</font></i>: 
 <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>8</sub></font> <a href="group_2.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="group_1.html#K3">"</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="topgrp_1.html#E18:104_3"><i><font color="Green">E64</font></i></a>, <a class="ref" href="group_2.html#T34">GROUP_2:34</a></i>;<br/>
<a NAME="E26:104_3"/>
<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> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="ref" href="group_1.html#D5">GROUP_1:def 5</a></i>;<br/>
<a NAME="E27:104_3"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="group_2.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="group_1.html#K3">"</a> </span>)</span></span>)</span> <a href="group_2.html#K4">*</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="topgrp_1.html#E25:104_3"><i><font color="Green">E66</font></i></a>, <a class="ref" href="group_2.html#T34">GROUP_2:34</a></i>;<br/>
<b>hence </b><a NAME="E28:104_3"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="ref" href="group_2.html#T40">GROUP_2:40</a></i>;<br/>

<a NAME="E29:104_3"/>
<font color="Maroon">c<sub>8</sub></font> <a href="group_2.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>7</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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="104_3_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>13</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="E1:104_3_3"/>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> <a href="group_2.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="group_1.html#K3">"</a> </span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>14</sub></font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:104_3_3"/><i><font color="Green">E67</font></i>: 
( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="group_1.html#K3">"</a> </span>)</span> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> )
 <b>by </b><i><a class="ref" href="group_2.html#T34">GROUP_2:34</a></i>;<br/>
<a NAME="E4:104_3_3"/>
<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#E12:104_3"><i><font color="Green">E62</font></i></a>, <a class="txt" href="topgrp_1.html#E18:104_3"><i><font color="Green">E64</font></i></a>, <a class="ref" href="group_2.html#T5">GROUP_2:5</a></i>;<br/>
<b>hence </b><a NAME="E5:104_3_3"/>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</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>
 <b>by </b><i><a class="txt" href="topgrp_1.html#E12:104_3"><i><font color="Green">E62</font></i></a>, <a class="txt" href="topgrp_1.html#E3:104_3_3"><i><font color="Green">E67</font></i></a>, <a class="ref" href="group_2.html#T12">GROUP_2:12</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E30:104_3"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font> <a href="group_2.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font> <a href="group_2.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="group_1.html#K3">"</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="topgrp_1.html#E9:104_3"><i><font color="Green">E61</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E31:104_3"/><b>then </b><i><font color="Green">E67</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="group_2.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="group_1.html#K3">"</a> </span>)</span></span>)</span> <a href="group_2.html#K4">*</a> <font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="group_2.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="group_1.html#K3">"</a> </span>)</span></span>)</span> <a href="group_2.html#K4">*</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="ref" href="topgrp_1.html#T5" target="_self">Th5</a></i>;<br/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="group_2.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="group_1.html#K3">"</a> </span>)</span></span>)</span> <a href="group_2.html#K4">*</a> <font color="Maroon">c<sub>6</sub></font> = 
<font color="Maroon">c<sub>5</sub></font> <a href="group_2.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="group_2.html#T40">GROUP_2:40</a></i>
<br/>.= 
<font color="Maroon">c<sub>5</sub></font> <a href="group_2.html#K4">*</a> <span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#D6">GROUP_1:def 6</a></i>
<br/>.= 
<font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="ref" href="group_2.html#T43">GROUP_2:43</a></i>
;<br/>
<b>hence </b><a NAME="E33:104_3"/>
<font color="Maroon">c<sub>11</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="topgrp_1.html#E31:104_3"><i><font color="Green">E67</font></i></a>, <a class="ref" href="group_2.html#T40">GROUP_2:40</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:104"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="group_2.html#K4">*</a> <font color="Olive">b<sub>2</sub></font></span>)</span> where B is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>, B is   <a href="pre_topc.html#NM2">Point</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>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> ) </span>}</span>  is    <a href="cantor_1.html#M1">Basis</a> of <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="topgrp_1.html#E2:104"><i><font color="Green">E55</font></i></a>, <a class="txt" href="topgrp_1.html#E3:104"><i><font color="Green">E56</font></i></a>, <a class="ref" href="yellow_9.html#T32">YELLOW_9:32</a></i>;<br/>


</div>
