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

<span class="kw">let </span><font color="Maroon" title="c1">G</font> be   <a href="topgrp_1.html#NM5" title="TOPGRP_1:NM.5">TopologicalGroup</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">B</font> being  <a href="yellow_8.html#NM1" title="YELLOW_8:NM.1">Basis</a> of  <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">G</font><br/> for <font color="Olive" title="b2">M</font> being  <a href="tops_1.html#V1" title="TOPS_1:attr.1">dense</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">G</font> holds  <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b3">V</font> <a href="group_2.html#K5" title="GROUP_2:func.5">*</a> <font color="Olive" title="b4">x</font></span>)</span> where <font color="Olive" title="b3">V</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">G</font>, <font color="Olive" title="b4">x</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">G</font> : ( <font color="Olive" title="b3">V</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">B</font> &amp; <font color="Olive" title="b4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">M</font> ) </span>}</span>  is   <a href="cantor_1.html#NM1" title="CANTOR_1:NM.1">Basis</a> of <font color="Maroon" title="c1">G</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">B</font> be   <a href="yellow_8.html#NM1" title="YELLOW_8:NM.1">Basis</a> of  <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">G</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">M</font> being  <a href="tops_1.html#V1" title="TOPS_1:attr.1">dense</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">G</font> holds  <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b2">V</font> <a href="group_2.html#K5" title="GROUP_2:func.5">*</a> <font color="Olive" title="b3">x</font></span>)</span> where <font color="Olive" title="b2">V</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">G</font>, <font color="Olive" title="b3">x</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">G</font> : ( <font color="Olive" title="b2">V</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B</font> &amp; <font color="Olive" title="b3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">M</font> ) </span>}</span>  is   <a href="cantor_1.html#NM1" title="CANTOR_1:NM.1">Basis</a> of <font color="Maroon" title="c1">G</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">M</font> be   <a href="tops_1.html#V1" title="TOPS_1:attr.1">dense</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">G</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">V</font> <a href="group_2.html#K5" title="GROUP_2:func.5">*</a> <font color="Olive" title="b2">x</font></span>)</span> where <font color="Olive" title="b1">V</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">G</font>, <font color="Olive" title="b2">x</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">G</font> : ( <font color="Olive" title="b1">V</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B</font> &amp; <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">M</font> ) </span>}</span>  is   <a href="cantor_1.html#NM1" title="CANTOR_1:NM.1">Basis</a> of <font color="Maroon" title="c1">G</font></span><br/>





<span class="kw">set </span><font color="Maroon" title="c4">Z</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">V</font> <a href="group_2.html#K5" title="GROUP_2:func.5">*</a> <font color="Olive" title="b2">x</font></span>)</span> where <font color="Olive" title="b1">V</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">G</font>, <font color="Olive" title="b2">x</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">G</font> : ( <font color="Olive" title="b1">V</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B</font> &amp; <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">M</font> ) </span>}</span> ;<br/>
<a NAME="E1:106"/><span class="lab"><font color="Green" title="E56">A1</font></span>: 
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">V</font> <a href="group_2.html#K5" title="GROUP_2:func.5">*</a> <font color="Olive" title="b2">x</font></span>)</span> where <font color="Olive" title="b1">V</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">G</font>, <font color="Olive" title="b2">x</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">G</font> : ( <font color="Olive" title="b1">V</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B</font> &amp; <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">M</font> ) </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">G</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="106_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">V</font> <a href="group_2.html#K5" title="GROUP_2:func.5">*</a> <font color="Olive" title="b2">x</font></span>)</span> where <font color="Olive" title="b1">V</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">G</font>, <font color="Olive" title="b2">x</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">G</font> : ( <font color="Olive" title="b1">V</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B</font> &amp; <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">M</font> ) </span>}</span>  or <font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">G</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:106_1"/>
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">V</font> <a href="group_2.html#K5" title="GROUP_2:func.5">*</a> <font color="Olive" title="b2">x</font></span>)</span> where <font color="Olive" title="b1">V</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">G</font>, <font color="Olive" title="b2">x</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">G</font> : ( <font color="Olive" title="b1">V</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B</font> &amp; <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">M</font> ) </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">G</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">V</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">G</font>, <font color="Maroon" title="c7">x</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">G</font><span class="kw"> such that </span><br/><a NAME="E3:106_1"/><span class="lab"><font color="Green" title="E57">A2</font></span>: 
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">V</font> <a href="group_2.html#K5" title="GROUP_2:func.5">*</a> <font color="Maroon" title="c7">x</font>
 <span class="kw">and </span><br/><a NAME="E4:106_1"/><span class="lab"><font color="Green" title="E58">A3</font></span>: 
<font color="Maroon" title="c6">V</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B</font>
 <span class="kw">and </span><br/><a NAME="E5:106_1"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">M</font>
 ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c8">V</font> = <font color="Maroon" title="c6">V</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">G</font> ;<br/>
<a NAME="E7:106_1"/>
<font color="Maroon" title="c8">V</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:106_1"><span class="lab"><font color="Green" title="E58">A3</font></span></a>, <a class="ref" href="yellow_8.html#T21" title="YELLOW_8:th.21">YELLOW_8:21</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:106_1"/>
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:106_1"><span class="lab"><font color="Green" title="E57">A2</font></span></a>, <a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E2:106"/><span class="lab"><font color="Green" title="E57">A4</font></span>: 
for <font color="Olive" title="b1">W</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">G</font>  st <font color="Olive" title="b1">W</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  holds <br/>for <font color="Olive" title="b2">a</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">G</font>  st <font color="Olive" title="b2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">W</font> holds <br/> ex <font color="Olive" title="b3">V</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">G</font> st <br/>( <font color="Olive" title="b3">V</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b4">V</font> <a href="group_2.html#K5" title="GROUP_2:func.5">*</a> <font color="Olive" title="b5">x</font></span>)</span> where <font color="Olive" title="b4">V</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">G</font>, <font color="Olive" title="b5">x</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">G</font> : ( <font color="Olive" title="b4">V</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B</font> &amp; <font color="Olive" title="b5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">M</font> ) </span>}</span>  &amp; <font color="Olive" title="b2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">V</font> &amp; <font color="Olive" title="b3">V</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">W</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="106_2"><span class="kw">proof </span></a><div class="add">

<span class="lab"><font color="Green" title="E58">A5</font></span>: <span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> </span>)</span> = 
<span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">G</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_1.html#T16" title="GROUP_1:th.16">GROUP_1:16</a></span>
<br/>.= 
 <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">G</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_1.html#D5" title="GROUP_1:def.5">GROUP_1:def 5</a></span>
;<br/>
<span class="kw">let </span><font color="Maroon" title="c5">W</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">G</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">W</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  implies for <font color="Olive" title="b1">a</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">G</font>  st <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">W</font> holds <br/> ex <font color="Olive" title="b2">V</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">G</font> st <br/>( <font color="Olive" title="b2">V</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b3">V</font> <a href="group_2.html#K5" title="GROUP_2:func.5">*</a> <font color="Olive" title="b4">x</font></span>)</span> where <font color="Olive" title="b3">V</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">G</font>, <font color="Olive" title="b4">x</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">G</font> : ( <font color="Olive" title="b3">V</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B</font> &amp; <font color="Olive" title="b4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">M</font> ) </span>}</span>  &amp; <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">V</font> &amp; <font color="Olive" title="b2">V</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">W</font> ) )</span><br/>

<span class="kw">assume </span><a NAME="E2:106_2"/><span class="lab"><font color="Green" title="E59">A6</font></span>: 
<font color="Maroon" title="c5">W</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">a</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">G</font>  st <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">W</font> holds <br/> ex <font color="Olive" title="b2">V</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">G</font> st <br/>( <font color="Olive" title="b2">V</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b3">V</font> <a href="group_2.html#K5" title="GROUP_2:func.5">*</a> <font color="Olive" title="b4">x</font></span>)</span> where <font color="Olive" title="b3">V</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">G</font>, <font color="Olive" title="b4">x</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">G</font> : ( <font color="Olive" title="b3">V</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B</font> &amp; <font color="Olive" title="b4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">M</font> ) </span>}</span>  &amp; <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">V</font> &amp; <font color="Olive" title="b2">V</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">W</font> )</span><br/>

<span class="kw">let </span><font color="Maroon" title="c6">a</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">G</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">W</font> implies  ex <font color="Olive" title="b1">V</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">G</font> st <br/>( <font color="Olive" title="b1">V</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b2">V</font> <a href="group_2.html#K5" title="GROUP_2:func.5">*</a> <font color="Olive" title="b3">x</font></span>)</span> where <font color="Olive" title="b2">V</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">G</font>, <font color="Olive" title="b3">x</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">G</font> : ( <font color="Olive" title="b2">V</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B</font> &amp; <font color="Olive" title="b3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">M</font> ) </span>}</span>  &amp; <font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">V</font> &amp; <font color="Olive" title="b1">V</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">W</font> ) )</span><br/>

<span class="kw">assume </span><a NAME="E3:106_2"/><span class="lab"><font color="Green" title="E60">A7</font></span>: 
<font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">W</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">V</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">G</font> st <br/>( <font color="Olive" title="b1">V</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b2">V</font> <a href="group_2.html#K5" title="GROUP_2:func.5">*</a> <font color="Olive" title="b3">x</font></span>)</span> where <font color="Olive" title="b2">V</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">G</font>, <font color="Olive" title="b3">x</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">G</font> : ( <font color="Olive" title="b2">V</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B</font> &amp; <font color="Olive" title="b3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">M</font> ) </span>}</span>  &amp; <font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">V</font> &amp; <font color="Olive" title="b1">V</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">W</font> )</span><br/>

<a NAME="E4:106_2"/>
 <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">a</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> </span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_1.html#D6" title="GROUP_1:def.6">GROUP_1:def 6</a></span>;<br/>
<a NAME="E5:106_2"/><span class="kw">then </span>
 <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">G</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">W</font> <a href="group_2.html#K5" title="GROUP_2:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">a</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> </span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:106_2"><span class="lab"><font color="Green" title="E60">A7</font></span></a>, <a class="ref" href="group_2.html#T34" title="GROUP_2:th.34">GROUP_2:34</a></span>;<br/>
<a NAME="E6:106_2"/><span class="kw">then </span>
<font color="Maroon" title="c5">W</font> <a href="group_2.html#K5" title="GROUP_2:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">a</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> </span>)</span> is    <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> </span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:106_2"><span class="lab"><font color="Green" title="E59">A6</font></span></a>, <a class="txt" href="#E1:106_2"><span class="lab"><font color="Green" title="E58">A5</font></span></a>, <a class="ref" href="connsp_2.html#T5" title="CONNSP_2:th.5">CONNSP_2:5</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">V</font> being   <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>   <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of  <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">G</font><span class="kw"> such that </span><br/><a NAME="E8:106_2"/><span class="lab"><font color="Green" title="E61">A8</font></span>: 
<font color="Maroon" title="c7">V</font> <a href="group_2.html#K2" title="GROUP_2:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">V</font> <a href="group_2.html#K1" title="GROUP_2:func.1">"</a> </span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">W</font> <a href="group_2.html#K5" title="GROUP_2:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">a</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> </span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="topgrp_1.html#T55" target="_self" title="TOPGRP_1:th.55">Th55</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c8">E</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">G</font><span class="kw"> such that </span><br/><a NAME="E10:106_2"/><span class="lab"><font color="Green" title="E62">A9</font></span>: 
<font color="Maroon" title="c8">E</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B</font>
 <span class="kw">and </span><br/><a NAME="E11:106_2"/><span class="lab"><font color="Green" title="E63">A10</font></span>: 
<font color="Maroon" title="c8">E</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">V</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="connsp_2.html#T6" title="CONNSP_2:th.6">CONNSP_2:6</a>, <a class="ref" href="yellow_8.html#T22" title="YELLOW_8:th.22">YELLOW_8:22</a></span>;<br/>
<a NAME="E12:106_2"/>
( <font color="Maroon" title="c8">E</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Maroon" title="c8">E</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:106_2"><span class="lab"><font color="Green" title="E62">A9</font></span></a>, <a class="ref" href="yellow_8.html#T21" title="YELLOW_8:th.21">YELLOW_8:21</a></span>;<br/>
<a NAME="E13:106_2"/><span class="kw">then </span>
<font color="Maroon" title="c6">a</font> <a href="group_2.html#K4" title="GROUP_2:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="group_2.html#K1" title="GROUP_2:func.1">"</a> </span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c8">E</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tops_1.html#T80" title="TOPS_1:th.80">TOPS_1:80</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">d</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E15:106_2"/><span class="lab"><font color="Green" title="E64">A11</font></span>: 
<font color="Maroon" title="c9">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">a</font> <a href="group_2.html#K4" title="GROUP_2:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="group_2.html#K1" title="GROUP_2:func.1">"</a> </span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c8">E</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#T4" title="XBOOLE_0:th.4">XBOOLE_0:4</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c10">d</font> = <font color="Maroon" title="c9">d</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">G</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:106_2"><span class="lab"><font color="Green" title="E64">A11</font></span></a></span>;<br/>
<span class="kw">take </span><font color="Maroon" title="c11">I</font> = <font color="Maroon" title="c8">E</font> <a href="group_2.html#K5" title="GROUP_2:func.5">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">d</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> </span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c6">a</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c11">I</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">V</font> <a href="group_2.html#K5" title="GROUP_2:func.5">*</a> <font color="Olive" title="b2">x</font></span>)</span> where <font color="Olive" title="b1">V</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">G</font>, <font color="Olive" title="b2">x</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">G</font> : ( <font color="Olive" title="b1">V</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B</font> &amp; <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">M</font> ) </span>}</span>  &amp; <font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">I</font> &amp; <font color="Maroon" title="c11">I</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">W</font> )</span><br/>

<a NAME="E17:106_2"/>
<font color="Maroon" title="c10">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">a</font> <a href="group_2.html#K4" title="GROUP_2:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="group_2.html#K1" title="GROUP_2:func.1">"</a> </span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:106_2"><span class="lab"><font color="Green" title="E64">A11</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c12">m</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">G</font><span class="kw"> such that </span><br/><a NAME="E19:106_2"/><span class="lab"><font color="Green" title="E65">A12</font></span>: 
<font color="Maroon" title="c10">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c12">m</font>
 <span class="kw">and </span><br/><a NAME="E20:106_2"/><span class="lab"><font color="Green" title="E66">A13</font></span>: 
<font color="Maroon" title="c12">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">M</font> <a href="group_2.html#K1" title="GROUP_2:func.1">"</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="group_2.html#T33" title="GROUP_2:th.33">GROUP_2:33</a></span>;<br/>
<span class="p1">(<span class="default"><font color="Maroon" title="c10">d</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> </span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c6">a</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">d</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> </span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c6">a</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">G</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_1.html#D5" title="GROUP_1:def.5">GROUP_1:def 5</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">d</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> </span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c6">a</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> </span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_1.html#D6" title="GROUP_1:def.6">GROUP_1:def 6</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">d</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> </span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c6">a</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c12">m</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> </span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_1.html#D4" title="GROUP_1:def.4">GROUP_1:def 4</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">d</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> </span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c10">d</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> </span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:106_2"><span class="lab"><font color="Green" title="E65">A12</font></span></a>, <a class="ref" href="group_1.html#D4" title="GROUP_1:def.4">GROUP_1:def 4</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> </span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_1.html#D6" title="GROUP_1:def.6">GROUP_1:def 6</a></span>
<br/>.= 
<font color="Maroon" title="c12">m</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_1.html#D5" title="GROUP_1:def.5">GROUP_1:def 5</a></span>
;<br/>
<a NAME="E22:106_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c10">d</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> </span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">M</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:106_2"><span class="lab"><font color="Green" title="E66">A13</font></span></a>, <a class="ref" href="topgrp_1.html#T7" target="_self" title="TOPGRP_1:th.7">Th7</a></span>;<br/>
<span class="kw">hence </span><a NAME="E23:106_2"/>
<font color="Maroon" title="c11">I</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">V</font> <a href="group_2.html#K5" title="GROUP_2:func.5">*</a> <font color="Olive" title="b2">x</font></span>)</span> where <font color="Olive" title="b1">V</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">G</font>, <font color="Olive" title="b2">x</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">G</font> : ( <font color="Olive" title="b1">V</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B</font> &amp; <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">M</font> ) </span>}</span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:106_2"><span class="lab"><font color="Green" title="E62">A9</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">I</font> &amp; <font color="Maroon" title="c11">I</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">W</font> )</span><br/>

<a NAME="E24:106_2"/><span class="lab"><font color="Green" title="E67">A14</font></span>: 
<span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c6">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_1.html#D5" title="GROUP_1:def.5">GROUP_1:def 5</a></span>;<br/>
<a NAME="E25:106_2"/><span class="lab"><font color="Green" title="E68">A15</font></span>: 
<font color="Maroon" title="c10">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">E</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:106_2"><span class="lab"><font color="Green" title="E64">A11</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E26:106_2"/>
<font color="Maroon" title="c8">E</font> <a href="group_2.html#K5" title="GROUP_2:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">d</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> </span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">V</font> <a href="group_2.html#K2" title="GROUP_2:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">V</font> <a href="group_2.html#K1" title="GROUP_2:func.1">"</a> </span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="106_2_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c13">q</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c13">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">E</font> <a href="group_2.html#K5" title="GROUP_2:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">d</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> </span>)</span> or <font color="Maroon" title="c13">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">V</font> <a href="group_2.html#K2" title="GROUP_2:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">V</font> <a href="group_2.html#K1" title="GROUP_2:func.1">"</a> </span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:106_2_3"/>
<font color="Maroon" title="c13">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">E</font> <a href="group_2.html#K5" title="GROUP_2:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">d</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> </span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c13">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">V</font> <a href="group_2.html#K2" title="GROUP_2:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">V</font> <a href="group_2.html#K1" title="GROUP_2:func.1">"</a> </span>)</span></span><br/>

<a NAME="E2:106_2_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E69">A16</font></span>: 
 ex <font color="Olive" title="b1">v</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">G</font> st <br/>( <font color="Maroon" title="c13">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">v</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">d</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> </span>)</span> &amp; <font color="Olive" title="b1">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">E</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_2.html#T34" title="GROUP_2:th.34">GROUP_2:34</a></span>;<br/>
<a NAME="E3:106_2_3"/>
<font color="Maroon" title="c10">d</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">V</font> <a href="group_2.html#K1" title="GROUP_2:func.1">"</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:106_2"><span class="lab"><font color="Green" title="E63">A10</font></span></a>, <a class="txt" href="#E25:106_2"><span class="lab"><font color="Green" title="E68">A15</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:106_2_3"/>
<font color="Maroon" title="c13">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">V</font> <a href="group_2.html#K2" title="GROUP_2:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">V</font> <a href="group_2.html#K1" title="GROUP_2:func.1">"</a> </span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:106_2"><span class="lab"><font color="Green" title="E63">A10</font></span></a>, <a class="txt" href="#E2:106_2_3"><span class="lab"><font color="Green" title="E69">A16</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E27:106_2"/><span class="kw">then </span>
<font color="Maroon" title="c8">E</font> <a href="group_2.html#K5" title="GROUP_2:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">d</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> </span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">W</font> <a href="group_2.html#K5" title="GROUP_2:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">a</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> </span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:106_2"><span class="lab"><font color="Green" title="E61">A8</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<a NAME="E28:106_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E69">A17</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">E</font> <a href="group_2.html#K5" title="GROUP_2:func.5">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">d</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> </span>)</span></span>)</span> <a href="group_2.html#K5" title="GROUP_2:func.5">*</a> <font color="Maroon" title="c6">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">W</font> <a href="group_2.html#K5" title="GROUP_2:func.5">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">a</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> </span>)</span></span>)</span> <a href="group_2.html#K5" title="GROUP_2:func.5">*</a> <font color="Maroon" title="c6">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="topgrp_1.html#T5" target="_self" title="TOPGRP_1:th.5">Th5</a></span>;<br/>
<a NAME="E29:106_2"/>
<font color="Maroon" title="c10">d</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">d</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">G</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_1.html#D6" title="GROUP_1:def.6">GROUP_1:def 6</a></span>;<br/>
<a NAME="E30:106_2"/><span class="kw">then </span>
 <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">G</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">E</font> <a href="group_2.html#K5" title="GROUP_2:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">d</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> </span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:106_2"><span class="lab"><font color="Green" title="E68">A15</font></span></a>, <a class="ref" href="group_2.html#T34" title="GROUP_2:th.34">GROUP_2:34</a></span>;<br/>
<a NAME="E31:106_2"/><span class="kw">then </span>
<font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">E</font> <a href="group_2.html#K5" title="GROUP_2:func.5">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">d</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> </span>)</span></span>)</span> <a href="group_2.html#K5" title="GROUP_2:func.5">*</a> <font color="Maroon" title="c6">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:106_2"><span class="lab"><font color="Green" title="E67">A14</font></span></a>, <a class="ref" href="group_2.html#T34" title="GROUP_2:th.34">GROUP_2:34</a></span>;<br/>
<span class="kw">hence </span><a NAME="E32:106_2"/>
<font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">I</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="group_2.html#T40" title="GROUP_2:th.40">GROUP_2:40</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c11">I</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">W</font></span><br/>

<span class="p1">(<span class="default"><font color="Maroon" title="c5">W</font> <a href="group_2.html#K5" title="GROUP_2:func.5">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">a</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> </span>)</span></span>)</span> <a href="group_2.html#K5" title="GROUP_2:func.5">*</a> <font color="Maroon" title="c6">a</font> = 
<font color="Maroon" title="c5">W</font> <a href="group_2.html#K5" title="GROUP_2:func.5">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">a</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> </span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c6">a</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_2.html#T40" title="GROUP_2:th.40">GROUP_2:40</a></span>
<br/>.= 
<font color="Maroon" title="c5">W</font> <a href="group_2.html#K5" title="GROUP_2:func.5">*</a> <span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">G</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_1.html#D6" title="GROUP_1:def.6">GROUP_1:def 6</a></span>
<br/>.= 
<font color="Maroon" title="c5">W</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_2.html#T43" title="GROUP_2:th.43">GROUP_2:43</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E34:106_2"/>
<font color="Maroon" title="c11">I</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">W</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:106_2"><span class="lab"><font color="Green" title="E69">A17</font></span></a>, <a class="ref" href="group_2.html#T40" title="GROUP_2:th.40">GROUP_2:40</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E3:106"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">V</font> <a href="group_2.html#K5" title="GROUP_2:func.5">*</a> <font color="Olive" title="b2">x</font></span>)</span> where <font color="Olive" title="b1">V</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">G</font>, <font color="Olive" title="b2">x</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">G</font> : ( <font color="Olive" title="b1">V</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B</font> &amp; <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">M</font> ) </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="106_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">V</font> <a href="group_2.html#K5" title="GROUP_2:func.5">*</a> <font color="Olive" title="b2">x</font></span>)</span> where <font color="Olive" title="b1">V</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">G</font>, <font color="Olive" title="b2">x</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">G</font> : ( <font color="Olive" title="b1">V</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B</font> &amp; <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">M</font> ) </span>}</span>  or <font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:106_3"/>
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">V</font> <a href="group_2.html#K5" title="GROUP_2:func.5">*</a> <font color="Olive" title="b2">x</font></span>)</span> where <font color="Olive" title="b1">V</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">G</font>, <font color="Olive" title="b2">x</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">G</font> : ( <font color="Olive" title="b1">V</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B</font> &amp; <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">M</font> ) </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font></span><br/>

<a NAME="E2:106_3"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">V</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">G</font> ex <font color="Olive" title="b2">x</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">G</font> st <br/>( <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">V</font> <a href="group_2.html#K5" title="GROUP_2:func.5">*</a> <font color="Olive" title="b2">x</font> &amp; <font color="Olive" title="b1">V</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B</font> &amp; <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">M</font> )
 
;<br/>
<span class="kw">hence </span><a NAME="E3:106_3"/>
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E4:106"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">V</font> <a href="group_2.html#K5" title="GROUP_2:func.5">*</a> <font color="Olive" title="b2">x</font></span>)</span> where <font color="Olive" title="b1">V</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">G</font>, <font color="Olive" title="b2">x</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">G</font> : ( <font color="Olive" title="b1">V</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B</font> &amp; <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">M</font> ) </span>}</span>  is   <a href="cantor_1.html#NM1" title="CANTOR_1:NM.1">Basis</a> of <font color="Maroon" title="c1">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:106"><span class="lab"><font color="Green" title="E56">A1</font></span></a>, <a class="txt" href="#E2:106"><span class="lab"><font color="Green" title="E57">A4</font></span></a>, <a class="ref" href="yellow_9.html#T32" title="YELLOW_9:th.32">YELLOW_9:32</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
