<?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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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><br/>  for <font color="Olive" title="b2">A</font> being    <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Olive" title="b1">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a></span>)</span>  ex <font color="Olive" title="b3">B</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 <font color="Olive" title="b1">a</font> st <font color="Olive" title="b3">B</font> <a href="group_2.html#K2" title="GROUP_2:func.2">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">B</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="Olive" title="b2">A</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">A</font> being    <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c2">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a></span>)</span>  ex <font color="Olive" title="b2">B</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 <font color="Maroon" title="c2">a</font> st <font color="Olive" title="b2">B</font> <a href="group_2.html#K2" title="GROUP_2:func.2">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">B</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="Olive" title="b1">A</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">A</font> be    <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c2">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a></span>)</span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">B</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 <font color="Maroon" title="c2">a</font> st <font color="Olive" title="b1">B</font> <a href="group_2.html#K2" title="GROUP_2:func.2">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">B</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="c3">A</font></span><br/>





<span class="kw">consider </span><font color="Maroon" title="c4">X</font>, <font color="Maroon" title="c5">Y</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 <font color="Maroon" title="c2">a</font><span class="kw"> such that </span><br/><a NAME="E2:99"/><span class="lab"><font color="Green" title="E53">A1</font></span>: 
<font color="Maroon" title="c4">X</font> <a href="group_2.html#K2" title="GROUP_2:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">Y</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="c3">A</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="topgrp_1.html#T41" target="_self" onmouseover="rs('topgrp_1/T41')" onmouseout="rh()">Th40</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c6">B</font> = <font color="Maroon" title="c4">X</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">Y</font> as   <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 <font color="Maroon" title="c2">a</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="connsp_2.html#T2" onmouseover="rs('connsp_2/T2')" onmouseout="rh()">CONNSP_2:2</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c6">B</font>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">B</font> <a href="group_2.html#K2" title="GROUP_2:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">B</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="c3">A</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c7">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">B</font> <a href="group_2.html#K2" title="GROUP_2:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">B</font> <a href="group_2.html#K1" title="GROUP_2:func.1">"</a></span>)</span> or <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> )</span><br/>

<span class="kw">assume </span><a NAME="E4:99"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">B</font> <a href="group_2.html#K2" title="GROUP_2:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">B</font> <a href="group_2.html#K1" title="GROUP_2:func.1">"</a></span>)</span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">g</font>, <font color="Maroon" title="c9">h</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="E6:99"/><span class="lab"><font color="Green" title="E54">A2</font></span>: 
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">g</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c9">h</font>
 <span class="kw">and </span><br/><a NAME="E7:99"/><span class="lab"><font color="Green" title="E55">A3</font></span>: 
<font color="Maroon" title="c8">g</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">B</font>
 <span class="kw">and </span><br/><a NAME="E8:99"/><span class="lab"><font color="Green" title="E56">A4</font></span>: 
<font color="Maroon" title="c9">h</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">B</font> <a href="group_2.html#K1" title="GROUP_2:func.1">"</a> 
 ;<br/>
<a NAME="E9:99"/>
<font color="Maroon" title="c9">h</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a>  <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">B</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:99"><span class="lab"><font color="Green" title="E56">A4</font></span></a>, <a class="ref" href="topgrp_1.html#T7" target="_self" onmouseover="rs('topgrp_1/T7')" onmouseout="rh()">Th7</a></span>;<br/>
<a NAME="E10:99"/><span class="kw">then </span>
<font color="Maroon" title="c9">h</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a>  <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">Y</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E11:99"/><span class="kw">then </span><span class="lab"><font color="Green" title="E57">A5</font></span>: 
<font color="Maroon" title="c9">h</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">Y</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="topgrp_1.html#T7" target="_self" onmouseover="rs('topgrp_1/T7')" onmouseout="rh()">Th7</a></span>;<br/>
<a NAME="E12:99"/>
<font color="Maroon" title="c8">g</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:99"><span class="lab"><font color="Green" title="E55">A3</font></span></a>, <a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E13:99"/><span class="kw">then </span>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> <a href="group_2.html#K2" title="GROUP_2:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">Y</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="#E6:99"><span class="lab"><font color="Green" title="E54">A2</font></span></a>, <a class="txt" href="#E11:99"><span class="lab"><font color="Green" title="E57">A5</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E14:99"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:99"><span class="lab"><font color="Green" title="E53">A1</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
