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<b>let </b><font color="Maroon" title="c1">G</font> be  <a href="topgrp_1.html#V4" title="TOPGRP_1:attr.4">BinContinuous</a> <a href="topgrp_1.html#NM4" title="TOPGRP_1:NM.4">TopGroup</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">A</font>, <font color="Olive" title="b2">O</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="b2">O</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> holds <br/><font color="Olive" title="b2">O</font> <a href="group_2.html#K2" title="GROUP_2:func.2">*</a> <font color="Olive" title="b1">A</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a></span><br/><b>let </b><font color="Maroon" title="c2">A</font>, <font color="Maroon" title="c3">O</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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">O</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> implies <font color="Maroon" title="c3">O</font> <a href="group_2.html#K2" title="GROUP_2:func.2">*</a> <font color="Maroon" title="c2">A</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> )</span><br/>



<b>assume </b><a NAME="E1:97"/><i><font color="Green" title="E50">A1</font></i>: 
<font color="Maroon" title="c3">O</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">O</font> <a href="group_2.html#K2" title="GROUP_2:func.2">*</a> <font color="Maroon" title="c2">A</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a></span><br/>

<a NAME="E2:97"/>
 <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">O</font> <a href="group_2.html#K2" title="GROUP_2:func.2">*</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">O</font> <a href="group_2.html#K2" title="GROUP_2:func.2">*</a> <font color="Maroon" title="c2">A</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="97_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:97_1"/>
 <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">O</font> <a href="group_2.html#K2" title="GROUP_2:func.2">*</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">O</font> <a href="group_2.html#K2" title="GROUP_2:func.2">*</a> <font color="Maroon" title="c2">A</font>
 <b>by </b><i><a class="ref" href="tops_1.html#T44" title="TOPS_1:th.44">TOPS_1:44</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">O</font> <a href="group_2.html#K2" title="GROUP_2:func.2">*</a> <font color="Maroon" title="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">O</font> <a href="group_2.html#K2" title="GROUP_2:func.2">*</a> <font color="Maroon" title="c2">A</font></span>)</span></span><br/>

<b>let </b><font color="Maroon" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">O</font> <a href="group_2.html#K2" title="GROUP_2:func.2">*</a> <font color="Maroon" title="c2">A</font> or <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">O</font> <a href="group_2.html#K2" title="GROUP_2:func.2">*</a> <font color="Maroon" title="c2">A</font></span>)</span> )</span><br/>

<b>assume </b><a NAME="E2:97_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">O</font> <a href="group_2.html#K2" title="GROUP_2:func.2">*</a> <font color="Maroon" title="c2">A</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">O</font> <a href="group_2.html#K2" title="GROUP_2:func.2">*</a> <font color="Maroon" title="c2">A</font></span>)</span></span><br/>

<b>then consider </b><font color="Maroon" title="c5">o</font>, <font color="Maroon" title="c6">a</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font><b> such that </b><br/><a NAME="E4:97_1"/><i><font color="Green" title="E51">A2</font></i>: 
( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">o</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c6">a</font> &amp; <font color="Maroon" title="c5">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">O</font> &amp; <font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> )
 <b>by </b><i><a class="ref" href="group_2.html#T12" title="GROUP_2:th.12">GROUP_2:12</a></i>;<br/>
<b>set </b><font color="Maroon" title="c7">Q</font> = <font color="Maroon" title="c3">O</font> <a href="group_2.html#K4" title="GROUP_2:func.4">*</a> <font color="Maroon" title="c6">a</font>;<br/>
<a NAME="E5:97_1"/><i><font color="Green" title="E52">A3</font></i>: 
<font color="Maroon" title="c3">O</font> <a href="group_2.html#K4" title="GROUP_2:func.4">*</a> <font color="Maroon" title="c6">a</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 
<b>by </b><i><a class="txt" href="#E1:97"><i><font color="Green" title="E50">A1</font></i></a></i>;<br/>
<a NAME="E6:97_1"/><i><font color="Green" title="E53">A4</font></i>: 
<font color="Maroon" title="c3">O</font> <a href="group_2.html#K4" title="GROUP_2:func.4">*</a> <font color="Maroon" title="c6">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">O</font> <a href="group_2.html#K2" title="GROUP_2:func.2">*</a> <font color="Maroon" title="c2">A</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="97_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">q</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c8">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">O</font> <a href="group_2.html#K4" title="GROUP_2:func.4">*</a> <font color="Maroon" title="c6">a</font> or <font color="Maroon" title="c8">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">O</font> <a href="group_2.html#K2" title="GROUP_2:func.2">*</a> <font color="Maroon" title="c2">A</font> )</span><br/>

<b>assume </b><a NAME="E1:97_1_1"/>
<font color="Maroon" title="c8">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">O</font> <a href="group_2.html#K4" title="GROUP_2:func.4">*</a> <font color="Maroon" title="c6">a</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c8">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">O</font> <a href="group_2.html#K2" title="GROUP_2:func.2">*</a> <font color="Maroon" title="c2">A</font></span><br/>

<b>then consider </b><font color="Maroon" title="c9">h</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font><b> such that </b><br/><a NAME="E3:97_1_1"/><i><font color="Green" title="E54">A5</font></i>: 
( <font color="Maroon" title="c8">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">h</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c6">a</font> &amp; <font color="Maroon" title="c9">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">O</font> )
 <b>by </b><i><a class="ref" href="group_2.html#T34" title="GROUP_2:th.34">GROUP_2:34</a></i>;<br/>
<b>thus </b><a NAME="E4:97_1_1"/>
<font color="Maroon" title="c8">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">O</font> <a href="group_2.html#K2" title="GROUP_2:func.2">*</a> <font color="Maroon" title="c2">A</font>
 <b>by </b><i><a class="txt" href="#E4:97_1"><i><font color="Green" title="E51">A2</font></i></a>, <a class="txt" href="#E3:97_1_1"><i><font color="Green" title="E54">A5</font></i></a>, <a class="ref" href="group_2.html#T12" title="GROUP_2:th.12">GROUP_2:12</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E7:97_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">O</font> <a href="group_2.html#K4" title="GROUP_2:func.4">*</a> <font color="Maroon" title="c6">a</font>
 
<b>by </b><i><a class="txt" href="#E4:97_1"><i><font color="Green" title="E51">A2</font></i></a>, <a class="ref" href="group_2.html#T34" title="GROUP_2:th.34">GROUP_2:34</a></i>;<br/>
<b>hence </b><a NAME="E8:97_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">O</font> <a href="group_2.html#K2" title="GROUP_2:func.2">*</a> <font color="Maroon" title="c2">A</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E5:97_1"><i><font color="Green" title="E52">A3</font></i></a>, <a class="txt" href="#E6:97_1"><i><font color="Green" title="E53">A4</font></i></a>, <a class="ref" href="tops_1.html#T54" title="TOPS_1:th.54">TOPS_1:54</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:97"/>
<font color="Maroon" title="c3">O</font> <a href="group_2.html#K2" title="GROUP_2:func.2">*</a> <font color="Maroon" title="c2">A</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
