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

<b>consider </b><font color="Maroon" title="c1">R</font> being   <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">0</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>;<br/>
<a NAME="E2:73_1_1"/>
 <a href="compts_1.html#K1" title="COMPTS_1:func.1">1TopSp</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">0</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">0</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<span class="p1">(<span class="default"><a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">0</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> #)
 
;<br/>
<b>then reconsider </b><font color="Maroon" title="c2">T</font> =  <a href="topgrp_1.html#G1" title="TOPGRP_1:aggr.1">TopGrStr</a>(# <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">0</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<font color="Maroon" title="c1">R</font>,<span class="p1">(<span class="default"><a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">0</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> #) as  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> <a href="pre_topc.html#V2" title="PRE_TOPC:attr.2">TopSpace-like</a> <a href="group_1.html#V3" title="GROUP_1:attr.3">Group-like</a> <a href="group_1.html#V4" title="GROUP_1:attr.4">associative</a> <a href="topgrp_1.html#V2" title="TOPGRP_1:attr.2">strict</a>  <a href="topgrp_1.html#L1" title="TOPGRP_1:struct.1">TopGrStr</a>  <b>by </b><i><a class="ref" href="tex_2.html#T12" title="TEX_2:th.12">TEX_2:12</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c2">T</font>
;<br/>

<b>thus </b><a NAME="E4:73_1_1"/>
( <font color="Maroon" title="c2">T</font> is <a href="topgrp_1.html#V2" title="TOPGRP_1:attr.2">strict</a> &amp; <font color="Maroon" title="c2">T</font> is <a href="group_1.html#V7" title="GROUP_1:attr.7">commutative</a> &amp; <font color="Maroon" title="c2">T</font> is <a href="realset2.html#V3" title="REALSET2:attr.3">trivial</a> )
 ;<br/>

<div><b>hereby </b> <i><font color="Red">:: according to </font></i><a class="ref" href="topgrp_1.html#D7" target="_self" title="TOPGRP_1:def.7">TOPGRP_1:def 7</a>
<div class="add"><b>set </b><font color="Maroon" title="c3">f</font> =  <a href="topgrp_1.html#K6" title="TOPGRP_1:func.6">inverse_op</a> <font color="Maroon" title="c2">T</font>;<br/><b>thus </b><a NAME="E1:73_1_1_1"/>
 <a href="topgrp_1.html#K6" title="TOPGRP_1:func.6">inverse_op</a> <font color="Maroon" title="c2">T</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">P1</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">T</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="pre_topc.html#D12" title="PRE_TOPC:def.12">PRE_TOPC:def 12</a><br/>

<b>assume </b><a NAME="E1:73_1_1_1_1"/>
<font color="Maroon" title="c4">P1</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="73_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:73_1_1_1_1_1"/>
( <span class="p1">(<span class="default"><a href="topgrp_1.html#K6" title="TOPGRP_1:func.6">inverse_op</a> <font color="Maroon" title="c2">T</font></span>)</span> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c4">P1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <span class="p1">(<span class="default"><a href="topgrp_1.html#K6" title="TOPGRP_1:func.6">inverse_op</a> <font color="Maroon" title="c2">T</font></span>)</span> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c4">P1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">0</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 </span><b>by </b><i><a class="ref" href="zfmisc_1.html#T39" title="ZFMISC_1:th.39">ZFMISC_1:39</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="73_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:73_1_1_1_1_1_1"/>
<span class="p1">(<span class="default"><a href="topgrp_1.html#K6" title="TOPGRP_1:func.6">inverse_op</a> <font color="Maroon" title="c2">T</font></span>)</span> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c4">P1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:73_1_1_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="topgrp_1.html#K6" title="TOPGRP_1:func.6">inverse_op</a> <font color="Maroon" title="c2">T</font></span>)</span> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c4">P1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K1" title="STRUCT_0:func.1">{}</a> <font color="Maroon" title="c2">T</font>
 ;<br/><b>hence </b><a NAME="E3:73_1_1_1_1_1_1"/>
<span class="p1">(<span class="default"><a href="topgrp_1.html#K6" title="TOPGRP_1:func.6">inverse_op</a> <font color="Maroon" title="c2">T</font></span>)</span> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c4">P1</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="73_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:73_1_1_1_1_1_2"/>
<span class="p1">(<span class="default"><a href="topgrp_1.html#K6" title="TOPGRP_1:func.6">inverse_op</a> <font color="Maroon" title="c2">T</font></span>)</span> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c4">P1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">0</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ;<br/><div class="add"><a NAME="E2:73_1_1_1_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="topgrp_1.html#K6" title="TOPGRP_1:func.6">inverse_op</a> <font color="Maroon" title="c2">T</font></span>)</span> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c4">P1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c2">T</font>
 ;<br/><b>hence </b><a NAME="E3:73_1_1_1_1_1_2"/>
<span class="p1">(<span class="default"><a href="topgrp_1.html#K6" title="TOPGRP_1:func.6">inverse_op</a> <font color="Maroon" title="c2">T</font></span>)</span> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c4">P1</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div>
<b>end;</b></div>

<b>let </b><font color="Maroon" title="c3">f</font> be   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c2">T</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>,<font color="Maroon" title="c2">T</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="topgrp_1.html#D8" target="_self" title="TOPGRP_1:def.8">TOPGRP_1:def 8</a><br/>

<b>assume </b><a NAME="E6:73_1_1"/>
<font color="Maroon" title="c3">f</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> the <a href="group_1.html#U1" title="GROUP_1:sel.1">mult</a> of <font color="Maroon" title="c2">T</font>
 ;<br/>

<i><font color="Green" title="E36">A1</font></i>: the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c2">T</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> = 
<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">0</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">0</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
<b>by </b><i><a class="ref" href="borsuk_1.html#D5" title="BORSUK_1:def.5">BORSUK_1:def 5</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">0,0</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
<b>by </b><i><a class="ref" href="zfmisc_1.html#T35" title="ZFMISC_1:th.35">ZFMISC_1:35</a></i>
;<br/>
<b>let </b><font color="Maroon" title="c4">P1</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">T</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="pre_topc.html#D12" title="PRE_TOPC:def.12">PRE_TOPC:def 12</a><br/>

<b>assume </b><a NAME="E8:73_1_1"/>
<font color="Maroon" title="c4">P1</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="73_1_1_3"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:73_1_1_3"/>
( <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c4">P1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c4">P1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">0,0</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 </span><b>by </b><i><a class="txt" href="topgrp_1.html#E7:73_1_1"><i><font color="Green" title="E36">A1</font></i></a>, <a class="ref" href="zfmisc_1.html#T39" title="ZFMISC_1:th.39">ZFMISC_1:39</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="73_1_1_3_1"><b>suppose </b></a><a NAME="E1:73_1_1_3_1"/>
<font color="Maroon" title="c3">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c4">P1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:73_1_1_3_1"/><b>then </b>
<font color="Maroon" title="c3">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c4">P1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K1" title="STRUCT_0:func.1">{}</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c2">T</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>
 ;<br/><b>hence </b><a NAME="E3:73_1_1_3_1"/>
<font color="Maroon" title="c3">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c4">P1</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="73_1_1_3_2"><b>suppose </b></a><a NAME="E1:73_1_1_3_2"/>
<font color="Maroon" title="c3">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c4">P1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">0,0</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ;<br/><div class="add"><a NAME="E2:73_1_1_3_2"/><b>then </b>
<font color="Maroon" title="c3">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c4">P1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c2">T</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>
 <b>by </b><i><a class="txt" href="topgrp_1.html#E7:73_1_1"><i><font color="Green" title="E36">A1</font></i></a></i>;<br/><b>hence </b><a NAME="E3:73_1_1_3_2"/>
<font color="Maroon" title="c3">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c4">P1</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
