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

<b>let </b><font color="Maroon" title="c11">A</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="topalg_2.html#K3" title="TOPALG_2:func.3">R^1</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><a href="toprealb.html#K5" title="TOPREALB:func.5">R^1</a> <span class="p3">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p4">(<span class="default"><a href="jordan16.html#K1" title="JORDAN16:func.1">AffineMap</a> <font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c10">b</font></span>)</span></span>)</span></span>)</span></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="t_0topsp.html#D2" title="T_0TOPSP:def.2">T_0TOPSP:def 2</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="c11">A</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> or <span class="p1">(<span class="default"><a href="toprealb.html#K6" title="TOPREALB:func.6">R^1</a> <span class="p2">(<span class="default"><a href="jordan16.html#K1" title="JORDAN16:func.1">AffineMap</a> <font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c10">b</font></span>)</span></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c11">A</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> )</span><br/>

<a NAME="E1:31_1_1"/><i><font color="Green" title="E20">A1</font></i>: 
( <font color="Maroon" title="c9">a</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> &amp; <font color="Maroon" title="c10">b</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> )
 
<b>by </b><i><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></i>;<br/>
<a NAME="E2:31_1_1"/><i><font color="Green" title="E21">A2</font></i>: 
(  <a href="topalg_2.html#K3" title="TOPALG_2:func.3">R^1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="topalg_2.html#K3" title="TOPALG_2:func.3">R^1</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p1">(<span class="default"><a href="toprealb.html#K5" title="TOPREALB:func.5">R^1</a> <span class="p2">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p3">(<span class="default"><a href="jordan16.html#K1" title="JORDAN16:func.1">AffineMap</a> <font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c10">b</font></span>)</span></span>)</span></span>)</span> &amp;  <a href="topalg_2.html#K3" title="TOPALG_2:func.3">R^1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="topalg_2.html#K3" title="TOPALG_2:func.3">R^1</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p1">(<span class="default"><a href="toprealb.html#K5" title="TOPREALB:func.5">R^1</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p3">(<span class="default"><a href="jordan16.html#K1" title="JORDAN16:func.1">AffineMap</a> <font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c10">b</font></span>)</span></span>)</span></span>)</span> )
 
<b>by </b><i><a class="txt" href="toprealb.html#E27"><i><font color="Green" title="E19">Lm13</font></i></a></i>;<br/>
<a NAME="E3:31_1_1"/><b>then </b>
 <a href="toprealb.html#K6" title="TOPREALB:func.6">R^1</a> <span class="p1">(<span class="default"><a href="jordan16.html#K1" title="JORDAN16:func.1">AffineMap</a> <font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c10">b</font></span>)</span> <a href="tops_2.html#NR1" title="TOPS_2:NR.1">is_homeomorphism</a> 
 
<b>by </b><i><a class="txt" href="toprealb.html#E1:31_1_1"><i><font color="Green" title="E20">A1</font></i></a>, <a class="ref" href="jordan16.html#T35" title="JORDAN16:th.35">JORDAN16:35</a></i>;<br/>
<b>hence </b><a NAME="E4:31_1_1"/>
(  not <font color="Maroon" title="c11">A</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> or <span class="p1">(<span class="default"><a href="toprealb.html#K6" title="TOPREALB:func.6">R^1</a> <span class="p2">(<span class="default"><a href="jordan16.html#K1" title="JORDAN16:func.1">AffineMap</a> <font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c10">b</font></span>)</span></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c11">A</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> )
 <b>by </b><i><a class="txt" href="toprealb.html#E2:31_1_1"><i><font color="Green" title="E21">A2</font></i></a>, <a class="ref" href="topgrp_1.html#T25" title="TOPGRP_1:th.25">TOPGRP_1:25</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>
