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<span class="kw">let </span><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#K2" title="TOPALG_2:func.2">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#K1" title="RELSET_1:func.1">dom</a> <span class="p4">(<span class="default"><a href="fcont_1.html#K1" title="FCONT_1: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>; <span class="comment"><font color="firebrick">:: according to </font></span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="fcont_1.html#K1" title="FCONT_1: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#K7" title="RELSET_1:func.7">.:</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"/><span class="lab"><font color="Green" title="E19">A1</font></span>: 
<font color="Maroon" title="c10">b</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></span>;<br/>
<a NAME="E2:31_1_1"/><span class="lab"><font color="Green" title="E20">A2</font></span>: 
 <a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="topalg_2.html#K2" title="TOPALG_2:func.2">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#K1" title="RELSET_1:func.1">dom</a> <span class="p3">(<span class="default"><a href="fcont_1.html#K1" title="FCONT_1: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 class="kw">by</span> <span class="lab"><a class="txt" href="toprealb.html#E26"><span class="lab"><font color="Green" title="E18">Lm12</font></span></a></span>;<br/>
<a NAME="E3:31_1_1"/><span class="lab"><font color="Green" title="E21">A3</font></span>: 
 <a href="topalg_2.html#K2" title="TOPALG_2:func.2">R^1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="topalg_2.html#K2" title="TOPALG_2:func.2">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#K2" title="RELSET_1:func.2">rng</a> <span class="p3">(<span class="default"><a href="fcont_1.html#K1" title="FCONT_1: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 class="kw">by</span> <span class="lab"><a class="txt" href="toprealb.html#E26"><span class="lab"><font color="Green" title="E18">Lm12</font></span></a></span>;<br/>
<a NAME="E4:31_1_1"/>
<font color="Maroon" title="c9">a</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></span>;<br/>
<a NAME="E5:31_1_1"/><span class="kw">then </span>
 <a href="toprealb.html#K6" title="TOPREALB:func.6">R^1</a> <span class="p1">(<span class="default"><a href="fcont_1.html#K1" title="FCONT_1:func.1">AffineMap</a> (<font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c10">b</font>)</span>)</span> is  <a href="tops_2.html#V3" title="TOPS_2:attr.3">being_homeomorphism</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31_1_1"><span class="lab"><font color="Green" title="E19">A1</font></span></a>, <a class="txt" href="#E2:31_1_1"><span class="lab"><font color="Green" title="E20">A2</font></span></a>, <a class="txt" href="#E3:31_1_1"><span class="lab"><font color="Green" title="E21">A3</font></span></a>, <a class="ref" href="jordan16.html#T20" title="JORDAN16:th.20">JORDAN16:20</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:31_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="fcont_1.html#K1" title="FCONT_1: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#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c11">A</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="#E2:31_1_1"><span class="lab"><font color="Green" title="E20">A2</font></span></a>, <a class="txt" href="#E3:31_1_1"><span class="lab"><font color="Green" title="E21">A3</font></span></a>, <a class="ref" href="topgrp_1.html#T25" title="TOPGRP_1:th.25">TOPGRP_1:25</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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