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

<b>let </b><font color="Maroon" title="c1">X</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>;<br/><b>let </b><font color="Maroon" title="c2">X0</font> be   <a href="tsp_2.html#NM1" title="TSP_2:NM.1">maximal_Kolmogorov_subspace</a> of <font color="Maroon" title="c1">X</font>;<br/>



<b>reconsider </b><font color="Maroon" title="c3">M</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">X0</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> <b>by </b><i><a class="txt" href="tsp_2.html#E19"><i><font color="Green" title="E15">Lm1</font></i></a></i>;<br/>
<a NAME="E2:25"/><i><font color="Green" title="E18">A1</font></i>: 
<font color="Maroon" title="c3">M</font> is <a href="tsp_2.html#V1" title="TSP_2:attr.1">maximal_T_0</a>
 
<b>by </b><i><a class="ref" href="tsp_2.html#T11" target="_self" title="TSP_2:th.11">Th11</a></i>;<br/>
<b>let </b><font color="Maroon" title="c4">G</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>;<br/><b>let </b><font color="Maroon" title="c5">G0</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">X0</font>;<br/>



<b>assume </b><a NAME="E3:25"/><i><font color="Green" title="E19">A2</font></i>: 
<font color="Maroon" title="c5">G0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">G</font>
 ;<br/>

<b>thus </b><a NAME="E4:25"/>
( <font color="Maroon" title="c5">G0</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> implies (  <a href="tex_4.html#K3" title="TEX_4:func.3">MaxADSet</a> <font color="Maroon" title="c4">G</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c5">G0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="tex_4.html#K3" title="TEX_4:func.3">MaxADSet</a> <font color="Maroon" title="c4">G</font></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">X0</font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:25_1"/>
<font color="Maroon" title="c5">G0</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c6">H</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font><b> such that </b><br/><a NAME="E3:25_1"/><i><font color="Green" title="E20">A3</font></i>: 
<font color="Maroon" title="c6">H</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>and </b><br/><a NAME="E4:25_1"/><i><font color="Green" title="E21">A4</font></i>: 
<font color="Maroon" title="c5">G0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">H</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">M</font>
 <b>by </b><i><a class="ref" href="tsp_1.html#D1" title="TSP_1:def.1">TSP_1:def 1</a></i>;<br/>
<b>thus </b><a NAME="E5:25_1"/>
 <a href="tex_4.html#K3" title="TEX_4:func.3">MaxADSet</a> <font color="Maroon" title="c4">G</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>by </b><i><a class="txt" href="tsp_2.html#E2:25"><i><font color="Green" title="E18">A1</font></i></a>, <a class="txt" href="tsp_2.html#E3:25"><i><font color="Green" title="E19">A2</font></i></a>, <a class="txt" href="tsp_2.html#E3:25_1"><i><font color="Green" title="E20">A3</font></i></a>, <a class="txt" href="tsp_2.html#E4:25_1"><i><font color="Green" title="E21">A4</font></i></a>, <a class="ref" href="tsp_2.html#T6" target="_self" title="TSP_2:th.6">Th6</a></i>;<br/>

<b>thus </b><a NAME="E6:25_1"/>
<font color="Maroon" title="c5">G0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="tex_4.html#K3" title="TEX_4:func.3">MaxADSet</a> <font color="Maroon" title="c4">G</font></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">X0</font>
 <b>by </b><i><a class="txt" href="tsp_2.html#E2:25"><i><font color="Green" title="E18">A1</font></i></a>, <a class="txt" href="tsp_2.html#E3:25"><i><font color="Green" title="E19">A2</font></i></a>, <a class="txt" href="tsp_2.html#E3:25_1"><i><font color="Green" title="E20">A3</font></i></a>, <a class="txt" href="tsp_2.html#E4:25_1"><i><font color="Green" title="E21">A4</font></i></a>, <a class="ref" href="tsp_2.html#T6" target="_self" title="TSP_2:th.6">Th6</a></i>;<br/>


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

<b>assume </b><a NAME="E5:25"/><i><font color="Green" title="E20">A5</font></i>: 
 <a href="tex_4.html#K3" title="TEX_4:func.3">MaxADSet</a> <font color="Maroon" title="c4">G</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 ;<br/>

<b>assume </b><a NAME="E6:25"/>
<font color="Maroon" title="c5">G0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="tex_4.html#K3" title="TEX_4:func.3">MaxADSet</a> <font color="Maroon" title="c4">G</font></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">X0</font>
 ;<br/>

<b>hence </b><a NAME="E7:25"/>
<font color="Maroon" title="c5">G0</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>by </b><i><a class="txt" href="tsp_2.html#E5:25"><i><font color="Green" title="E20">A5</font></i></a>, <a class="ref" href="tsp_1.html#D1" title="TSP_1:def.1">TSP_1:def 1</a></i>;<br/>


</div>
