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<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>; <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">X0</font> being non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="tsp_2.html#NM1" title="TSP_2:NM.1">maximal_Kolmogorov_subspace</a> of <font color="Maroon" title="c1">X</font><br/> for <font color="Olive" title="b2">A</font>, <font color="Olive" title="b3">B</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>  st ( <font color="Olive" title="b2">A</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> or <font color="Olive" title="b3">B</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> ) holds <br/><span class="p1">(<span class="default"><a href="tsp_2.html#K4" title="TSP_2:func.4">Stone-retraction</a> <font color="Maroon" title="c1">X</font>,<font color="Olive" title="b1">X0</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Olive" title="b3">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="tsp_2.html#K4" title="TSP_2:func.4">Stone-retraction</a> <font color="Maroon" title="c1">X</font>,<font color="Olive" title="b1">X0</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Olive" title="b2">A</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="tsp_2.html#K4" title="TSP_2:func.4">Stone-retraction</a> <font color="Maroon" title="c1">X</font>,<font color="Olive" title="b1">X0</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Olive" title="b3">B</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c2">X0</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="tsp_2.html#NM1" title="TSP_2:NM.1">maximal_Kolmogorov_subspace</a> of <font color="Maroon" title="c1">X</font>; <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">B</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>  st ( <font color="Olive" title="b1">A</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> or <font color="Olive" title="b2">B</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> ) holds <br/><span class="p1">(<span class="default"><a href="tsp_2.html#K4" title="TSP_2:func.4">Stone-retraction</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">X0</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Olive" title="b2">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="tsp_2.html#K4" title="TSP_2:func.4">Stone-retraction</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">X0</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Olive" title="b1">A</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="tsp_2.html#K4" title="TSP_2:func.4">Stone-retraction</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">X0</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Olive" title="b2">B</font></span>)</span></span><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="ref" href="tsep_1.html#T1" title="TSEP_1:th.1">TSEP_1:1</a></i>;<br/>
<b>set </b><font color="Maroon" title="c4">r</font> =  <a href="tsp_2.html#K4" title="TSP_2:func.4">Stone-retraction</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">X0</font>;<br/>
<b>let </b><font color="Maroon" title="c5">A</font>, <font color="Maroon" title="c6">B</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( <font color="Maroon" title="c5">A</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> or <font color="Maroon" title="c6">B</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> ) implies <span class="p1">(<span class="default"><a href="tsp_2.html#K4" title="TSP_2:func.4">Stone-retraction</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">X0</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c6">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="tsp_2.html#K4" title="TSP_2:func.4">Stone-retraction</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">X0</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c5">A</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="tsp_2.html#K4" title="TSP_2:func.4">Stone-retraction</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">X0</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c6">B</font></span>)</span> )</span><br/>

<b>assume </b><a NAME="E2:52"/><i><font color="Green" title="E36">A1</font></i>: 
( <font color="Maroon" title="c5">A</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> or <font color="Maroon" title="c6">B</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"> <span class="p1">(<span class="default"><a href="tsp_2.html#K4" title="TSP_2:func.4">Stone-retraction</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">X0</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c6">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="tsp_2.html#K4" title="TSP_2:func.4">Stone-retraction</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">X0</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c5">A</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="tsp_2.html#K4" title="TSP_2:func.4">Stone-retraction</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">X0</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c6">B</font></span>)</span></span><br/>

<span class="p1">(<span class="default"><a href="tsp_2.html#K4" title="TSP_2:func.4">Stone-retraction</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">X0</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c6">B</font></span>)</span> = 
<font color="Maroon" title="c3">M</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="tex_4.html#K3" title="TEX_4:func.3">MaxADSet</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c6">B</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="tsp_2.html#D12" target="_self" title="TSP_2:def.12">Def12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">M</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="tex_4.html#K3" title="TEX_4:func.3">MaxADSet</a> <font color="Maroon" title="c5">A</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="tex_4.html#K3" title="TEX_4:func.3">MaxADSet</a> <font color="Maroon" title="c6">B</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E2:52"><i><font color="Green" title="E36">A1</font></i></a>, <a class="ref" href="tex_4.html#T67" title="TEX_4:th.67">TEX_4:67</a></i>
<br/>.= 
<font color="Maroon" title="c3">M</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="tex_4.html#K3" title="TEX_4:func.3">MaxADSet</a> <font color="Maroon" title="c5">A</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3">(<span class="default"><a href="tex_4.html#K3" title="TEX_4:func.3">MaxADSet</a> <font color="Maroon" title="c6">B</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T16" title="XBOOLE_1:th.16">XBOOLE_1:16</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3">(<span class="default"><a href="tex_4.html#K3" title="TEX_4:func.3">MaxADSet</a> <font color="Maroon" title="c5">A</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="tex_4.html#K3" title="TEX_4:func.3">MaxADSet</a> <font color="Maroon" title="c6">B</font></span>)</span></span>)</span> <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="xboole_1.html#T16" title="XBOOLE_1:th.16">XBOOLE_1:16</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="tex_4.html#K3" title="TEX_4:func.3">MaxADSet</a> <font color="Maroon" title="c5">A</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="tex_4.html#K3" title="TEX_4:func.3">MaxADSet</a> <font color="Maroon" title="c6">B</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T16" title="XBOOLE_1:th.16">XBOOLE_1:16</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="tsp_2.html#K4" title="TSP_2:func.4">Stone-retraction</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">X0</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c5">A</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="tex_4.html#K3" title="TEX_4:func.3">MaxADSet</a> <font color="Maroon" title="c6">B</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="tsp_2.html#D12" target="_self" title="TSP_2:def.12">Def12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="tsp_2.html#K4" title="TSP_2:func.4">Stone-retraction</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">X0</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c5">A</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="tsp_2.html#K4" title="TSP_2:func.4">Stone-retraction</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">X0</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c6">B</font></span>)</span>
<b>by </b><i><a class="ref" href="tsp_2.html#D12" target="_self" title="TSP_2:def.12">Def12</a></i>
;<br/>
<b>hence </b><a NAME="E4:52"/>
<span class="p1">(<span class="default"><a href="tsp_2.html#K4" title="TSP_2:func.4">Stone-retraction</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">X0</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c6">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="tsp_2.html#K4" title="TSP_2:func.4">Stone-retraction</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">X0</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c5">A</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="tsp_2.html#K4" title="TSP_2:func.4">Stone-retraction</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">X0</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c6">B</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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