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

<span class="kw">let </span><font color="Maroon" title="c1">X</font> be   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   non  <a href="tdlat_3.html#V3" title="TDLAT_3:attr.3">almost_discrete</a>  <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="tex_3.html#V4" title="TEX_3:attr.4">nowhere_dense</a>   <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c1">X</font>  ex <font color="Olive" title="b2">X1</font> being   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="pre_topc.html#V1" title="PRE_TOPC:attr.1">strict</a>   <a href="tsep_1.html#V1" title="TSEP_1:attr.1">open</a>   <a href="tex_2.html#V1" title="TEX_2:attr.1">proper</a>   <a href="tex_3.html#V1" title="TEX_3:attr.1">dense</a>   <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c1">X</font> ex <font color="Olive" title="b3">X2</font> being   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="pre_topc.html#V1" title="PRE_TOPC:attr.1">strict</a>   <a href="borsuk_1.html#V1" title="BORSUK_1:attr.1">closed</a>   <a href="tex_3.html#V3" title="TEX_3:attr.3">boundary</a>   <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c1">X</font> st <br/>( <font color="Olive" title="b2">X1</font>,<font color="Olive" title="b3">X2</font> <a href="tsep_2.html#R4" title="TSEP_2:pred.4">constitute_a_decomposition</a>  &amp; <font color="Olive" title="b1">X0</font> is    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Olive" title="b3">X2</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">X0</font> be   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="tex_3.html#V4" title="TEX_3:attr.4">nowhere_dense</a>   <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">X1</font> being   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="pre_topc.html#V1" title="PRE_TOPC:attr.1">strict</a>   <a href="tsep_1.html#V1" title="TSEP_1:attr.1">open</a>   <a href="tex_2.html#V1" title="TEX_2:attr.1">proper</a>   <a href="tex_3.html#V1" title="TEX_3:attr.1">dense</a>   <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c1">X</font> ex <font color="Olive" title="b2">X2</font> being   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="pre_topc.html#V1" title="PRE_TOPC:attr.1">strict</a>   <a href="borsuk_1.html#V1" title="BORSUK_1:attr.1">closed</a>   <a href="tex_3.html#V3" title="TEX_3:attr.3">boundary</a>   <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c1">X</font> st <br/>( <font color="Olive" title="b1">X1</font>,<font color="Olive" title="b2">X2</font> <a href="tsep_2.html#R4" title="TSEP_2:pred.4">constitute_a_decomposition</a>  &amp; <font color="Maroon" title="c2">X0</font> is    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Olive" title="b2">X2</font> )</span><br/>



<span class="kw">reconsider </span><font color="Maroon" title="c3">D</font> =  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">X0</font> as   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="tsep_1.html#T1" title="TSEP_1:th.1">TSEP_1:1</a></span>;<br/>
<a NAME="E2:88"/>
<font color="Maroon" title="c3">D</font> is  <a href="tops_1.html#V3" title="TOPS_1:attr.3">nowhere_dense</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tex_3.html#T35" target="_self" title="TEX_3:th.35">Th35</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c4">C</font>, <font color="Maroon" title="c5">B</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font><span class="kw"> such that </span><br/><a NAME="E4:88"/><span class="lab"><font color="Green" title="E29">A1</font></span>: 
( <font color="Maroon" title="c4">C</font> is  <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>  &amp; <font color="Maroon" title="c4">C</font> is  <a href="tops_1.html#V2" title="TOPS_1:attr.2">boundary</a>  )
 <span class="kw">and </span><br/><a NAME="E5:88"/><span class="lab"><font color="Green" title="E30">A2</font></span>: 
( <font color="Maroon" title="c5">B</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Maroon" title="c5">B</font> is  <a href="tops_1.html#V1" title="TOPS_1:attr.1">dense</a>  )
 <span class="kw">and </span><br/><a NAME="E6:88"/><span class="lab"><font color="Green" title="E31">A3</font></span>: 
<font color="Maroon" title="c4">C</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">D</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c5">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">D</font>
 <span class="kw">and </span><br/><a NAME="E7:88"/><span class="lab"><font color="Green" title="E32">A4</font></span>: 
<font color="Maroon" title="c4">C</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c5">B</font>
 <span class="kw">and </span><br/><a NAME="E8:88"/><span class="lab"><font color="Green" title="E33">A5</font></span>: 
<font color="Maroon" title="c4">C</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c5">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tops_3.html#T52" title="TOPS_3:th.52">TOPS_3:52</a></span>;<br/>
<a NAME="E9:88"/>
<font color="Maroon" title="c5">B</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:88"><span class="lab"><font color="Green" title="E30">A2</font></span></a>, <a class="ref" href="tops_3.html#T17" title="TOPS_3:th.17">TOPS_3:17</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">X1</font> being   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="pre_topc.html#V1" title="PRE_TOPC:attr.1">strict</a>   <a href="tsep_1.html#V1" title="TSEP_1:attr.1">open</a>   <a href="tex_3.html#V1" title="TEX_3:attr.1">dense</a>   <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c1">X</font><span class="kw"> such that </span><br/><a NAME="E11:88"/><span class="lab"><font color="Green" title="E34">A6</font></span>: 
<font color="Maroon" title="c5">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c6">X1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:88"><span class="lab"><font color="Green" title="E30">A2</font></span></a>, <a class="ref" href="tex_3.html#T23" target="_self" title="TEX_3:th.23">Th23</a></span>;<br/>
<a NAME="E12:88"/><span class="lab"><font color="Green" title="E35">A7</font></span>: 
<font color="Maroon" title="c4">C</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:88"><span class="lab"><font color="Green" title="E31">A3</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">X2</font> being   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="pre_topc.html#V1" title="PRE_TOPC:attr.1">strict</a>   <a href="borsuk_1.html#V1" title="BORSUK_1:attr.1">closed</a>   <a href="tex_3.html#V3" title="TEX_3:attr.3">boundary</a>   <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c1">X</font><span class="kw"> such that </span><br/><a NAME="E14:88"/><span class="lab"><font color="Green" title="E36">A8</font></span>: 
<font color="Maroon" title="c4">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">X2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:88"><span class="lab"><font color="Green" title="E29">A1</font></span></a>, <a class="ref" href="tex_3.html#T67" target="_self" title="TEX_3:th.67">Th67</a></span>;<br/>
<a NAME="E15:88"/><span class="lab"><font color="Green" title="E37">A9</font></span>: 
<font color="Maroon" title="c4">C</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:88"><span class="lab"><font color="Green" title="E32">A4</font></span></a>, <a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="88_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:88_1"/>
 not <font color="Maroon" title="c5">B</font> is  <a href="subset_1.html#V1" title="SUBSET_1:attr.1">proper</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:88_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="subset_1.html#D6" title="SUBSET_1:def.6">SUBSET_1:def 6</a></span>;<br/><span class="kw">hence </span><a NAME="E3:88_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:88"><span class="lab"><font color="Green" title="E35">A7</font></span></a>, <a class="txt" href="#E15:88"><span class="lab"><font color="Green" title="E37">A9</font></span></a>, <a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</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/></div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c8">X1</font> = <font color="Maroon" title="c6">X1</font> as   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="pre_topc.html#V1" title="PRE_TOPC:attr.1">strict</a>   <a href="tsep_1.html#V1" title="TSEP_1:attr.1">open</a>   <a href="tex_2.html#V1" title="TEX_2:attr.1">proper</a>   <a href="tex_3.html#V1" title="TEX_3:attr.1">dense</a>   <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c1">X</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:88"><span class="lab"><font color="Green" title="E34">A6</font></span></a>, <a class="ref" href="tex_2.html#T8" title="TEX_2:th.8">TEX_2:8</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c8">X1</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">X2</font> being   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="pre_topc.html#V1" title="PRE_TOPC:attr.1">strict</a>   <a href="borsuk_1.html#V1" title="BORSUK_1:attr.1">closed</a>   <a href="tex_3.html#V3" title="TEX_3:attr.3">boundary</a>   <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c1">X</font> st <br/>( <font color="Maroon" title="c8">X1</font>,<font color="Olive" title="b1">X2</font> <a href="tsep_2.html#R4" title="TSEP_2:pred.4">constitute_a_decomposition</a>  &amp; <font color="Maroon" title="c2">X0</font> is    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Olive" title="b1">X2</font> )</span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c7">X2</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c8">X1</font>,<font color="Maroon" title="c7">X2</font> <a href="tsep_2.html#R4" title="TSEP_2:pred.4">constitute_a_decomposition</a>  &amp; <font color="Maroon" title="c2">X0</font> is    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c7">X2</font> )</span><br/>

<a NAME="E18:88"/>
 for <font color="Olive" title="b1">P</font>, <font color="Olive" title="b2">Q</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">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">X1</font> &amp; <font color="Olive" title="b2">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">X2</font> holds <br/><font color="Olive" title="b1">P</font>,<font color="Olive" title="b2">Q</font> <a href="tsep_2.html#R2" title="TSEP_2:pred.2">constitute_a_decomposition</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:88"><span class="lab"><font color="Green" title="E32">A4</font></span></a>, <a class="txt" href="#E8:88"><span class="lab"><font color="Green" title="E33">A5</font></span></a>, <a class="txt" href="#E11:88"><span class="lab"><font color="Green" title="E34">A6</font></span></a>, <a class="txt" href="#E14:88"><span class="lab"><font color="Green" title="E36">A8</font></span></a>, <a class="ref" href="tsep_2.html#D1" title="TSEP_2:def.1">TSEP_2:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E19:88"/>
<font color="Maroon" title="c8">X1</font>,<font color="Maroon" title="c7">X2</font> <a href="tsep_2.html#R4" title="TSEP_2:pred.4">constitute_a_decomposition</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tsep_2.html#D2" title="TSEP_2:def.2">TSEP_2:def 2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">X0</font> is    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c7">X2</font></span><br/>

<a NAME="E20:88"/>
<font color="Maroon" title="c3">D</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">C</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:88"><span class="lab"><font color="Green" title="E31">A3</font></span></a>, <a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></span>;<br/>
<span class="kw">hence </span><a NAME="E21:88"/>
<font color="Maroon" title="c2">X0</font> is    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c7">X2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:88"><span class="lab"><font color="Green" title="E36">A8</font></span></a>, <a class="ref" href="tsep_1.html#T4" title="TSEP_1:th.4">TSEP_1:4</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/>


</div>
