<?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="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">X0</font> being  <a href="tex_2.html#V1" title="TEX_2:attr.1">proper</a>   <a href="tex_3.html#V2" title="TEX_3:attr.2">everywhere_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  <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  <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#R3" title="TSEP_2:pred.3">constitute_a_decomposition</a>  &amp; <font color="Olive" title="b2">X1</font> is    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Olive" title="b1">X0</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">X0</font> be   <a href="tex_2.html#V1" title="TEX_2:attr.1">proper</a>   <a href="tex_3.html#V2" title="TEX_3:attr.2">everywhere_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="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">X1</font> being  <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  <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#R3" title="TSEP_2:pred.3">constitute_a_decomposition</a>  &amp; <font color="Olive" title="b1">X1</font> is    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c2">X0</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   <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="E2:89"/>
<font color="Maroon" title="c3">D</font> is  <a href="tops_3.html#V1" title="TOPS_3:attr.1">everywhere_dense</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tex_3.html#T16" target="_self" title="TEX_3:th.16">Th16</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="E4:89"/><span class="lab"><font color="Green" title="E29">A1</font></span>: 
( <font color="Maroon" title="c4">C</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Maroon" title="c4">C</font> is  <a href="tops_1.html#V1" title="TOPS_1:attr.1">dense</a>  )
 <span class="kw">and </span><br/><a NAME="E5:89"/><span class="lab"><font color="Green" title="E30">A2</font></span>: 
( <font color="Maroon" title="c5">B</font> is  <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>  &amp; <font color="Maroon" title="c5">B</font> is  <a href="tops_1.html#V2" title="TOPS_1:attr.2">boundary</a>  )
 <span class="kw">and </span><br/><a NAME="E6:89"/><span class="lab"><font color="Green" title="E31">A3</font></span>: 
<font color="Maroon" title="c4">C</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">D</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</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="E7:89"/><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="E8:89"/><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#T50" title="TOPS_3:th.50">TOPS_3:50</a></span>;<br/>
<a NAME="E9:89"/>
<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="#E4:89"><span class="lab"><font color="Green" title="E29">A1</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="E11:89"/><span class="lab"><font color="Green" title="E34">A6</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="c6">X1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:89"><span class="lab"><font color="Green" title="E29">A1</font></span></a>, <a class="ref" href="tex_3.html#T23" target="_self" title="TEX_3:th.23">Th23</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E35">A7</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="89_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:89_1"/>
not <font color="Maroon" title="c4">C</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="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:89_1"/><span class="kw">then </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="c1">X</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="subset_1.html#D7" title="SUBSET_1:def.7">SUBSET_1:def 7</a></span>;<br/><a NAME="E3:89_1"/><span class="kw">then </span>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">D</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:89"><span class="lab"><font color="Green" title="E31">A3</font></span></a>, <a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></span>;<br/><a NAME="E4:89_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">D</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="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></span>;<br/><a NAME="E5:89_1"/><span class="kw">then </span>
not <font color="Maroon" title="c3">D</font> is  <a href="subset_1.html#V1" title="SUBSET_1:attr.1">proper</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="subset_1.html#D7" title="SUBSET_1:def.7">SUBSET_1:def 7</a></span>;<br/><span class="kw">hence </span><a NAME="E6:89_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tex_2.html#T13" title="TEX_2:th.13">TEX_2:13</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E13:89"/><span class="kw">then </span>
not <font color="Maroon" title="c5">B</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:89"><span class="lab"><font color="Green" title="E33">A5</font></span></a>, <a class="ref" href="subset_1.html#D7" title="SUBSET_1:def.7">SUBSET_1:def 7</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="E15:89"/><span class="lab"><font color="Green" title="E36">A8</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="c7">X2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:89"><span class="lab"><font color="Green" title="E30">A2</font></span></a>, <a class="ref" href="tex_3.html#T67" target="_self" title="TEX_3:th.67">Th67</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c8">X1</font> = <font color="Maroon" title="c6">X1</font> as   <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="#E11:89"><span class="lab"><font color="Green" title="E34">A6</font></span></a>, <a class="txt" href="#E12:89"><span class="lab"><font color="Green" title="E35">A7</font></span></a>, <a class="ref" href="tex_2.html#T13" title="TEX_2:th.13">TEX_2:13</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="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">X2</font> being  <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#R3" title="TSEP_2:pred.3">constitute_a_decomposition</a>  &amp; <font color="Maroon" title="c8">X1</font> is    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c2">X0</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="Red">::  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#R3" title="TSEP_2:pred.3">constitute_a_decomposition</a>  &amp; <font color="Maroon" title="c8">X1</font> is    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c2">X0</font> )</span><br/>

<a NAME="E17:89"/>
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="#E7:89"><span class="lab"><font color="Green" title="E32">A4</font></span></a>, <a class="txt" href="#E8:89"><span class="lab"><font color="Green" title="E33">A5</font></span></a>, <a class="txt" href="#E11:89"><span class="lab"><font color="Green" title="E34">A6</font></span></a>, <a class="txt" href="#E15:89"><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="E18:89"/>
<font color="Maroon" title="c8">X1</font>,<font color="Maroon" title="c7">X2</font> <a href="tsep_2.html#R3" title="TSEP_2:pred.3">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="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c8">X1</font> is    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c2">X0</font></span><br/>

<a NAME="E19:89"/>
<font color="Maroon" title="c4">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">D</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:89"><span class="lab"><font color="Green" title="E31">A3</font></span></a>, <a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></span>;<br/>
<span class="kw">hence </span><a NAME="E20:89"/>
<font color="Maroon" title="c8">X1</font> is    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c2">X0</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:89"><span class="lab"><font color="Green" title="E34">A6</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="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
