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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>;<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="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c1">X</font>;<br/>



<b>reconsider </b><font color="Maroon" title="c3">A</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/>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:95_1"/>
<font color="Maroon" title="c2">X0</font> is <a href="tex_2.html#V5" title="TEX_2:attr.5">maximal_discrete</a>
 ;<br/><a NAME="E2:95_1"/><b>then </b><i><font color="Green" title="E41">A1</font></i>: 
<font color="Maroon" title="c3">A</font> is <a href="tex_2.html#V4" title="TEX_2:attr.4">maximal_discrete</a>
 <b>by </b><i><a class="ref" href="tex_2.html#T51" target="_self" title="TEX_2:th.51">Th51</a></i>;<br/><a NAME="E3:95_1"/><b>then </b>
<font color="Maroon" title="c3">A</font> is <a href="tex_2.html#V3" title="TEX_2:attr.3">discrete</a>
 <b>by </b><i><a class="ref" href="tex_2.html#D7" target="_self" title="TEX_2:def.7">Def7</a></i>;<br/><b>hence </b><a NAME="E4:95_1"/>
<font color="Maroon" title="c2">X0</font> is <a href="tdlat_3.html#V1" title="TDLAT_3:attr.1">discrete</a>
 <b>by </b><i><a class="ref" href="tex_2.html#T26" target="_self" title="TEX_2:th.26">Th26</a></i>;<br/><b>thus </b><a NAME="E5:95_1"/>
for <font color="Olive" title="b1">Y0</font> being non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="tdlat_3.html#V1" title="TDLAT_3:attr.1">discrete</a>  <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c1">X</font>  st <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">Y0</font> holds <br/> <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">X0</font>,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c2">X0</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">Y0</font>,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Olive" title="b1">Y0</font> #)
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="95_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">Y0</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="tdlat_3.html#V1" title="TDLAT_3:attr.1">discrete</a>  <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c1">X</font>;<br/>

<b>reconsider </b><font color="Maroon" title="c5">D</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">Y0</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>assume </b><a NAME="E2:95_1_1"/>
<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="c4">Y0</font>
 ;<br/>

<a NAME="E3:95_1_1"/><b>then </b>
( <font color="Maroon" title="c3">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">D</font> &amp; <font color="Maroon" title="c5">D</font> is <a href="tex_2.html#V3" title="TEX_2:attr.3">discrete</a> )
 
<b>by </b><i><a class="ref" href="tex_2.html#T26" target="_self" title="TEX_2:th.26">Th26</a>, <a class="ref" href="tsep_1.html#T4" title="TSEP_1:th.4">TSEP_1:4</a></i>;<br/>
<a NAME="E4:95_1_1"/><b>then </b>
<font color="Maroon" title="c3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">D</font>
 
<b>by </b><i><a class="txt" href="tex_2.html#E2:95_1"><i><font color="Green" title="E41">A1</font></i></a>, <a class="ref" href="tex_2.html#D7" target="_self" title="TEX_2:def.7">Def7</a></i>;<br/>
<b>hence </b><a NAME="E5:95_1_1"/>
 <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">X0</font>,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c2">X0</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">Y0</font>,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c4">Y0</font> #)
 <b>by </b><i><a class="ref" href="tsep_1.html#T5" title="TSEP_1:th.5">TSEP_1:5</a></i>;<br/>


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

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:95_2"/>
<font color="Maroon" title="c2">X0</font> is <a href="tdlat_3.html#V1" title="TDLAT_3:attr.1">discrete</a>
 ;<br/><a NAME="E2:95_2"/><b>then </b><i><font color="Green" title="E41">A2</font></i>: 
<font color="Maroon" title="c3">A</font> is <a href="tex_2.html#V3" title="TEX_2:attr.3">discrete</a>
 <b>by </b><i><a class="ref" href="tex_2.html#T26" target="_self" title="TEX_2:th.26">Th26</a></i>;<br/><b>assume </b><a NAME="E3:95_2"/><i><font color="Green" title="E42">A3</font></i>: 
for <font color="Olive" title="b1">Y0</font> being non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="tdlat_3.html#V1" title="TDLAT_3:attr.1">discrete</a>  <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c1">X</font>  st <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">Y0</font> holds <br/> <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">X0</font>,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c2">X0</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">Y0</font>,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Olive" title="b1">Y0</font> #)
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="95_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">D</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>assume </b><a NAME="E1:95_2_1"/><i><font color="Green" title="E43">A4</font></i>: 
<font color="Maroon" title="c4">D</font> is <a href="tex_2.html#V3" title="TEX_2:attr.3">discrete</a>
 ;<br/><b>assume </b><a NAME="E2:95_2_1"/><i><font color="Green" title="E44">A5</font></i>: 
<font color="Maroon" title="c3">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">D</font>
 ;<br/><a NAME="E3:95_2_1"/><b>then </b>
<font color="Maroon" title="c4">D</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> 
 <b>by </b><i><a class="ref" href="xboole_1.html#T3" title="XBOOLE_1:th.3">XBOOLE_1:3</a></i>;<br/><b>then consider </b><font color="Maroon" title="c5">Y0</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="tdlat_3.html#V1" title="TDLAT_3:attr.1">discrete</a>  <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c1">X</font><b> such that </b><br/><a NAME="E5:95_2_1"/><i><font color="Green" title="E45">A6</font></i>: 
<font color="Maroon" title="c4">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="c5">Y0</font>
 <b>by </b><i><a class="txt" href="tex_2.html#E1:95_2_1"><i><font color="Green" title="E43">A4</font></i></a>, <a class="ref" href="tex_2.html#T34" target="_self" title="TEX_2:th.34">Th34</a></i>;<br/><a NAME="E6:95_2_1"/>
<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="c5">Y0</font>
 <b>by </b><i><a class="txt" href="tex_2.html#E2:95_2_1"><i><font color="Green" title="E44">A5</font></i></a>, <a class="txt" href="tex_2.html#E5:95_2_1"><i><font color="Green" title="E45">A6</font></i></a>, <a class="ref" href="tsep_1.html#T4" title="TSEP_1:th.4">TSEP_1:4</a></i>;<br/><b>hence </b><a NAME="E7:95_2_1"/>
<font color="Maroon" title="c3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">D</font>
 <b>by </b><i><a class="txt" href="tex_2.html#E3:95_2"><i><font color="Green" title="E42">A3</font></i></a>, <a class="txt" href="tex_2.html#E5:95_2_1"><i><font color="Green" title="E45">A6</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E5:95_2"/><b>then </b>
<font color="Maroon" title="c3">A</font> is <a href="tex_2.html#V4" title="TEX_2:attr.4">maximal_discrete</a>
 <b>by </b><i><a class="txt" href="tex_2.html#E2:95_2"><i><font color="Green" title="E41">A2</font></i></a>, <a class="ref" href="tex_2.html#D7" target="_self" title="TEX_2:def.7">Def7</a></i>;<br/><b>hence </b><a NAME="E6:95_2"/>
<font color="Maroon" title="c2">X0</font> is <a href="tex_2.html#V5" title="TEX_2:attr.5">maximal_discrete</a>
 <b>by </b><i><a class="ref" href="tex_2.html#T51" target="_self" title="TEX_2:th.51">Th51</a></i>;<br/></div>
<b>end;</b></div>


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