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<div class="add">

<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/>
<b>thus </b><a NAME="E2:19_1_1"/>
( <font color="Maroon" title="c2">X0</font> is <a href="tsp_2.html#V2" title="TSP_2:attr.2">maximal_T_0</a> implies ( <font color="Maroon" title="c2">X0</font> is <a href="t_0topsp.html#V2" title="T_0TOPSP:attr.2">T_0</a> &amp; ( for <font color="Olive" title="b1">Y0</font> being non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> <a href="t_0topsp.html#V2" title="T_0TOPSP:attr.2">V80</a>  <a href="tsp_1.html#M2" title="TSP_1:mode.2">SubSpace</a> of <font color="Maroon" title="c1">X</font>  st <font color="Maroon" title="c2">X0</font> is    <a href="tsp_1.html#M2" title="TSP_1:mode.2">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="19_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:19_1_1_1"/>
<font color="Maroon" title="c2">X0</font> is <a href="tsp_2.html#V2" title="TSP_2:attr.2">maximal_T_0</a>
 ;<br/>

<a NAME="E2:19_1_1_1"/><b>then </b><i><font color="Green" title="E16">A1</font></i>: 
<font color="Maroon" title="c3">A</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/>
<a NAME="E3:19_1_1_1"/><b>then </b>
<font color="Maroon" title="c3">A</font> is <a href="tsp_1.html#V1" title="TSP_1:attr.1">T_0</a>
 
<b>by </b><i><a class="ref" href="tsp_2.html#D4" target="_self" title="TSP_2:def.4">Def4</a></i>;<br/>
<b>hence </b><a NAME="E4:19_1_1_1"/>
<font color="Maroon" title="c2">X0</font> is <a href="t_0topsp.html#V2" title="T_0TOPSP:attr.2">T_0</a>
 <b>by </b><i><a class="ref" href="tsp_1.html#T15" title="TSP_1:th.15">TSP_1:15</a></i>;<br/>

<b>thus </b><a NAME="E5:19_1_1_1"/>
for <font color="Olive" title="b1">Y0</font> being non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> <a href="t_0topsp.html#V2" title="T_0TOPSP:attr.2">V80</a>  <a href="tsp_1.html#M2" title="TSP_1:mode.2">SubSpace</a> of <font color="Maroon" title="c1">X</font>  st <font color="Maroon" title="c2">X0</font> is    <a href="tsp_1.html#M2" title="TSP_1:mode.2">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="19_1_1_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#V3" title="STRUCT_0:attr.3">empty</a> <a href="t_0topsp.html#V2" title="T_0TOPSP:attr.2">V80</a>  <a href="tsp_1.html#M2" title="TSP_1:mode.2">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="E2:19_1_1_1_1"/>
<font color="Maroon" title="c2">X0</font> is    <a href="tsp_1.html#M2" title="TSP_1:mode.2">SubSpace</a> of <font color="Maroon" title="c4">Y0</font>
 ;<br/>

<a NAME="E3:19_1_1_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="tsp_1.html#V1" title="TSP_1:attr.1">T_0</a> )
 
<b>by </b><i><a class="ref" href="tsep_1.html#T4" title="TSEP_1:th.4">TSEP_1:4</a>, <a class="ref" href="tsp_1.html#T15" title="TSP_1:th.15">TSP_1:15</a></i>;<br/>
<a NAME="E4:19_1_1_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="tsp_2.html#E2:19_1_1_1"><i><font color="Green" title="E16">A1</font></i></a>, <a class="ref" href="tsp_2.html#D4" target="_self" title="TSP_2:def.4">Def4</a></i>;<br/>
<b>hence </b><a NAME="E5:19_1_1_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>

<b>assume </b><a NAME="E3:19_1_1"/>
<font color="Maroon" title="c2">X0</font> is <a href="t_0topsp.html#V2" title="T_0TOPSP:attr.2">T_0</a>
 ;<br/>

<a NAME="E4:19_1_1"/><b>then </b><i><font color="Green" title="E16">A2</font></i>: 
<font color="Maroon" title="c3">A</font> is <a href="tsp_1.html#V1" title="TSP_1:attr.1">T_0</a>
 
<b>by </b><i><a class="ref" href="tsp_1.html#T15" title="TSP_1:th.15">TSP_1:15</a></i>;<br/>
<b>assume </b><a NAME="E5:19_1_1"/><i><font color="Green" title="E17">A3</font></i>: 
for <font color="Olive" title="b1">Y0</font> being non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> <a href="t_0topsp.html#V2" title="T_0TOPSP:attr.2">V80</a>  <a href="tsp_1.html#M2" title="TSP_1:mode.2">SubSpace</a> of <font color="Maroon" title="c1">X</font>  st <font color="Maroon" title="c2">X0</font> is    <a href="tsp_1.html#M2" title="TSP_1:mode.2">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="19_1_1_2"><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="E1:19_1_1_2"/><i><font color="Green" title="E18">A4</font></i>: 
<font color="Maroon" title="c4">D</font> is <a href="tsp_1.html#V1" title="TSP_1:attr.1">T_0</a>
 ;<br/><b>assume </b><a NAME="E2:19_1_1_2"/><i><font color="Green" title="E19">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="E3:19_1_1_2"/><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#V3" title="STRUCT_0:attr.3">empty</a> <a href="pre_topc.html#V1" title="PRE_TOPC:attr.1">strict</a> <a href="t_0topsp.html#V2" title="T_0TOPSP:attr.2">V80</a>  <a href="tsp_1.html#M2" title="TSP_1:mode.2">SubSpace</a> of <font color="Maroon" title="c1">X</font><b> such that </b><br/><a NAME="E5:19_1_1_2"/><i><font color="Green" title="E20">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="tsp_2.html#E1:19_1_1_2"><i><font color="Green" title="E18">A4</font></i></a>, <a class="ref" href="tsp_1.html#T20" title="TSP_1:th.20">TSP_1:20</a></i>;<br/><a NAME="E6:19_1_1_2"/>
<font color="Maroon" title="c2">X0</font> is    <a href="tsp_1.html#M2" title="TSP_1:mode.2">SubSpace</a> of <font color="Maroon" title="c5">Y0</font>
 <b>by </b><i><a class="txt" href="tsp_2.html#E2:19_1_1_2"><i><font color="Green" title="E19">A5</font></i></a>, <a class="txt" href="tsp_2.html#E5:19_1_1_2"><i><font color="Green" title="E20">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="E7:19_1_1_2"/>
<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="tsp_2.html#E5:19_1_1"><i><font color="Green" title="E17">A3</font></i></a>, <a class="txt" href="tsp_2.html#E5:19_1_1_2"><i><font color="Green" title="E20">A6</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E7:19_1_1"/><b>then </b>
<font color="Maroon" title="c3">A</font> is <a href="tsp_2.html#V1" title="TSP_2:attr.1">maximal_T_0</a>
 
<b>by </b><i><a class="txt" href="tsp_2.html#E4:19_1_1"><i><font color="Green" title="E16">A2</font></i></a>, <a class="ref" href="tsp_2.html#D4" target="_self" title="TSP_2:def.4">Def4</a></i>;<br/>
<b>hence </b><a NAME="E8:19_1_1"/>
<font color="Maroon" title="c2">X0</font> is <a href="tsp_2.html#V2" title="TSP_2:attr.2">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/>


</div>
