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<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be  non <a href="struct_0.html#V3">empty</a>  <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<b>reconsider </b><font color="Maroon">c<sub>3</sub></font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="tsep_1.html#T1">TSEP_1:1</a></i>;<br/>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:96_1"/>
<font color="Maroon">c<sub>2</sub></font> is <a href="tex_2.html#V5" target="_self">maximal_discrete</a>
 ;<br/><a NAME="E2:96_1"/><b>then </b><i><font color="Green">E41</font></i>: 
<font color="Maroon">c<sub>3</sub></font> is <a href="tex_2.html#V4" target="_self">maximal_discrete</a>
 <b>by </b><i><a class="ref" href="tex_2.html#T51" target="_self">Th51</a></i>;<br/><a NAME="E3:96_1"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> is <a href="tex_2.html#V3" target="_self">discrete</a>
 <b>by </b><i><a class="ref" href="tex_2.html#D7" target="_self">Def7</a></i>;<br/><b>hence </b><a NAME="E4:96_1"/>
<font color="Maroon">c<sub>2</sub></font> is <a href="tdlat_3.html#V1">discrete</a>
 <b>by </b><i><a class="ref" href="tex_2.html#T26" target="_self">Th26</a></i>;<br/><b>thus </b><a NAME="E5:96_1"/>
for <font color="Olive">b<sub>1</sub></font> being non <a href="struct_0.html#V3">empty</a> <a href="tdlat_3.html#V1">discrete</a>  <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Maroon">c<sub>2</sub></font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Olive">b<sub>1</sub></font> holds <br/> <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>2</sub></font> #) <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">b<sub>1</sub></font>,the <a href="pre_topc.html#U1">topology</a> of <font color="Olive">b<sub>1</sub></font> #)
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="96_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font> be  non <a href="struct_0.html#V3">empty</a> <a href="tdlat_3.html#V1">discrete</a>  <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>reconsider </b><font color="Maroon">c<sub>5</sub></font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>4</sub></font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="tsep_1.html#T1">TSEP_1:1</a></i>;<br/>
<b>assume </b><a NAME="E2:96_1_1"/>
<font color="Maroon">c<sub>2</sub></font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<a NAME="E3:96_1_1"/><b>then </b>
( <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> is <a href="tex_2.html#V3" target="_self">discrete</a> )
 
<b>by </b><i><a class="ref" href="tex_2.html#T26" target="_self">Th26</a>, <a class="ref" href="tsep_1.html#T4">TSEP_1:4</a></i>;<br/>
<a NAME="E4:96_1_1"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="tex_2.html#E2:96_1"><i><font color="Green">E41</font></i></a>, <a class="ref" href="tex_2.html#D7" target="_self">Def7</a></i>;<br/>
<b>hence </b><a NAME="E5:96_1_1"/>
 <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>2</sub></font> #) <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>4</sub></font>,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>4</sub></font> #)
 <b>by </b><i><a class="ref" href="tsep_1.html#T5">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="E1:96_2"/>
<font color="Maroon">c<sub>2</sub></font> is <a href="tdlat_3.html#V1">discrete</a>
 ;<br/><a NAME="E2:96_2"/><b>then </b><i><font color="Green">E41</font></i>: 
<font color="Maroon">c<sub>3</sub></font> is <a href="tex_2.html#V3" target="_self">discrete</a>
 <b>by </b><i><a class="ref" href="tex_2.html#T26" target="_self">Th26</a></i>;<br/><b>assume </b><a NAME="E3:96_2"/><i><font color="Green">E42</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being non <a href="struct_0.html#V3">empty</a> <a href="tdlat_3.html#V1">discrete</a>  <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Maroon">c<sub>2</sub></font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Olive">b<sub>1</sub></font> holds <br/> <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>2</sub></font> #) <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">b<sub>1</sub></font>,the <a href="pre_topc.html#U1">topology</a> of <font color="Olive">b<sub>1</sub></font> #)
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="96_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>assume </b><a NAME="E1:96_2_1"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">c<sub>4</sub></font> is <a href="tex_2.html#V3" target="_self">discrete</a>
 ;<br/><b>assume </b><a NAME="E2:96_2_1"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><a NAME="E3:96_2_1"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="xboole_1.html#T3">XBOOLE_1:3</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>5</sub></font> being  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#V1">strict</a> <a href="tdlat_3.html#V1">discrete</a>  <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E5:96_2_1"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="tex_2.html#E1:96_2_1"><i><font color="Green">E43</font></i></a>, <a class="ref" href="tex_2.html#T34" target="_self">Th34</a></i>;<br/><a NAME="E6:96_2_1"/>
<font color="Maroon">c<sub>2</sub></font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="tex_2.html#E2:96_2_1"><i><font color="Green">E44</font></i></a>, <a class="txt" href="tex_2.html#E5:96_2_1"><i><font color="Green">E45</font></i></a>, <a class="ref" href="tsep_1.html#T4">TSEP_1:4</a></i>;<br/><b>hence </b><a NAME="E7:96_2_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="tex_2.html#E3:96_2"><i><font color="Green">E42</font></i></a>, <a class="txt" href="tex_2.html#E5:96_2_1"><i><font color="Green">E45</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E5:96_2"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> is <a href="tex_2.html#V4" target="_self">maximal_discrete</a>
 <b>by </b><i><a class="txt" href="tex_2.html#E2:96_2"><i><font color="Green">E41</font></i></a>, <a class="ref" href="tex_2.html#D7" target="_self">Def7</a></i>;<br/><b>hence </b><a NAME="E6:96_2"/>
<font color="Maroon">c<sub>2</sub></font> is <a href="tex_2.html#V5" target="_self">maximal_discrete</a>
 <b>by </b><i><a class="ref" href="tex_2.html#T51" target="_self">Th51</a></i>;<br/></div>
<b>end;</b></div>


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