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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="tdlat_3.html#V1" title="TDLAT_3:attr.1">discrete</a> <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>;<br/><b>let </b><font color="Maroon" title="c2">A</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>;<br/>



<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:89_1"/><i><font color="Green" title="E38">A1</font></i>: 
<font color="Maroon" title="c2">A</font> is <a href="tex_2.html#V4" title="TEX_2:attr.4">maximal_discrete</a>
 ;<br/><a NAME="E2:89_1"/>
<font color="Maroon" title="c1">X</font> is    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="ref" href="tsep_1.html#T2" title="TSEP_1:th.2">TSEP_1:2</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c3">C</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</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/><a NAME="E4:89_1"/>
( <font color="Maroon" title="c3">C</font> is <a href="tex_2.html#V3" title="TEX_2:attr.3">discrete</a> &amp; <font color="Maroon" title="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">C</font> )
 <b>by </b><i><a class="ref" href="tex_2.html#T27" target="_self" title="TEX_2:th.27">Th27</a></i>;<br/><a NAME="E5:89_1"/><b>then </b>
<font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">C</font>
 <b>by </b><i><a class="txt" href="tex_2.html#E1:89_1"><i><font color="Green" title="E38">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="E6:89_1"/>
not <font color="Maroon" title="c2">A</font> is <a href="tex_2.html#V1" title="TEX_2:attr.1">proper</a>
 <b>by </b><i><a class="ref" href="tex_2.html#T5" target="_self" title="TEX_2:th.5">Th5</a></i>;<br/></div>
<b>end;</b></div>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:89_2"/>
not <font color="Maroon" title="c2">A</font> is <a href="tex_2.html#V1" title="TEX_2:attr.1">proper</a>
 ;<br/><a NAME="E2:89_2"/><b>then </b><i><font color="Green" title="E38">A2</font></i>: 
<font color="Maroon" title="c2">A</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>
 <b>by </b><i><a class="ref" href="tex_2.html#T5" target="_self" title="TEX_2:th.5">Th5</a></i>;<br/><a NAME="E3:89_2"/><b>then </b><i><font color="Green" title="E39">A3</font></i>: 
<font color="Maroon" title="c2">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#T27" target="_self" title="TEX_2:th.27">Th27</a></i>;<br/><a NAME="E4:89_2"/>
for <font color="Olive" title="b1">D</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">D</font> is <a href="tex_2.html#V3" title="TEX_2:attr.3">discrete</a> &amp; <font color="Maroon" title="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">D</font> holds <br/><font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">D</font>
 <b>by </b><i><a class="txt" href="tex_2.html#E2:89_2"><i><font color="Green" title="E38">A2</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/><b>hence </b><a NAME="E5:89_2"/>
<font color="Maroon" title="c2">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#E3:89_2"><i><font color="Green" title="E39">A3</font></i></a>, <a class="ref" href="tex_2.html#D7" target="_self" title="TEX_2:def.7">Def7</a></i>;<br/></div>
<b>end;</b></div>


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