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<div class="add">

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:51_1_1_1"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is <a href="tdlat_3.html#V1">discrete</a>
 ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>assume </b><a NAME="E2:51_1_1_1"/>
<font color="Maroon">c<sub>2</sub></font> is <a href="tops_1.html#V2">boundary</a>
 ;<br/><a NAME="E3:51_1_1_1"/><b>then </b>
 <a href="tops_1.html#K1">Int</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="tops_1.html#T84">TOPS_1:84</a></i>;<br/><a NAME="E4:51_1_1_1"/><b>then </b>
not <font color="Maroon">c<sub>2</sub></font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i><a class="ref" href="tops_1.html#T55">TOPS_1:55</a></i>;<br/><b>hence </b><a NAME="E5:51_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="tex_1.html#E1:51_1_1_1"><i><font color="Green">E22</font></i></a>, <a class="ref" href="tdlat_3.html#T17">TDLAT_3:17</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:51_1_1"/><i><font color="Green">E22</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> holds not <font color="Olive">b<sub>1</sub></font> is <a href="tops_1.html#V2">boundary</a>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>assume </b><a NAME="E1:51_1_1_2"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="domain_1.html#K6">}</a></span>
 ;<br/><div><b>hereby </b>
<div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:51_1_1_2_1_1"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:51_1_1_2_1_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:51_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K1">{}</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><b>hence </b><a NAME="E3:51_1_1_2_1_1_1"/>
<font color="Maroon">c<sub>2</sub></font> is <a href="pre_topc.html#V3">open</a>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:51_1_1_2_1_1_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:51_1_1_2_1_1_2"/><b>then </b>
not <font color="Maroon">c<sub>2</sub></font> is <a href="tops_1.html#V2">boundary</a>
 <b>by </b><i><a class="txt" href="tex_1.html#E2:51_1_1"><i><font color="Green">E22</font></i></a></i>;<br/><a NAME="E3:51_1_1_2_1_1_2"/><b>then </b>
(  <a href="tops_1.html#K1">Int</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp;  <a href="tops_1.html#K1">Int</a> <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="domain_1.html#K6">}</a></span> )
 <b>by </b><i><a class="txt" href="tex_1.html#E1:51_1_1_2"><i><font color="Green">E23</font></i></a>, <a class="ref" href="tops_1.html#T44">TOPS_1:44</a>, <a class="ref" href="tops_1.html#T84">TOPS_1:84</a></i>;<br/><b>hence </b><a NAME="E4:51_1_1_2_1_1_2"/>
<font color="Maroon">c<sub>2</sub></font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i><a class="txt" href="tex_1.html#E1:51_1_1_2"><i><font color="Green">E23</font></i></a>, <a class="ref" href="zfmisc_1.html#T39">ZFMISC_1:39</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div>
<b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E4:51_1_1"/>
<font color="Maroon">c<sub>1</sub></font> is <a href="tdlat_3.html#V1">discrete</a>
 <b>by </b><i><a class="ref" href="tdlat_3.html#T19">TDLAT_3:19</a></i>;<br/>


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