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<div class="add">

<b>let </b><font color="Maroon">c<sub>2</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>set </b><font color="Maroon">c<sub>3</sub></font> = <a href="arytm_3.html#K13">{}</a>  <a href="topgen_3.html#K7" target="_self">-DiscreteTop</a> <font color="Maroon">c<sub>2</sub></font>;<br/>
<a NAME="E1:66"/><i><font color="Green">E48</font></i>: 
 <a href="tex_1.html#K2">ADTS</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#G1">TopStruct</a>(# <font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><a href="tex_1.html#K1">cobool</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> #)
 
<b>by </b><i><a class="ref" href="tex_1.html#D3">TEX_1:def 3</a></i>;<br/>
<a NAME="E2:66"/><i><font color="Green">E49</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="arytm_3.html#K13">{}</a>  <a href="topgen_3.html#K7" target="_self">-DiscreteTop</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="topgen_3.html#D8" target="_self">Def8</a></i>;<br/>
<a NAME="E3:66"/><i><font color="Green">E50</font></i>: 
 <a href="tex_1.html#K1">cobool</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><a href="arytm_3.html#K13">{}</a> ,<font color="Maroon">c<sub>2</sub></font></span><a href="tarski.html#K2">}</a></span>
 
<b>by </b><i><a class="ref" href="tex_1.html#D2">TEX_1:def 2</a></i>;<br/>
<a NAME="E4:66"/><i><font color="Green">E51</font></i>: 
(  <a href="pre_topc.html#K1">{}</a> <span class="p1">(<span class="default"><a href="arytm_3.html#K13">{}</a>  <a href="topgen_3.html#K7" target="_self">-DiscreteTop</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="arytm_3.html#K13">{}</a>  &amp;  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><a href="arytm_3.html#K13">{}</a>  <a href="topgen_3.html#K7" target="_self">-DiscreteTop</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="txt" href="topgen_3.html#E2:66"><i><font color="Green">E49</font></i></a></i>;<br/>
<a NAME="E5:66"/>
the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><a href="arytm_3.html#K13">{}</a>  <a href="topgen_3.html#K7" target="_self">-DiscreteTop</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="tex_1.html#K1">cobool</a> <font color="Maroon">c<sub>2</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:66_1"/>
the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><a href="arytm_3.html#K13">{}</a>  <a href="topgen_3.html#K7" target="_self">-DiscreteTop</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="tex_1.html#K1">cobool</a> <font color="Maroon">c<sub>2</sub></font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:66_1_1"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><a href="arytm_3.html#K13">{}</a>  <a href="topgen_3.html#K7" target="_self">-DiscreteTop</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</sub></font> as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="arytm_3.html#K13">{}</a>  <a href="topgen_3.html#K7" target="_self">-DiscreteTop</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> ;<br/>
<a NAME="E3:66_1_1"/>
<font color="Maroon">c<sub>5</sub></font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="txt" href="topgen_3.html#E1:66_1_1"><i><font color="Green">E52</font></i></a>, <a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/>
<a NAME="E4:66_1_1"/><b>then </b>
( ( <font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> is <a href="xboole_0.html#V1">empty</a> ) or  not <font color="Maroon">c<sub>5</sub></font> is <a href="tex_2.html#V1">proper</a> or <font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a>  <a href="arytm_3.html#K13">{}</a>  )
 
<b>by </b><i><a class="txt" href="topgen_3.html#E2:66"><i><font color="Green">E49</font></i></a>, <a class="ref" href="topgen_3.html#T43" target="_self">Th43</a></i>;<br/>
<a NAME="E5:66_1_1"/><b>then </b>
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="arytm_3.html#K13">{}</a>  or <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="txt" href="topgen_3.html#E2:66"><i><font color="Green">E49</font></i></a>, <a class="ref" href="tex_2.html#T5">TEX_2:5</a>, <a class="ref" href="xboole_1.html#T3">XBOOLE_1:3</a></i>;<br/>
<b>hence </b><a NAME="E6:66_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="tex_1.html#K1">cobool</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="topgen_3.html#E3:66"><i><font color="Green">E50</font></i></a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>


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

<a NAME="E2:66_1"/>
(  <a href="arytm_3.html#K13">{}</a>  <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><a href="arytm_3.html#K13">{}</a>  <a href="topgen_3.html#K7" target="_self">-DiscreteTop</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><a href="arytm_3.html#K13">{}</a>  <a href="topgen_3.html#K7" target="_self">-DiscreteTop</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> )
 
<b>by </b><i><a class="txt" href="topgen_3.html#E4:66"><i><font color="Green">E51</font></i></a>, <a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/>
<b>hence </b><a NAME="E3:66_1"/>
 <a href="tex_1.html#K1">cobool</a> <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><a href="arytm_3.html#K13">{}</a>  <a href="topgen_3.html#K7" target="_self">-DiscreteTop</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="topgen_3.html#E3:66"><i><font color="Green">E50</font></i></a>, <a class="ref" href="zfmisc_1.html#T38">ZFMISC_1:38</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:66"/>
 <a href="tex_1.html#K2">ADTS</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <a href="arytm_3.html#K13">{}</a>  <a href="topgen_3.html#K7" target="_self">-DiscreteTop</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="topgen_3.html#E1:66"><i><font color="Green">E48</font></i></a>, <a class="txt" href="topgen_3.html#E2:66"><i><font color="Green">E49</font></i></a></i>;<br/>


</div>
