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<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:44"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<b>set </b><font color="Maroon">c<sub>3</sub></font> =  <a href="topgen_2.html#K5" target="_self">DiscrWithInfin</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>;<br/>
<a NAME="E2:44"/>
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="topgen_2.html#K5" target="_self">DiscrWithInfin</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="topgen_2.html#D5" target="_self">Def5</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>4</sub></font> =  <a href="topgen_2.html#K5" target="_self">DiscrWithInfin</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> as  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a> <b>by </b><i><a class="txt" href="topgen_2.html#E1:44"><i><font color="Green">E29</font></i></a>, <a class="ref" href="struct_0.html#D1">STRUCT_0:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>2</sub></font> as   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>4</sub></font> <b>by </b><i><a class="txt" href="topgen_2.html#E1:44"><i><font color="Green">E29</font></i></a>, <a class="ref" href="topgen_2.html#D5" target="_self">Def5</a></i>;<br/>
<b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topgen_2.html#K5" target="_self">DiscrWithInfin</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>;<br/>

<b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K6">}</a></span> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>4</sub></font> ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>6</sub></font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>4</sub></font> ;<br/>
<a NAME="E7:44"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E8:44"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E9:44"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>7</sub></font> is <a href="pre_topc.html#V4">closed</a>
 
<b>by </b><i><a class="ref" href="topgen_2.html#T22" target="_self">Th22</a></i>;<br/>
<a NAME="E10:44"/><b>then </b><i><font color="Green">E30</font></i>: 
 <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T52">PRE_TOPC:52</a></i>;<br/>
<b>assume </b><a NAME="E11:44"/>
not <font color="Maroon">c<sub>6</sub></font> is <a href="finset_1.html#V1">finite</a>
 ;<br/>

<a NAME="E12:44"/><b>then </b>
(  not <font color="Maroon">c<sub>8</sub></font> is <a href="pre_topc.html#V4">closed</a> or <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> )
 
<b>by </b><i><a class="txt" href="topgen_2.html#E1:44"><i><font color="Green">E29</font></i></a>, <a class="ref" href="topgen_2.html#T22" target="_self">Th22</a></i>;<br/>
<b>hence </b><a NAME="E13:44"/>
 <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="topgen_2.html#E10:44"><i><font color="Green">E30</font></i></a>, <a class="ref" href="topgen_2.html#T27" target="_self">Th27</a>, <a class="ref" href="zfmisc_1.html#T46">ZFMISC_1:46</a></i>;<br/>


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