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<b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="pre_topc.html#NM1">TopSpace</a>;<br/>

<b>set </b><font color="Maroon">c<sub>6</sub></font> = <span class="p1"><a href="tarski.html#K1">{</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>5</sub></font></span><a href="tarski.html#K1">}</a></span>;<br/>
<a NAME="E1:14"/>
<span class="p1"><a href="tarski.html#K1">{</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>5</sub></font></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="pcomps_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>5</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>7</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="E1:14_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>5</sub></font></span><a href="tarski.html#K1">}</a></span>
 ;<br/>

<a NAME="E2:14_1"/><b>then </b>
<font color="Maroon">c<sub>7</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="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E3:14_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="pcomps_1.html#K1">bool</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="ref" href="zfmisc_1.html#D1">ZFMISC_1:def 1</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>7</sub></font> = <span class="p1"><a href="tarski.html#K1">{</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>5</sub></font></span><a href="tarski.html#K1">}</a></span> as   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">c<sub>5</sub></font> ;<br/>
<b>take </b>
<font color="Maroon">c<sub>7</sub></font>
;<br/>

<b>thus </b><a NAME="E3:14"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>5</sub></font></span><a href="tarski.html#K1">}</a></span>
 ;<br/>

<a NAME="E4:14"/><i><font color="Green">E6</font></i>: 
 <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>5</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>
 
;<br/>
<b>thus </b><a NAME="E5:14"/>
<font color="Maroon">c<sub>7</sub></font> <a href="pre_topc.html#R1">is_a_cover_of</a> <font color="Maroon">c<sub>5</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:14_2"/>
 <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T31">ZFMISC_1:31</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="pre_topc.html#D8">PRE_TOPC:def 8</a><br/>


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

<b>let </b><font color="Maroon">c<sub>8</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>5</sub></font>; <i><font color="Red">:: according to </font></i><a class="ref" href="tops_2.html#D1">TOPS_2:def 1</a><br/>

<b>thus </b><a NAME="E6:14"/>
(  not <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> or <font color="Maroon">c<sub>8</sub></font> is <a href="pre_topc.html#V3">open</a> )
 <b>by </b><i><a class="txt" href="topalg_3.html#E4:14"><i><font color="Green">E6</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>


</div>
