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<b>set </b><font color="Maroon">c<sub>2</sub></font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="openlatt.html#K9" target="_self">StoneS</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> : verum </span>}</span> ;<br/>
<b>set </b><font color="Maroon">c<sub>3</sub></font> =  <a href="openlatt.html#K7" target="_self">F_primeSet</a> <font color="Maroon">c<sub>1</sub></font>;<br/>
<a NAME="E1:51_1_1"/><i><font color="Green">E41</font></i>: 
 <a href="openlatt.html#K9" target="_self">StoneS</a> <font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a>  <a href="zfmisc_1.html#K1">bool</a> <span class="p1">(<span class="default"><a href="openlatt.html#K7" target="_self">F_primeSet</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="openlatt.html#T35" target="_self">Th35</a></i>;<br/>
<a NAME="E2:51_1_1"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="openlatt.html#K9" target="_self">StoneS</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> : verum </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="zfmisc_1.html#K1">bool</a> <span class="p1">(<span class="default"><a href="openlatt.html#K7" target="_self">F_primeSet</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_1_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="E1:51_1_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="openlatt.html#K9" target="_self">StoneS</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> : verum </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="openlatt.html#K9" target="_self">StoneS</a> <font color="Maroon">c<sub>1</sub></font></span>)</span><b> such that </b><br/><a NAME="E3:51_1_1_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/>
<a NAME="E4:51_1_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a>  <a href="zfmisc_1.html#K1">bool</a> <span class="p1">(<span class="default"><a href="openlatt.html#K7" target="_self">F_primeSet</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="openlatt.html#E1:51_1_1"><i><font color="Green">E41</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E5:51_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> <span class="p2">(<span class="default"><a href="openlatt.html#K7" target="_self">F_primeSet</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="openlatt.html#E3:51_1_1_1"><i><font color="Green">E42</font></i></a>, <a class="ref" href="zfmisc_1.html#T95">ZFMISC_1:95</a></i>;<br/>
<a NAME="E6:51_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> is   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="openlatt.html#K7" target="_self">F_primeSet</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T99">ZFMISC_1:99</a></i>;<br/>
<b>hence </b><a NAME="E7:51_1_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> <span class="p1">(<span class="default"><a href="openlatt.html#K7" target="_self">F_primeSet</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>4</sub></font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="openlatt.html#K9" target="_self">StoneS</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> : verum </span>}</span>  as   <a href="setfam_1.html#NM1">Subset-Family</a> of <span class="p1">(<span class="default"><a href="openlatt.html#K7" target="_self">F_primeSet</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> ;<br/>
<b>take </b>
 <a href="pre_topc.html#G1">TopStruct</a>(# <span class="p1">(<span class="default"><a href="openlatt.html#K7" target="_self">F_primeSet</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font> #)
;<br/>

<b>thus </b><a NAME="E4:51_1_1"/>
( the <a href="struct_0.html#U1">carrier</a> of <a href="pre_topc.html#G1">TopStruct</a>(# <span class="p1">(<span class="default"><a href="openlatt.html#K7" target="_self">F_primeSet</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font> #) <a href="hidden.html#R1">=</a>  <a href="openlatt.html#K7" target="_self">F_primeSet</a> <font color="Maroon">c<sub>1</sub></font> &amp; the <a href="pre_topc.html#U1">topology</a> of <a href="pre_topc.html#G1">TopStruct</a>(# <span class="p1">(<span class="default"><a href="openlatt.html#K7" target="_self">F_primeSet</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font> #) <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="openlatt.html#K9" target="_self">StoneS</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> : verum </span>}</span>  )
 ;<br/>


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