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<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="realset2.html#V3">trivial</a> <a href="lattices.html#NM8">B_Lattice</a>;<br/>

<b>consider </b><font color="Maroon">c<sub>2</sub></font> being    <a href="subset_1.html#M2">Element</a> of  <a href="lopclset.html#K1" target="_self">OpenClosedSet</a> <span class="p1">(<span class="default"><a href="lopclset.html#K11" target="_self">StoneSpace</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>3</sub></font> = <font color="Maroon">c<sub>2</sub></font> as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="lopclset.html#K11" target="_self">StoneSpace</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> ;<br/>
<a NAME="E3:33"/><i><font color="Green">E29</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">c<sub>3</sub></font> is <a href="pre_topc.html#V4">closed</a> )
 
<b>by </b><i><a class="ref" href="lopclset.html#T2" target="_self">Th2</a>, <a class="ref" href="lopclset.html#T3" target="_self">Th3</a></i>;<br/>
<a NAME="E4:33"/><b>then </b>
<font color="Maroon">c<sub>3</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="lopclset.html#K11" target="_self">StoneSpace</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/>
<a NAME="E5:33"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="setfam_1.html#K5">union</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="setfam_1.html#NM1">Subset-Family</a> of <span class="p1">(<span class="default"><a href="lopclset.html#K7" target="_self">ultraset</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> : <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1">c=</a>  <a href="lopclset.html#K10" target="_self">StoneR</a> <font color="Maroon">c<sub>1</sub></font> </span>}</span> 
 
<b>by </b><i><a class="ref" href="lopclset.html#D8" target="_self">Def8</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>4</sub></font> being   <a href="setfam_1.html#NM1">Subset-Family</a> of <span class="p1">(<span class="default"><a href="lopclset.html#K7" target="_self">ultraset</a> <font color="Maroon">c<sub>1</sub></font></span>)</span><b> such that </b><br/><a NAME="E7:33"/><i><font color="Green">E30</font></i>: 
(  <a href="setfam_1.html#K5">union</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a>  <a href="lopclset.html#K10" target="_self">StoneR</a> <font color="Maroon">c<sub>1</sub></font> )
 ;<br/>
<a NAME="E8:33"/>
<font color="Maroon">c<sub>3</sub></font> <a href="subset_1.html#K3">`</a>  is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="txt" href="lopclset.html#E3:33"><i><font color="Green">E29</font></i></a>, <a class="ref" href="tops_1.html#T29">TOPS_1:29</a></i>;<br/>
<a NAME="E9:33"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="subset_1.html#K3">`</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="lopclset.html#K11" target="_self">StoneSpace</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/>
<a NAME="E10:33"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="setfam_1.html#K5">union</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="setfam_1.html#NM1">Subset-Family</a> of <span class="p1">(<span class="default"><a href="lopclset.html#K7" target="_self">ultraset</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> : <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1">c=</a>  <a href="lopclset.html#K10" target="_self">StoneR</a> <font color="Maroon">c<sub>1</sub></font> </span>}</span> 
 
<b>by </b><i><a class="ref" href="lopclset.html#D8" target="_self">Def8</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="setfam_1.html#NM1">Subset-Family</a> of <span class="p1">(<span class="default"><a href="lopclset.html#K7" target="_self">ultraset</a> <font color="Maroon">c<sub>1</sub></font></span>)</span><b> such that </b><br/><a NAME="E12:33"/><i><font color="Green">E31</font></i>: 
(  <a href="setfam_1.html#K5">union</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="subset_1.html#K3">`</a>  &amp; <font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a>  <a href="lopclset.html#K10" target="_self">StoneR</a> <font color="Maroon">c<sub>1</sub></font> )
 ;<br/>
<a NAME="E13:33"/>
<font color="Maroon">c<sub>4</sub></font> <a href="finsub_1.html#K1">\/</a> <font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a>  <a href="lopclset.html#K10" target="_self">StoneR</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="lopclset.html#E7:33"><i><font color="Green">E30</font></i></a>, <a class="txt" href="lopclset.html#E12:33"><i><font color="Green">E31</font></i></a>, <a class="ref" href="xboole_1.html#T8">XBOOLE_1:8</a></i>;<br/>
<a NAME="E14:33"/><b>then </b>
 <a href="setfam_1.html#K5">union</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finsub_1.html#K1">\/</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="lopclset.html#K10" target="_self">StoneR</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T95">ZFMISC_1:95</a></i>;<br/>
<a NAME="E15:33"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="finsub_1.html#K1">\/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="tarski.html#R1">c=</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="lopclset.html#K10" target="_self">StoneR</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="lopclset.html#E7:33"><i><font color="Green">E30</font></i></a>, <a class="txt" href="lopclset.html#E12:33"><i><font color="Green">E31</font></i></a>, <a class="ref" href="zfmisc_1.html#T96">ZFMISC_1:96</a></i>;<br/>
<a NAME="E16:33"/><b>then </b>
 <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><a href="lopclset.html#K11" target="_self">StoneSpace</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="lopclset.html#K10" target="_self">StoneR</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T18">PRE_TOPC:18</a></i>;<br/>
<a NAME="E17:33"/><b>then </b>
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="lopclset.html#K11" target="_self">StoneSpace</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="lopclset.html#K10" target="_self">StoneR</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
;<br/>
<a NAME="E18:33"/><b>then </b><i><font color="Green">E32</font></i>: 
 <a href="lopclset.html#K7" target="_self">ultraset</a> <font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="lopclset.html#K10" target="_self">StoneR</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="lopclset.html#D8" target="_self">Def8</a></i>;<br/>
<a NAME="E19:33"/>
 <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="lopclset.html#K10" target="_self">StoneR</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <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="lopclset.html#K7" target="_self">ultraset</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T95">ZFMISC_1:95</a></i>;<br/>
<a NAME="E20:33"/><b>then </b>
 <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="lopclset.html#K10" target="_self">StoneR</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="lopclset.html#K7" target="_self">ultraset</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T99">ZFMISC_1:99</a></i>;<br/>
<b>hence </b><a NAME="E21:33"/>
 <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="lopclset.html#K10" target="_self">StoneR</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="lopclset.html#K7" target="_self">ultraset</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="lopclset.html#E18:33"><i><font color="Green">E32</font></i></a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


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