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<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<b>assume </b><a NAME="E1:116"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is <a href="tdlat_2.html#V3" target="_self">open-domains-family</a>
 ;<br/>

<a NAME="E2:116"/><b>then </b><i><font color="Green">E72</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is <a href="tdlat_2.html#V1" target="_self">domains-family</a>
 
<b>by </b><i><a class="ref" href="tdlat_2.html#T80" target="_self">Th80</a></i>;<br/>
<a NAME="E3:116"/>
<font color="Maroon">c<sub>2</sub></font> is <a href="tops_2.html#V1">open</a>
 
<b>by </b><i><a class="txt" href="tdlat_2.html#E1:116"><i><font color="Green">E71</font></i></a>, <a class="ref" href="tdlat_2.html#T81" target="_self">Th81</a></i>;<br/>
<a NAME="E4:116"/><b>then </b><i><font color="Green">E73</font></i>: 
 <a href="setfam_1.html#K5">union</a> <font color="Maroon">c<sub>2</sub></font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="ref" href="tops_2.html#T26">TOPS_2:26</a></i>;<br/>
<a NAME="E5:116"/>
 <a href="setfam_1.html#K5">union</a> <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5">union</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T48">PRE_TOPC:48</a></i>;<br/>
<a NAME="E6:116"/><b>then </b>
 <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5">union</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p2">(<span class="default"><a href="setfam_1.html#K5">union</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="tops_1.html#T48">TOPS_1:48</a></i>;<br/>
<a NAME="E7:116"/><b>then </b>
 <a href="setfam_1.html#K5">union</a> <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a>  <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p2">(<span class="default"><a href="setfam_1.html#K5">union</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="tdlat_2.html#E4:116"><i><font color="Green">E73</font></i></a>, <a class="ref" href="tops_1.html#T55">TOPS_1:55</a></i>;<br/>
<a NAME="E8:116"/><b>then </b><i><font color="Green">E74</font></i>: 
<span class="p1">(<span class="default"><a href="setfam_1.html#K5">union</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1">Int</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p3">(<span class="default"><a href="setfam_1.html#K5">union</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p2">(<span class="default"><a href="setfam_1.html#K5">union</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T12">XBOOLE_1:12</a></i>;<br/>
<b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="tdlat_1.html#K4">Domains_Lattice</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/>

<b>assume </b><a NAME="E9:116"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<b>hence </b><a NAME="E10:116"/>
 <a href="lattice3.html#K15">"\/"</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p1">(<span class="default"><a href="tdlat_1.html#K4">Domains_Lattice</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p2">(<span class="default"><a href="setfam_1.html#K5">union</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="tdlat_2.html#E2:116"><i><font color="Green">E72</font></i></a>, <a class="txt" href="tdlat_2.html#E8:116"><i><font color="Green">E74</font></i></a>, <a class="ref" href="tdlat_2.html#T92" target="_self">Th92</a></i>;<br/>


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