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<b>let </b><font color="Maroon">X</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:24"/>
 <a href="tarski.html#K3">union</a> <font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">a</font> =  <a href="tarski.html#K3">union</a> <font color="Maroon">X</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2" target="_self">InclPoset</a> <font color="Maroon">X</font></span>)</span> ;<br/>
<a NAME="E3:24"/><i><font color="Green">E21</font></i>: 
 <a href="yellow_0.html#K2">"/\"</a> <a href="xboole_0.html#K1">{}</a> ,<span class="p1">(<span class="default"><a href="yellow_1.html#K2" target="_self">InclPoset</a> <font color="Maroon">X</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">a</font>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T92">ZFMISC_1:92</a></i>;<br/>
<a NAME="E4:24"/><i><font color="Green">E24</font></i>: 
 <a href="xboole_0.html#K1">{}</a>  <a href="lattice3.html#NR3" target="_self">is_&gt;=_than</a> <font color="Maroon">a</font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#T6">YELLOW_0:6</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2" target="_self">InclPoset</a> <font color="Maroon">X</font></span>)</span>;<br/><b>assume </b><a NAME="E1:24_1"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/><a NAME="E2:24_1"/>
<font color="Maroon">b</font> <a href="tarski.html#R1">c=</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T92">ZFMISC_1:92</a></i>;<br/><b>hence </b><a NAME="E3:24_1"/>
<font color="Maroon">b</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">a</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<a NAME="E6:24"/><b>then </b>
<font color="Maroon">a</font> <a href="lattice3.html#NR4" target="_self">is_&gt;=_than</a> <font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="lattice3.html#D9">LATTICE3:def 9</a></i>;<br/>
<a NAME="E7:24"/><b>then </b>
<font color="Maroon">a</font> <a href="lattice3.html#NR4" target="_self">is_&gt;=_than</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2" target="_self">InclPoset</a> <font color="Maroon">X</font></span>)</span>
 
;<br/>
<a NAME="E8:24"/><b>then </b>
 <a href="yellow_1.html#K2" target="_self">InclPoset</a> <font color="Maroon">X</font> is <a href="yellow_0.html#V2">upper-bounded</a>
 
<b>by </b><i><a class="ref" href="yellow_0.html#D5">YELLOW_0:def 5</a></i>;<br/>
<a NAME="E9:24"/><b>then </b>
 <a href="yellow_0.html#R2">ex_inf_of</a>  <a href="xboole_0.html#K1">{}</a> , <a href="yellow_1.html#K2" target="_self">InclPoset</a> <font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#T43">YELLOW_0:43</a></i>;<br/>
<a NAME="E10:24"/><b>then </b>
<font color="Maroon">a</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#K2">"/\"</a> <a href="xboole_0.html#K1">{}</a> ,<span class="p1">(<span class="default"><a href="yellow_1.html#K2" target="_self">InclPoset</a> <font color="Maroon">X</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="yellow_0.html#D10">YELLOW_0:def 10</a></i>;<br/>
<a NAME="E11:24"/><b>then </b><i><font color="Green">E25</font></i>: 
<font color="Maroon">a</font> <a href="tarski.html#R1">c=</a>  <a href="yellow_0.html#K2">"/\"</a> <a href="xboole_0.html#K1">{}</a> ,<span class="p1">(<span class="default"><a href="yellow_1.html#K2" target="_self">InclPoset</a> <font color="Maroon">X</font></span>)</span>
 
<b>by </b><i/>;<br/>
<b>thus </b> <a href="yellow_0.html#K4">Top</a> <span class="p1">(<span class="default"><a href="yellow_1.html#K2" target="_self">InclPoset</a> <font color="Maroon">X</font></span>)</span> = 
 <a href="yellow_0.html#K2">"/\"</a> <a href="xboole_0.html#K1">{}</a> ,<span class="p1">(<span class="default"><a href="yellow_1.html#K2" target="_self">InclPoset</a> <font color="Maroon">X</font></span>)</span>
<b>by </b><i><a class="ref" href="yellow_0.html#D12">YELLOW_0:def 12</a></i>
<br/>.= 
 <a href="tarski.html#K3">union</a> <font color="Maroon">X</font>
<b>by </b><i>, , <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>
;<br/>


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