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<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:23"/>
 <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>2</sub></font> =  <a href="xboole_0.html#K1">{}</a>  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">c<sub>1</sub></font></span>)</span> ;<br/>
<a NAME="E3:23"/><i><font color="Green">E7</font></i>: 
 <a href="xboole_0.html#K1">{}</a>  <a href="lattice3.html#R2">is_&lt;=_than</a> <font color="Maroon">c<sub>2</sub></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="23_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>3</sub></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">c<sub>1</sub></font></span>)</span>;<br/><b>assume </b><a NAME="E1:23_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><a NAME="E2:23_1"/>
 <a href="xboole_0.html#K1">{}</a>  <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T2">XBOOLE_1:2</a></i>;<br/><b>hence </b><a NAME="E3:23_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="yellow_1.html#T3" target="_self">Th3</a></i>;<br/></div><b>end;</b></div>
<a NAME="E5:23"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#R1">is_&lt;=_than</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="lattice3.html#D8">LATTICE3:def 8</a></i>;<br/>
<a NAME="E6:23"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#R1">is_&lt;=_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">c<sub>1</sub></font></span>)</span>
 
;<br/>
<a NAME="E7:23"/><b>then </b>
 <a href="yellow_1.html#K2" target="_self">InclPoset</a> <font color="Maroon">c<sub>1</sub></font> is <a href="yellow_0.html#V1">lower-bounded</a>
 
<b>by </b><i><a class="ref" href="yellow_0.html#D4">YELLOW_0:def 4</a></i>;<br/>
<a NAME="E8:23"/><b>then </b>
 <a href="yellow_0.html#R1">ex_sup_of</a>  <a href="xboole_0.html#K1">{}</a> , <a href="yellow_1.html#K2" target="_self">InclPoset</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#T42">YELLOW_0:42</a></i>;<br/>
<a NAME="E9:23"/><b>then </b>
 <a href="yellow_0.html#K1">"\/"</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">c<sub>1</sub></font></span>)</span> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="yellow_1.html#E3:23"><i><font color="Green">E7</font></i></a>, <a class="ref" href="yellow_0.html#D9">YELLOW_0:def 9</a></i>;<br/>
<a NAME="E10:23"/><b>then </b><i><font color="Green">E8</font></i>: 
 <a href="yellow_0.html#K1">"\/"</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">c<sub>1</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="yellow_1.html#T3" target="_self">Th3</a></i>;<br/>
<b>thus </b> <a href="yellow_0.html#K3">Bottom</a> <span class="p1">(<span class="default"><a href="yellow_1.html#K2" target="_self">InclPoset</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> = 
 <a href="yellow_0.html#K1">"\/"</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">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="yellow_0.html#D11">YELLOW_0:def 11</a></i>
<br/>.= 
 <a href="xboole_0.html#K1">{}</a> 
<b>by </b><i><a class="txt" href="yellow_1.html#E10:23"><i><font color="Green">E8</font></i></a>, <a class="ref" href="xboole_1.html#T3">XBOOLE_1:3</a></i>
;<br/>


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