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<span class="kw">let </span><font color="Maroon" title="c1">X</font> be   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">X</font> implies  <a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <span class="p1">(<span class="default"><a href="yellow_1.html#K1" title="YELLOW_1:func.1">InclPoset</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:22"/>
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">X</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <span class="p1">(<span class="default"><a href="yellow_1.html#K1" title="YELLOW_1:func.1">InclPoset</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c2">a</font> =  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K1" title="YELLOW_1:func.1">InclPoset</a> <font color="Maroon" title="c1">X</font></span>)</span> ;<br/>
<a NAME="E3:22"/>
 for <font color="Olive" title="b1">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K1" title="YELLOW_1:func.1">InclPoset</a> <font color="Maroon" title="c1">X</font></span>)</span>  st <font color="Olive" title="b1">b</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">X</font> holds <br/><font color="Maroon" title="c2">a</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b1">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_1.html#T3" target="_self" onmouseover="rs('yellow_1/T3')" onmouseout="rh()">Th3</a>, <a class="ref" href="xboole_1.html#T2" onmouseover="rs('xboole_1/T2')" onmouseout="rh()">XBOOLE_1:2</a></span>;<br/>
<a NAME="E4:22"/><span class="kw">then </span>
<font color="Maroon" title="c2">a</font> <a href="lattice3.html#R1" title="LATTICE3:pred.1">is_&lt;=_than</a> <font color="Maroon" title="c1">X</font>
 
;<br/>
<a NAME="E5:22"/><span class="kw">then </span>
 <a href="yellow_1.html#K1" title="YELLOW_1:func.1">InclPoset</a> <font color="Maroon" title="c1">X</font> is  <a href="yellow_0.html#V1" title="YELLOW_0:attr.1">lower-bounded</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_0.html#D4" onmouseover="rs('yellow_0/D4')" onmouseout="rh()">YELLOW_0:def 4</a></span>;<br/>
<a NAME="E6:22"/><span class="kw">then </span>
(  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a> <font color="Maroon" title="c2">a</font> &amp;  <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> , <a href="yellow_1.html#K1" title="YELLOW_1:func.1">InclPoset</a> <font color="Maroon" title="c1">X</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_0.html#T42" onmouseover="rs('yellow_0/T42')" onmouseout="rh()">YELLOW_0:42</a></span>;<br/>
<a NAME="E7:22"/><span class="kw">then </span>
 <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> (<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>,<span class="p1">(<span class="default"><a href="yellow_1.html#K1" title="YELLOW_1:func.1">InclPoset</a> <font color="Maroon" title="c1">X</font></span>)</span>) <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c2">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_0.html#D9" onmouseover="rs('yellow_0/D9')" onmouseout="rh()">YELLOW_0:def 9</a></span>;<br/>
<a NAME="E8:22"/><span class="kw">then </span><span class="lab"><font color="Green" title="E7">A1</font></span>: 
 <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> (<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>,<span class="p1">(<span class="default"><a href="yellow_1.html#K1" title="YELLOW_1:func.1">InclPoset</a> <font color="Maroon" title="c1">X</font></span>)</span>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_1.html#T3" target="_self" onmouseover="rs('yellow_1/T3')" onmouseout="rh()">Th3</a></span>;<br/>
<span class="kw">thus </span> <a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <span class="p1">(<span class="default"><a href="yellow_1.html#K1" title="YELLOW_1:func.1">InclPoset</a> <font color="Maroon" title="c1">X</font></span>)</span> = 
 <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> (<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>,<span class="p1">(<span class="default"><a href="yellow_1.html#K1" title="YELLOW_1:func.1">InclPoset</a> <font color="Maroon" title="c1">X</font></span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_0.html#D11" onmouseover="rs('yellow_0/D11')" onmouseout="rh()">YELLOW_0:def 11</a></span>
<br/>.= 
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:22"><span class="lab"><font color="Green" title="E7">A1</font></span></a></span>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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