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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="orders_2.html#L1">RelStr</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:132"/><i><font color="Green">E79</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="waybel_1.html#NR13" target="_self">is_a_Heyting_algebra</a> 
 <b>and </b><br/><a NAME="E2:132"/><i><font color="Green">E80</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is <a href="yellow_0.html#V1">lower-bounded</a>
 ;<br/>

<a NAME="E3:132"/><i><font color="Green">E81</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is   <a href="waybel_0.html#NM8">LATTICE</a>
 
<b>by </b><i><a class="txt" href="waybel_1.html#E1:132"><i><font color="Green">E79</font></i></a>, <a class="ref" href="waybel_1.html#D19" target="_self">Def19</a></i>;<br/>
<a NAME="E4:132"/><b>then </b>
 <a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">c<sub>1</sub></font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="lattice3.html#K11">"/\"</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K4">Top</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="yellow_0.html#T23">YELLOW_0:23</a></i>;<br/>
<a NAME="E5:132"/><b>then </b>
 <a href="yellow_0.html#K4">Top</a> <font color="Maroon">c<sub>1</sub></font> <a href="orders_2.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="waybel_1.html#K6" target="_self">=&gt;</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel_1.html#E1:132"><i><font color="Green">E79</font></i></a>, <a class="ref" href="waybel_1.html#T70" target="_self">Th70</a></i>;<br/>
<a NAME="E6:132"/><b>then </b><i><font color="Green">E82</font></i>: 
 <a href="waybel_1.html#K7" target="_self">'not'</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="orders_2.html#NR2" target="_self">&gt;=</a>  <a href="yellow_0.html#K4">Top</a> <font color="Maroon">c<sub>1</sub></font>
 
;<br/>
<a NAME="E7:132"/><i><font color="Green">E83</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is <a href="yellow_0.html#V2">upper-bounded</a>
 
<b>by </b><i><a class="txt" href="waybel_1.html#E1:132"><i><font color="Green">E79</font></i></a>, <a class="ref" href="waybel_1.html#T71" target="_self">Th71</a></i>;<br/>
<a NAME="E8:132"/><b>then </b>
 <a href="waybel_1.html#K7" target="_self">'not'</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="orders_2.html#R1">&lt;=</a>  <a href="yellow_0.html#K4">Top</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="waybel_1.html#E3:132"><i><font color="Green">E81</font></i></a>, <a class="ref" href="yellow_0.html#T45">YELLOW_0:45</a></i>;<br/>
<b>hence </b><a NAME="E9:132"/>
 <a href="yellow_0.html#K4">Top</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="waybel_1.html#K7" target="_self">'not'</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="waybel_1.html#E3:132"><i><font color="Green">E81</font></i></a>, <a class="txt" href="waybel_1.html#E6:132"><i><font color="Green">E82</font></i></a>, <a class="ref" href="orders_2.html#T25">ORDERS_2:25</a></i>;<br/>

<a NAME="E10:132"/>
<span class="p1">(<span class="default"><a href="yellow_0.html#K4">Top</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="waybel_1.html#K6" target="_self">=&gt;</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="orders_2.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K4">Top</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="waybel_1.html#K6" target="_self">=&gt;</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel_1.html#E3:132"><i><font color="Green">E81</font></i></a>, <a class="ref" href="orders_2.html#T24">ORDERS_2:24</a></i>;<br/>
<a NAME="E11:132"/><b>then </b>
 <a href="waybel_1.html#K7" target="_self">'not'</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K4">Top</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="orders_2.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K4">Top</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="waybel_1.html#K6" target="_self">=&gt;</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
;<br/>
<a NAME="E12:132"/><b>then </b>
(  <a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">c<sub>1</sub></font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K4">Top</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="lattice3.html#K11">"/\"</a> <span class="p1">(<span class="default"><a href="waybel_1.html#K7" target="_self">'not'</a> <span class="p2">(<span class="default"><a href="yellow_0.html#K4">Top</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> &amp;  <a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">c<sub>1</sub></font> <a href="orders_2.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K4">Top</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="lattice3.html#K11">"/\"</a> <span class="p1">(<span class="default"><a href="waybel_1.html#K7" target="_self">'not'</a> <span class="p2">(<span class="default"><a href="yellow_0.html#K4">Top</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="txt" href="waybel_1.html#E1:132"><i><font color="Green">E79</font></i></a>, <a class="txt" href="waybel_1.html#E2:132"><i><font color="Green">E80</font></i></a>, <a class="txt" href="waybel_1.html#E3:132"><i><font color="Green">E81</font></i></a>, <a class="ref" href="waybel_1.html#T70" target="_self">Th70</a>, <a class="ref" href="yellow_0.html#T44">YELLOW_0:44</a></i>;<br/>
<a NAME="E13:132"/><b>then </b><i><font color="Green">E84</font></i>: 
 <a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K4">Top</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="lattice3.html#K11">"/\"</a> <span class="p1">(<span class="default"><a href="waybel_1.html#K7" target="_self">'not'</a> <span class="p2">(<span class="default"><a href="yellow_0.html#K4">Top</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel_1.html#E3:132"><i><font color="Green">E81</font></i></a>, <a class="ref" href="orders_2.html#T25">ORDERS_2:25</a></i>;<br/>
<a NAME="E14:132"/>
 <a href="waybel_1.html#K7" target="_self">'not'</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K4">Top</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="orders_2.html#R1">&lt;=</a>  <a href="yellow_0.html#K4">Top</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="waybel_1.html#E3:132"><i><font color="Green">E81</font></i></a>, <a class="txt" href="waybel_1.html#E7:132"><i><font color="Green">E83</font></i></a>, <a class="ref" href="yellow_0.html#T45">YELLOW_0:45</a></i>;<br/>
<b>hence </b> <a href="waybel_1.html#K7" target="_self">'not'</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K4">Top</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><a href="waybel_1.html#K7" target="_self">'not'</a> <span class="p2">(<span class="default"><a href="yellow_0.html#K4">Top</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="lattice3.html#K11">"/\"</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K4">Top</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="waybel_1.html#E3:132"><i><font color="Green">E81</font></i></a>, <a class="ref" href="yellow_0.html#T25">YELLOW_0:25</a></i>
<br/>.= 
 <a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="txt" href="waybel_1.html#E3:132"><i><font color="Green">E81</font></i></a>, <a class="txt" href="waybel_1.html#E13:132"><i><font color="Green">E84</font></i></a>, <a class="ref" href="lattice3.html#T15">LATTICE3:15</a></i>
;<br/><br/>


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