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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="yellow_0.html#V1">lower-bounded</a>  <a href="orders_2.html#L1">RelStr</a> ;<br/>

<b>assume </b><a NAME="E1:136"/><i><font color="Green">E81</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="waybel_1.html#NR13" target="_self">is_a_Heyting_algebra</a> 
 ;<br/>

<a NAME="E2:136"/><b>then </b><i><font color="Green">E82</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is <a href="waybel_1.html#V2" target="_self">distributive</a>
 
<b>by </b><i><a class="ref" href="waybel_1.html#T69" target="_self">Th69</a></i>;<br/>
<b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<a NAME="E3:136"/><i><font color="Green">E83</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#E1:136"><i><font color="Green">E81</font></i></a>, <a class="ref" href="waybel_1.html#D19" target="_self">Def19</a></i>;<br/>
<a NAME="E4:136"/><b>then </b><i><font color="Green">E84</font></i>: 
 <a href="waybel_1.html#K7" target="_self">'not'</a> <font color="Maroon">c<sub>2</sub></font> <a href="orders_2.html#R1">&lt;=</a>  <a href="waybel_1.html#K7" target="_self">'not'</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="orders_2.html#T24">ORDERS_2:24</a></i>;<br/>
<a NAME="E5:136"/><i><font color="Green">E85</font></i>: 
 <a href="waybel_1.html#K7" target="_self">'not'</a> <font color="Maroon">c<sub>3</sub></font> <a href="orders_2.html#R1">&lt;=</a>  <a href="waybel_1.html#K7" target="_self">'not'</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="waybel_1.html#E3:136"><i><font color="Green">E83</font></i></a>, <a class="ref" href="orders_2.html#T24">ORDERS_2:24</a></i>;<br/>
<a NAME="E6:136"/><i><font color="Green">E86</font></i>: 
 <a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">c<sub>1</sub></font> <a href="orders_2.html#R1">&lt;=</a>  <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#E3:136"><i><font color="Green">E83</font></i></a>, <a class="ref" href="orders_2.html#T24">ORDERS_2:24</a></i>;<br/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K11">"/\"</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="lattice3.html#K11">"/\"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_1.html#K7" target="_self">'not'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="lattice3.html#K10">"\/"</a> <span class="p2">(<span class="default"><a href="waybel_1.html#K7" target="_self">'not'</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K11">"/\"</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="lattice3.html#K11">"/\"</a> <span class="p2">(<span class="default"><a href="waybel_1.html#K7" target="_self">'not'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="lattice3.html#K10">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K11">"/\"</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="lattice3.html#K11">"/\"</a> <span class="p2">(<span class="default"><a href="waybel_1.html#K7" target="_self">'not'</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="waybel_1.html#E2:136"><i><font color="Green">E82</font></i></a>, <a class="ref" href="waybel_1.html#D3" target="_self">Def3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="lattice3.html#K11">"/\"</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="lattice3.html#K11">"/\"</a> <span class="p2">(<span class="default"><a href="waybel_1.html#K7" target="_self">'not'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="lattice3.html#K10">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K11">"/\"</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="lattice3.html#K11">"/\"</a> <span class="p2">(<span class="default"><a href="waybel_1.html#K7" target="_self">'not'</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="waybel_1.html#E3:136"><i><font color="Green">E83</font></i></a>, <a class="ref" href="lattice3.html#T15">LATTICE3:15</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="lattice3.html#K11">"/\"</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K11">"/\"</a> <span class="p3">(<span class="default"><a href="waybel_1.html#K7" target="_self">'not'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span> <a href="lattice3.html#K10">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K11">"/\"</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="lattice3.html#K11">"/\"</a> <span class="p2">(<span class="default"><a href="waybel_1.html#K7" target="_self">'not'</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="waybel_1.html#E3:136"><i><font color="Green">E83</font></i></a>, <a class="ref" href="lattice3.html#T16">LATTICE3:16</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="lattice3.html#K11">"/\"</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K11">"/\"</a> <span class="p3">(<span class="default"><a href="waybel_1.html#K7" target="_self">'not'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span> <a href="lattice3.html#K10">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K11">"/\"</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="lattice3.html#K11">"/\"</a> <span class="p3">(<span class="default"><a href="waybel_1.html#K7" target="_self">'not'</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="waybel_1.html#E3:136"><i><font color="Green">E83</font></i></a>, <a class="ref" href="lattice3.html#T16">LATTICE3:16</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="lattice3.html#K11">"/\"</a> <span class="p2">(<span class="default"><a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="lattice3.html#K10">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K11">"/\"</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="lattice3.html#K11">"/\"</a> <span class="p3">(<span class="default"><a href="waybel_1.html#K7" target="_self">'not'</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="waybel_1.html#E1:136"><i><font color="Green">E81</font></i></a>, <a class="txt" href="waybel_1.html#E4:136"><i><font color="Green">E84</font></i></a>, <a class="ref" href="waybel_1.html#T85" target="_self">Th85</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="lattice3.html#K11">"/\"</a> <span class="p2">(<span class="default"><a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="lattice3.html#K10">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K11">"/\"</a> <span class="p2">(<span class="default"><a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="waybel_1.html#E1:136"><i><font color="Green">E81</font></i></a>, <a class="txt" href="waybel_1.html#E5:136"><i><font color="Green">E85</font></i></a>, <a class="ref" href="waybel_1.html#T85" target="_self">Th85</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<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> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="lattice3.html#K10">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K11">"/\"</a> <span class="p2">(<span class="default"><a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="waybel_1.html#E3:136"><i><font color="Green">E83</font></i></a>, <a class="ref" href="lattice3.html#T15">LATTICE3:15</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<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> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="lattice3.html#K10">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<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> <font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="waybel_1.html#E3:136"><i><font color="Green">E83</font></i></a>, <a class="ref" href="lattice3.html#T15">LATTICE3:15</a></i>
<br/>.= 
<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#K10">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<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> <font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="waybel_1.html#E3:136"><i><font color="Green">E83</font></i></a>, <a class="ref" href="waybel_1.html#T4" target="_self">Th4</a></i>
<br/>.= 
<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#K10">"\/"</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#E3:136"><i><font color="Green">E83</font></i></a>, <a class="ref" href="waybel_1.html#T4" target="_self">Th4</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#E3:136"><i><font color="Green">E83</font></i></a>, <a class="txt" href="waybel_1.html#E6:136"><i><font color="Green">E86</font></i></a>, <a class="ref" href="yellow_0.html#T24">YELLOW_0:24</a></i>
;<br/>
<b>hence </b><a NAME="E8:136"/>
 <a href="waybel_1.html#K7" target="_self">'not'</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K11">"/\"</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><a href="waybel_1.html#K7" target="_self">'not'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="lattice3.html#K10">"\/"</a> <span class="p1">(<span class="default"><a href="waybel_1.html#K7" target="_self">'not'</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="waybel_1.html#E1:136"><i><font color="Green">E81</font></i></a>, <a class="ref" href="waybel_1.html#T85" target="_self">Th85</a></i>;<br/>


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