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<div class="add">

<b>let </b><font color="Maroon">H</font> be  non <a href="struct_0.html#V3">empty</a>  <a href="orders_2.html#L1">RelStr</a> ;<br/>

<b>assume </b><a NAME="E1:129"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">H</font> <a href="waybel_1.html#NR13" target="_self">is_a_Heyting_algebra</a> 
 ;<br/>

<a NAME="E2:129"/><b>then </b><i><font color="Green">E35</font></i>: 
<font color="Maroon">H</font> is   <a href="waybel_0.html#NM8">LATTICE</a>
 
<b>by </b><i/>;<br/>
<a NAME="E3:129"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">H</font> is <a href="waybel_1.html#V2" target="_self">distributive</a>
 
<b>by </b><i>, </i>;<br/>
<b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">H</font>, <font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">H</font>, <font color="Maroon">c</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">H</font>;<br/>

<a NAME="E4:129"/>
( <font color="Maroon">a</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">a</font> <a href="lattice3.html#K10">"\/"</a> <font color="Maroon">b</font> &amp; <font color="Maroon">b</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">a</font> <a href="lattice3.html#K10">"\/"</a> <font color="Maroon">b</font> )
 
<b>by </b><i>, <a class="ref" href="yellow_0.html#T22">YELLOW_0:22</a></i>;<br/>
<a NAME="E5:129"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="lattice3.html#K10">"\/"</a> <font color="Maroon">b</font></span>)</span> <a href="waybel_1.html#K6" target="_self">=&gt;</a> <font color="Maroon">c</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">a</font> <a href="waybel_1.html#K6" target="_self">=&gt;</a> <font color="Maroon">c</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="lattice3.html#K10">"\/"</a> <font color="Maroon">b</font></span>)</span> <a href="waybel_1.html#K6" target="_self">=&gt;</a> <font color="Maroon">c</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">b</font> <a href="waybel_1.html#K6" target="_self">=&gt;</a> <font color="Maroon">c</font> )
 
<b>by </b><i>, <a class="ref" href="waybel_1.html#D10" target="_self">Def10</a></i>;<br/>
<a NAME="E6:129"/><b>then </b><i><font color="Green">E48</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="lattice3.html#K10">"\/"</a> <font color="Maroon">b</font></span>)</span> <a href="waybel_1.html#K6" target="_self">=&gt;</a> <font color="Maroon">c</font> <a href="orders_2.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="waybel_1.html#K6" target="_self">=&gt;</a> <font color="Maroon">c</font></span>)</span> <a href="lattice3.html#K11">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="waybel_1.html#K6" target="_self">=&gt;</a> <font color="Maroon">c</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="yellow_0.html#T23">YELLOW_0:23</a></i>;<br/>
<b>set </b><font color="Maroon">d</font> = <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="waybel_1.html#K6" target="_self">=&gt;</a> <font color="Maroon">c</font></span>)</span> <a href="lattice3.html#K11">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="waybel_1.html#K6" target="_self">=&gt;</a> <font color="Maroon">c</font></span>)</span>;<br/>
<i><font color="Green">E49</font></i>: <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="lattice3.html#K10">"\/"</a> <font color="Maroon">b</font></span>)</span> <a href="lattice3.html#K11">"/\"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="waybel_1.html#K6" target="_self">=&gt;</a> <font color="Maroon">c</font></span>)</span> <a href="lattice3.html#K11">"/\"</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="waybel_1.html#K6" target="_self">=&gt;</a> <font color="Maroon">c</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="waybel_1.html#K6" target="_self">=&gt;</a> <font color="Maroon">c</font></span>)</span> <a href="lattice3.html#K11">"/\"</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="waybel_1.html#K6" target="_self">=&gt;</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="lattice3.html#K11">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="lattice3.html#K10">"\/"</a> <font color="Maroon">b</font></span>)</span>
<b>by </b><i>, <a class="ref" href="lattice3.html#T15">LATTICE3:15</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="waybel_1.html#K6" target="_self">=&gt;</a> <font color="Maroon">c</font></span>)</span> <a href="lattice3.html#K11">"/\"</a> <span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="waybel_1.html#K6" target="_self">=&gt;</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="lattice3.html#K11">"/\"</a> <font color="Maroon">a</font></span>)</span> <a href="lattice3.html#K10">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="waybel_1.html#K6" target="_self">=&gt;</a> <font color="Maroon">c</font></span>)</span> <a href="lattice3.html#K11">"/\"</a> <span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="waybel_1.html#K6" target="_self">=&gt;</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="lattice3.html#K11">"/\"</a> <font color="Maroon">b</font></span>)</span>
<b>by </b><i><a class="ref" href="waybel_1.html#D11" target="_self">Def11</a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="lattice3.html#K11">"/\"</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="waybel_1.html#K6" target="_self">=&gt;</a> <font color="Maroon">c</font></span>)</span> <a href="lattice3.html#K11">"/\"</a> <span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="waybel_1.html#K6" target="_self">=&gt;</a> <font color="Maroon">c</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"><span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="waybel_1.html#K6" target="_self">=&gt;</a> <font color="Maroon">c</font></span>)</span> <a href="lattice3.html#K11">"/\"</a> <span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="waybel_1.html#K6" target="_self">=&gt;</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="lattice3.html#K11">"/\"</a> <font color="Maroon">b</font></span>)</span>
<b>by </b><i>, <a class="ref" href="lattice3.html#T15">LATTICE3:15</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="lattice3.html#K11">"/\"</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="waybel_1.html#K6" target="_self">=&gt;</a> <font color="Maroon">c</font></span>)</span> <a href="lattice3.html#K11">"/\"</a> <span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="waybel_1.html#K6" target="_self">=&gt;</a> <font color="Maroon">c</font></span>)</span></span>)</span></span>)</span> <a href="lattice3.html#K10">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="lattice3.html#K11">"/\"</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="waybel_1.html#K6" target="_self">=&gt;</a> <font color="Maroon">c</font></span>)</span> <a href="lattice3.html#K11">"/\"</a> <span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="waybel_1.html#K6" target="_self">=&gt;</a> <font color="Maroon">c</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="lattice3.html#T15">LATTICE3:15</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="lattice3.html#K11">"/\"</a> <span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="waybel_1.html#K6" target="_self">=&gt;</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="lattice3.html#K11">"/\"</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="waybel_1.html#K6" target="_self">=&gt;</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="lattice3.html#K10">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="lattice3.html#K11">"/\"</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="waybel_1.html#K6" target="_self">=&gt;</a> <font color="Maroon">c</font></span>)</span> <a href="lattice3.html#K11">"/\"</a> <span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="waybel_1.html#K6" target="_self">=&gt;</a> <font color="Maroon">c</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="lattice3.html#T16">LATTICE3:16</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="lattice3.html#K11">"/\"</a> <span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="waybel_1.html#K6" target="_self">=&gt;</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="lattice3.html#K11">"/\"</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="waybel_1.html#K6" target="_self">=&gt;</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="lattice3.html#K10">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="lattice3.html#K11">"/\"</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="waybel_1.html#K6" target="_self">=&gt;</a> <font color="Maroon">c</font></span>)</span> <a href="lattice3.html#K11">"/\"</a> <span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="waybel_1.html#K6" target="_self">=&gt;</a> <font color="Maroon">c</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="lattice3.html#T15">LATTICE3:15</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="lattice3.html#K11">"/\"</a> <span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="waybel_1.html#K6" target="_self">=&gt;</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="lattice3.html#K11">"/\"</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="waybel_1.html#K6" target="_self">=&gt;</a> <font color="Maroon">c</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">b</font> <a href="lattice3.html#K11">"/\"</a> <span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="waybel_1.html#K6" target="_self">=&gt;</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="lattice3.html#K11">"/\"</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="waybel_1.html#K6" target="_self">=&gt;</a> <font color="Maroon">c</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="lattice3.html#T16">LATTICE3:16</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="lattice3.html#K11">"/\"</a> <font color="Maroon">c</font></span>)</span> <a href="lattice3.html#K11">"/\"</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="waybel_1.html#K6" target="_self">=&gt;</a> <font color="Maroon">c</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">b</font> <a href="lattice3.html#K11">"/\"</a> <span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="waybel_1.html#K6" target="_self">=&gt;</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="lattice3.html#K11">"/\"</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="waybel_1.html#K6" target="_self">=&gt;</a> <font color="Maroon">c</font></span>)</span></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="lattice3.html#K11">"/\"</a> <font color="Maroon">c</font></span>)</span> <a href="lattice3.html#K11">"/\"</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="waybel_1.html#K6" target="_self">=&gt;</a> <font color="Maroon">c</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">b</font> <a href="lattice3.html#K11">"/\"</a> <font color="Maroon">c</font></span>)</span> <a href="lattice3.html#K11">"/\"</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="waybel_1.html#K6" target="_self">=&gt;</a> <font color="Maroon">c</font></span>)</span></span>)</span>
<b>by </b><i>, </i>
;<br/>
<a NAME="E8:129"/>
( <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="lattice3.html#K11">"/\"</a> <font color="Maroon">c</font></span>)</span> <a href="lattice3.html#K11">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="waybel_1.html#K6" target="_self">=&gt;</a> <font color="Maroon">c</font></span>)</span> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">a</font> <a href="lattice3.html#K11">"/\"</a> <font color="Maroon">c</font> &amp; <font color="Maroon">a</font> <a href="lattice3.html#K11">"/\"</a> <font color="Maroon">c</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c</font> )
 
<b>by </b><i>, <a class="ref" href="yellow_0.html#T23">YELLOW_0:23</a></i>;<br/>
<a NAME="E9:129"/><b>then </b><i><font color="Green">E50</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="lattice3.html#K11">"/\"</a> <font color="Maroon">c</font></span>)</span> <a href="lattice3.html#K11">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="waybel_1.html#K6" target="_self">=&gt;</a> <font color="Maroon">c</font></span>)</span> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c</font>
 
<b>by </b><i>, <a class="ref" href="orders_2.html#T26">ORDERS_2:26</a></i>;<br/>
<a NAME="E10:129"/>
( <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="lattice3.html#K11">"/\"</a> <font color="Maroon">c</font></span>)</span> <a href="lattice3.html#K11">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="waybel_1.html#K6" target="_self">=&gt;</a> <font color="Maroon">c</font></span>)</span> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">b</font> <a href="lattice3.html#K11">"/\"</a> <font color="Maroon">c</font> &amp; <font color="Maroon">b</font> <a href="lattice3.html#K11">"/\"</a> <font color="Maroon">c</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c</font> )
 
<b>by </b><i>, <a class="ref" href="yellow_0.html#T23">YELLOW_0:23</a></i>;<br/>
<a NAME="E11:129"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="lattice3.html#K11">"/\"</a> <font color="Maroon">c</font></span>)</span> <a href="lattice3.html#K11">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="waybel_1.html#K6" target="_self">=&gt;</a> <font color="Maroon">c</font></span>)</span> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c</font>
 
<b>by </b><i>, <a class="ref" href="orders_2.html#T26">ORDERS_2:26</a></i>;<br/>
<a NAME="E12:129"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="lattice3.html#K10">"\/"</a> <font color="Maroon">b</font></span>)</span> <a href="lattice3.html#K11">"/\"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="waybel_1.html#K6" target="_self">=&gt;</a> <font color="Maroon">c</font></span>)</span> <a href="lattice3.html#K11">"/\"</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="waybel_1.html#K6" target="_self">=&gt;</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c</font>
 
<b>by </b><i>, <a class="ref" href="waybel_1.html#T10" target="_self">Th10</a>, <a class="ref" href="waybel_1.html#T11" target="_self">Th11</a>, <a class="ref" href="yellow_0.html#T22">YELLOW_0:22</a></i>;<br/>
<a NAME="E13:129"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="lattice3.html#K10">"\/"</a> <font color="Maroon">b</font></span>)</span> <a href="waybel_1.html#K6" target="_self">=&gt;</a> <font color="Maroon">c</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="waybel_1.html#K6" target="_self">=&gt;</a> <font color="Maroon">c</font></span>)</span> <a href="lattice3.html#K11">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="waybel_1.html#K6" target="_self">=&gt;</a> <font color="Maroon">c</font></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E14:129"/>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="lattice3.html#K10">"\/"</a> <font color="Maroon">b</font></span>)</span> <a href="waybel_1.html#K6" target="_self">=&gt;</a> <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="waybel_1.html#K6" target="_self">=&gt;</a> <font color="Maroon">c</font></span>)</span> <a href="lattice3.html#K11">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="waybel_1.html#K6" target="_self">=&gt;</a> <font color="Maroon">c</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="waybel_1.html#D12" target="_self">Def12</a>, <a class="ref" href="orders_2.html#T25">ORDERS_2:25</a></i>;<br/>


</div>
