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<b>let </b><font color="Maroon" title="c1">R</font> be  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> <a href="lattice3.html#V1" title="LATTICE3:attr.1">with_suprema</a> <a href="orders_2.html#NM1" title="ORDERS_2:NM.1">Poset</a>;<br/><b>let </b><font color="Maroon" title="c2">l</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font>;<br/>



<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:15_1"/>
<font color="Maroon" title="c2">l</font> is <a href="waybel_6.html#V6" title="WAYBEL_6:attr.6">co-prime</a>
 ;<br/><a NAME="E2:15_1"/><b>then </b><i><font color="Green" title="E11">A1</font></i>: 
<font color="Maroon" title="c2">l</font> <a href="lattice3.html#K8" title="LATTICE3:func.8">~</a>  is <a href="waybel_6.html#V5" title="WAYBEL_6:attr.5">prime</a>
 <b>by </b><i><a class="ref" href="waybel_6.html#D8" title="WAYBEL_6:def.8">WAYBEL_6:def 8</a></i>;<br/><b>let </b><font color="Maroon" title="c3">x</font>, <font color="Maroon" title="c4">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font>;<br/><b>assume </b><a NAME="E3:15_1"/>
<font color="Maroon" title="c2">l</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c3">x</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c4">y</font>
 ;<br/><a NAME="E4:15_1"/><b>then </b><i><font color="Green" title="E12">A2</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c4">y</font></span>)</span> <a href="lattice3.html#K8" title="LATTICE3:func.8">~</a>  <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c2">l</font> <a href="lattice3.html#K8" title="LATTICE3:func.8">~</a> 
 <b>by </b><i><a class="ref" href="lattice3.html#T9" title="LATTICE3:th.9">LATTICE3:9</a></i>;<br/><span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c4">y</font></span>)</span> <a href="lattice3.html#K8" title="LATTICE3:func.8">~</a>  = 
<font color="Maroon" title="c3">x</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c4">y</font>
<b>by </b><i><a class="ref" href="lattice3.html#D6" title="LATTICE3:def.6">LATTICE3:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="lattice3.html#K8" title="LATTICE3:func.8">~</a> </span>)</span> <a href="lattice3.html#K11" title="LATTICE3:func.11">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">y</font> <a href="lattice3.html#K8" title="LATTICE3:func.8">~</a> </span>)</span>
<b>by </b><i><a class="ref" href="yellow_7.html#T23" title="YELLOW_7:th.23">YELLOW_7:23</a></i>
;<br/><a NAME="E6:15_1"/><b>then </b>
( <font color="Maroon" title="c3">x</font> <a href="lattice3.html#K8" title="LATTICE3:func.8">~</a>  <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c2">l</font> <a href="lattice3.html#K8" title="LATTICE3:func.8">~</a>  or <font color="Maroon" title="c4">y</font> <a href="lattice3.html#K8" title="LATTICE3:func.8">~</a>  <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c2">l</font> <a href="lattice3.html#K8" title="LATTICE3:func.8">~</a>  )
 <b>by </b><i><a class="txt" href="waybel14.html#E2:15_1"><i><font color="Green" title="E11">A1</font></i></a>, <a class="txt" href="waybel14.html#E4:15_1"><i><font color="Green" title="E12">A2</font></i></a>, <a class="ref" href="waybel_6.html#D6" title="WAYBEL_6:def.6">WAYBEL_6:def 6</a></i>;<br/><b>hence </b><a NAME="E7:15_1"/>
( <font color="Maroon" title="c2">l</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c3">x</font> or <font color="Maroon" title="c2">l</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c4">y</font> )
 <b>by </b><i><a class="ref" href="lattice3.html#T9" title="LATTICE3:th.9">LATTICE3:9</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:15"/><i><font color="Green" title="E11">A3</font></i>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font>  holds <br/>(  not <font color="Maroon" title="c2">l</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b1">x</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Olive" title="b2">y</font> or <font color="Maroon" title="c2">l</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b1">x</font> or <font color="Maroon" title="c2">l</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b2">y</font> )
 ;<br/>

<b>let </b><font color="Maroon" title="c3">x</font>, <font color="Maroon" title="c4">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">R</font> <a href="lattice3.html#K7" title="LATTICE3:func.7">~</a> </span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_6.html#D6" title="WAYBEL_6:def.6">WAYBEL_6:def 6</a>,<a class="ref" href="waybel_6.html#D8" title="WAYBEL_6:def.8">WAYBEL_6:def 8</a><br/>

<b>assume </b><a NAME="E3:15"/><i><font color="Green" title="E12">A4</font></i>: 
<font color="Maroon" title="c3">x</font> <a href="lattice3.html#K11" title="LATTICE3:func.11">"/\"</a> <font color="Maroon" title="c4">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c2">l</font> <a href="lattice3.html#K8" title="LATTICE3:func.8">~</a> 
 ;<br/>

 <a href="lattice3.html#K9" title="LATTICE3:func.9">~</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="lattice3.html#K11" title="LATTICE3:func.11">"/\"</a> <font color="Maroon" title="c4">y</font></span>)</span> = 
<font color="Maroon" title="c3">x</font> <a href="lattice3.html#K11" title="LATTICE3:func.11">"/\"</a> <font color="Maroon" title="c4">y</font>
<b>by </b><i><a class="ref" href="lattice3.html#D7" title="LATTICE3:def.7">LATTICE3:def 7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="lattice3.html#K9" title="LATTICE3:func.9">~</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><a href="lattice3.html#K9" title="LATTICE3:func.9">~</a> <font color="Maroon" title="c4">y</font></span>)</span>
<b>by </b><i><a class="ref" href="yellow_7.html#T24" title="YELLOW_7:th.24">YELLOW_7:24</a></i>
;<br/>
<a NAME="E5:15"/><b>then </b>
<font color="Maroon" title="c2">l</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><a href="lattice3.html#K9" title="LATTICE3:func.9">~</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><a href="lattice3.html#K9" title="LATTICE3:func.9">~</a> <font color="Maroon" title="c4">y</font></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel14.html#E3:15"><i><font color="Green" title="E12">A4</font></i></a>, <a class="ref" href="yellow_7.html#T2" title="YELLOW_7:th.2">YELLOW_7:2</a></i>;<br/>
<a NAME="E6:15"/><b>then </b>
( <font color="Maroon" title="c2">l</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a>  <a href="lattice3.html#K9" title="LATTICE3:func.9">~</a> <font color="Maroon" title="c3">x</font> or <font color="Maroon" title="c2">l</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a>  <a href="lattice3.html#K9" title="LATTICE3:func.9">~</a> <font color="Maroon" title="c4">y</font> )
 
<b>by </b><i><a class="txt" href="waybel14.html#E2:15"><i><font color="Green" title="E11">A3</font></i></a></i>;<br/>
<b>hence </b><a NAME="E7:15"/>
( <font color="Maroon" title="c3">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c2">l</font> <a href="lattice3.html#K8" title="LATTICE3:func.8">~</a>  or <font color="Maroon" title="c4">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c2">l</font> <a href="lattice3.html#K8" title="LATTICE3:func.8">~</a>  )
 <b>by </b><i><a class="ref" href="yellow_7.html#T2" title="YELLOW_7:th.2">YELLOW_7:2</a></i>;<br/>


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