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<b>let </b><font color="Maroon">c<sub>1</sub></font> be  <a href="lattice3.html#V3">complete</a> <a href="waybel_0.html#NM8">LATTICE</a>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:59"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> is <a href="waybel_1.html#V2">distributive</a> &amp; <font color="Maroon">c<sub>1</sub></font> is <a href="waybel_3.html#V3">continuous</a> )
 <b>and </b><br/><a NAME="E2:59"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="lattice3.html#K7">~</a>  is <a href="waybel_3.html#V3">continuous</a>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>reconsider </b><font color="Maroon">c<sub>3</sub></font> = <font color="Maroon">c<sub>1</sub></font> as  <a href="waybel_1.html#V2">distributive</a> <a href="lattice3.html#V3">complete</a> <a href="waybel_3.html#V3">continuous</a> <a href="waybel_0.html#NM8">LATTICE</a> <b>by </b><i><a class="txt" href="waybel_6.html#E1:59"><i><font color="Green">E39</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>2</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="lattice3.html#K7">~</a> </span>)</span> ;<br/>
<a NAME="E5:59"/>
 <a href="waybel_6.html#K4" target="_self">PRIME</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="lattice3.html#K7">~</a> </span>)</span> is <a href="waybel_6.html#V4" target="_self">order-generating</a>
 
<b>by </b><i><a class="txt" href="waybel_6.html#E2:59"><i><font color="Green">E40</font></i></a>, <a class="ref" href="waybel_6.html#T35" target="_self">Th35</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="waybel_6.html#K4" target="_self">PRIME</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="lattice3.html#K7">~</a> </span>)</span></span>)</span><b> such that </b><br/><a NAME="E7:59"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K2">"/\"</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="lattice3.html#K7">~</a> </span>)</span>
 <b>by </b><i><a class="ref" href="waybel_6.html#T15" target="_self">Th15</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>5</sub></font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>3</sub></font> <b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E9:59"/><i><font color="Green">E42</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>3</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> is <a href="waybel_6.html#V6" target="_self">co-prime</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>7</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>3</sub></font>;<br/>

<b>assume </b><a NAME="E1:59_1"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/>

<a NAME="E2:59_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="lattice3.html#K8">~</a>  is <a href="waybel_6.html#V5" target="_self">prime</a>
 
<b>by </b><i><a class="txt" href="waybel_6.html#E1:59_1"><i><font color="Green">E43</font></i></a>, <a class="ref" href="waybel_6.html#D7" target="_self">Def7</a></i>;<br/>
<b>hence </b><a NAME="E3:59_1"/>
<font color="Maroon">c<sub>7</sub></font> is <a href="waybel_6.html#V6" target="_self">co-prime</a>
 <b>by </b><i><a class="ref" href="waybel_6.html#D8" target="_self">Def8</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E10:59"/>
 <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#T17">YELLOW_0:17</a></i>;<br/>
<a NAME="E11:59"/><b>then </b>
 <a href="yellow_0.html#K1">"\/"</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K2">"/\"</a> <font color="Maroon">c<sub>6</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="lattice3.html#K7">~</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="yellow_7.html#T12">YELLOW_7:12</a></i>;<br/>
<b>hence </b><a NAME="E12:59"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Olive">b<sub>1</sub></font> &amp; ( for <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Olive">b<sub>2</sub></font> is <a href="waybel_6.html#V6" target="_self">co-prime</a> ) )
 <b>by </b><i><a class="txt" href="waybel_6.html#E7:59"><i><font color="Green">E41</font></i></a>, <a class="txt" href="waybel_6.html#E9:59"><i><font color="Green">E42</font></i></a></i>;<br/>


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