<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="waybel_0.html#NM7">sup-Semilattice</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>thus </b><a NAME="E1:42"/>
( <font color="Maroon">c<sub>2</sub></font> is <a href="waybel_6.html#V6" target="_self">co-prime</a> implies for <font color="Olive">b<sub>1</sub></font> being non <a href="xboole_0.html#V1">empty</a> <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Maroon">c<sub>2</sub></font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Olive">b<sub>1</sub></font> holds <br/>ex <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 <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">b<sub>2</sub></font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:42_1"/>
<font color="Maroon">c<sub>2</sub></font> is <a href="waybel_6.html#V6" target="_self">co-prime</a>
 ;<br/>

<a NAME="E2:42_1"/><b>then </b><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>2</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="ref" href="waybel_6.html#D8" target="_self">Def8</a></i>;<br/>
<b>let </b><font color="Maroon">c<sub>3</sub></font> be  non <a href="xboole_0.html#V1">empty</a> <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>assume </b><a NAME="E3:42_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<a NAME="E4:42_1"/><b>then </b><i><font color="Green">E26</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K8">~</a>  <a href="orders_2.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="lattice3.html#K8">~</a> 
 
<b>by </b><i><a class="ref" href="lattice3.html#T9">LATTICE3:9</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>3</sub></font> as  non <a href="xboole_0.html#V1">empty</a> <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="lattice3.html#K7">~</a> </span>)</span> ;<br/>
<a NAME="E6:42_1"/><i><font color="Green">E27</font></i>: 
 <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#T54">YELLOW_0:54</a></i>;<br/>
<a NAME="E7:42_1"/><b>then </b><i><font color="Green">E28</font></i>: 
 <a href="yellow_0.html#R2">ex_inf_of</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font> <a href="lattice3.html#K7">~</a> 
 
<b>by </b><i><a class="ref" href="yellow_7.html#T10">YELLOW_7:10</a></i>;<br/>
<a NAME="E8:42_1"/>
 <a href="yellow_0.html#K1">"\/"</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font> <a href="lattice3.html#NR4" target="_self">is_&gt;=_than</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="waybel_6.html#E6:42_1"><i><font color="Green">E27</font></i></a>, <a class="ref" href="yellow_0.html#D9">YELLOW_0:def 9</a></i>;<br/>
<a NAME="E9:42_1"/><b>then </b><i><font color="Green">E29</font></i>: 
<span class="p1">(<span class="default"><a href="yellow_0.html#K1">"\/"</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="lattice3.html#K8">~</a>  <a href="lattice3.html#R1">is_&lt;=_than</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="yellow_7.html#T8">YELLOW_7:8</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="lattice3.html#K7">~</a> </span>)</span>;<br/><b>assume </b><a NAME="E1:42_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="lattice3.html#R1">is_&lt;=_than</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><a NAME="E2:42_1_1"/><b>then </b>
 <a href="lattice3.html#K9">~</a> <font color="Maroon">c<sub>5</sub></font> <a href="lattice3.html#NR4" target="_self">is_&gt;=_than</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="yellow_7.html#T9">YELLOW_7:9</a></i>;<br/><a NAME="E3:42_1_1"/><b>then </b>
 <a href="lattice3.html#K9">~</a> <font color="Maroon">c<sub>5</sub></font> <a href="orders_2.html#NR2" target="_self">&gt;=</a>  <a href="yellow_0.html#K1">"\/"</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="waybel_6.html#E6:42_1"><i><font color="Green">E27</font></i></a>, <a class="ref" href="yellow_0.html#D9">YELLOW_0:def 9</a></i>;<br/><b>hence </b><a NAME="E4:42_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K1">"\/"</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="lattice3.html#K8">~</a> 
 <b>by </b><i><a class="ref" href="yellow_7.html#T2">YELLOW_7:2</a></i>;<br/></div><b>end;</b></div>
<a NAME="E11:42_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="lattice3.html#K8">~</a>  <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#NK4" target="_self">inf</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="waybel_6.html#E7:42_1"><i><font color="Green">E28</font></i></a>, <a class="txt" href="waybel_6.html#E9:42_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="yellow_0.html#D10">YELLOW_0:def 10</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="lattice3.html#K7">~</a> </span>)</span><b> such that </b><br/><a NAME="E13:42_1"/><i><font color="Green">E30</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K8">~</a>  <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="txt" href="waybel_6.html#E2:42_1"><i><font color="Green">E25</font></i></a>, <a class="txt" href="waybel_6.html#E4:42_1"><i><font color="Green">E26</font></i></a>, <a class="ref" href="waybel_6.html#T22" target="_self">Th22</a></i>;<br/>
<b>take </b>
 <a href="lattice3.html#K9">~</a> <font color="Maroon">c<sub>5</sub></font>
;<br/>

<b>thus </b><a NAME="E14:42_1"/>
(  <a href="lattice3.html#K9">~</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="orders_2.html#R3">&lt;=</a>  <a href="lattice3.html#K9">~</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="txt" href="waybel_6.html#E13:42_1"><i><font color="Green">E30</font></i></a>, <a class="ref" href="yellow_7.html#T2">YELLOW_7:2</a></i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E2:42"/>
( ( for <font color="Olive">b<sub>1</sub></font> being non <a href="xboole_0.html#V1">empty</a> <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Maroon">c<sub>2</sub></font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Olive">b<sub>1</sub></font> holds <br/>ex <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 <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">b<sub>2</sub></font> ) ) implies <font color="Maroon">c<sub>2</sub></font> is <a href="waybel_6.html#V6" target="_self">co-prime</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:42_2"/><i><font color="Green">E25</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being non <a href="xboole_0.html#V1">empty</a> <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Maroon">c<sub>2</sub></font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Olive">b<sub>1</sub></font> holds <br/>ex <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 <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">b<sub>2</sub></font> )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="lattice3.html#K7">~</a> </span>)</span>;<br/><b>assume </b><a NAME="E1:42_2_1"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">c<sub>4</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K8">~</a> 
 ;<br/><b>set </b><font color="Maroon">c<sub>5</sub></font> = <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><span class="p2">(<span class="default"><a href="lattice3.html#K9">~</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="lattice3.html#K9">~</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="domain_1.html#K7">}</a></span>;<br/><a NAME="E2:42_2_1"/><i><font color="Green">E27</font></i>: 
 <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><span class="p2">(<span class="default"><a href="lattice3.html#K9">~</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="lattice3.html#K9">~</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="domain_1.html#K7">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="lattice3.html#K9">~</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><a href="lattice3.html#K9">~</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="yellow_0.html#T41">YELLOW_0:41</a></i>;<br/><a NAME="E3:42_2_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="orders_2.html#R3">&lt;=</a>  <a href="lattice3.html#K9">~</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="waybel_6.html#E1:42_2_1"><i><font color="Green">E26</font></i></a>, <a class="ref" href="yellow_7.html#T2">YELLOW_7:2</a></i>;<br/><a NAME="E4:42_2_1"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><a href="lattice3.html#K9">~</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><a href="lattice3.html#K9">~</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="yellow_7.html#T24">YELLOW_7:24</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>6</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E6:42_2_1"/><i><font color="Green">E28</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><span class="p2">(<span class="default"><a href="lattice3.html#K9">~</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="lattice3.html#K9">~</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="domain_1.html#K7">}</a></span> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="txt" href="waybel_6.html#E1:42_2"><i><font color="Green">E25</font></i></a>, <a class="txt" href="waybel_6.html#E2:42_2_1"><i><font color="Green">E27</font></i></a></i>;<br/><a NAME="E7:42_2_1"/>
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="lattice3.html#K9">~</a> <font color="Maroon">c<sub>3</sub></font> or <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="lattice3.html#K9">~</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="txt" href="waybel_6.html#E6:42_2_1"><i><font color="Green">E28</font></i></a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><b>hence </b><a NAME="E8:42_2_1"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K8">~</a>  or <font color="Maroon">c<sub>4</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K8">~</a>  )
 <b>by </b><i><a class="txt" href="waybel_6.html#E6:42_2_1"><i><font color="Green">E28</font></i></a>, <a class="ref" href="yellow_7.html#T2">YELLOW_7:2</a></i>;<br/></div><b>end;</b></div>
<a NAME="E3:42_2"/><b>then </b>
<font color="Maroon">c<sub>2</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="ref" href="waybel_6.html#D6" target="_self">Def6</a></i>;<br/>
<b>hence </b><a NAME="E4:42_2"/>
<font color="Maroon">c<sub>2</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>


</div>
