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

<b>let </b><font color="Maroon" title="c1">L</font> be  <a href="waybel_1.html#V11" title="WAYBEL_1:attr.11">Boolean</a> <a href="waybel_0.html#NM8" title="WAYBEL_0:NM.8">LATTICE</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="waybel15.html#K1" title="WAYBEL15:func.1">ATOM</a> <font color="Maroon" title="c1">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="waybel_6.html#K4" title="WAYBEL_6:func.4">PRIME</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">L</font> <a href="yellow_7.html#NK1" title="YELLOW_7:NK.1">opp</a> </span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c1">L</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><br/>

<a NAME="E1:29"/><i><font color="Green" title="E21">A1</font></i>: 
 <a href="waybel15.html#K1" title="WAYBEL15:func.1">ATOM</a> <font color="Maroon" title="c1">L</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="waybel_6.html#K4" title="WAYBEL_6:func.4">PRIME</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">L</font> <a href="yellow_7.html#NK1" title="YELLOW_7:NK.1">opp</a> </span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c1">L</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel15.html#K1" title="WAYBEL15:func.1">ATOM</a> <font color="Maroon" title="c1">L</font> or <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="waybel_6.html#K4" title="WAYBEL_6:func.4">PRIME</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">L</font> <a href="yellow_7.html#NK1" title="YELLOW_7:NK.1">opp</a> </span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c1">L</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> )</span><br/>

<b>assume </b><a NAME="E1:29_1"/><i><font color="Green" title="E22">A2</font></i>: 
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel15.html#K1" title="WAYBEL15:func.1">ATOM</a> <font color="Maroon" title="c1">L</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="waybel_6.html#K4" title="WAYBEL_6:func.4">PRIME</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">L</font> <a href="yellow_7.html#NK1" title="YELLOW_7:NK.1">opp</a> </span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c1">L</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><br/>

<b>then reconsider </b><font color="Maroon" title="c3">x'</font> = <font color="Maroon" title="c2">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> ;<br/>
<a NAME="E3:29_1"/><i><font color="Green" title="E23">A3</font></i>: 
<font color="Maroon" title="c3">x'</font> is <a href="waybel15.html#V1" title="WAYBEL15:attr.1">atom</a>
 
<b>by </b><i><a class="txt" href="waybel15.html#E1:29_1"><i><font color="Green" title="E22">A2</font></i></a>, <a class="ref" href="waybel15.html#D2" target="_self" title="WAYBEL15:def.2">Def2</a></i>;<br/>
<a NAME="E4:29_1"/><b>then </b>
<font color="Maroon" title="c3">x'</font> is <a href="waybel_6.html#V6" title="WAYBEL_6:attr.6">co-prime</a>
 
<b>by </b><i><a class="ref" href="waybel15.html#T23" target="_self" title="WAYBEL15:th.23">Th23</a></i>;<br/>
<a NAME="E5:29_1"/><b>then </b>
<font color="Maroon" title="c3">x'</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/>
<a NAME="E6:29_1"/><b>then </b>
<font color="Maroon" title="c3">x'</font> <a href="lattice3.html#K8" title="LATTICE3:func.8">~</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_6.html#K4" title="WAYBEL_6:func.4">PRIME</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">L</font> <a href="yellow_7.html#NK1" title="YELLOW_7:NK.1">opp</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="waybel_6.html#D7" title="WAYBEL_6:def.7">WAYBEL_6:def 7</a></i>;<br/>
<a NAME="E7:29_1"/><b>then </b><i><font color="Green" title="E24">A4</font></i>: 
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_6.html#K4" title="WAYBEL_6:func.4">PRIME</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">L</font> <a href="yellow_7.html#NK1" title="YELLOW_7:NK.1">opp</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="lattice3.html#D6" title="LATTICE3:def.6">LATTICE3:def 6</a></i>;<br/>
<a NAME="E8:29_1"/>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c1">L</font>
 
<b>by </b><i><a class="txt" href="waybel15.html#E3:29_1"><i><font color="Green" title="E23">A3</font></i></a>, <a class="ref" href="waybel15.html#T23" target="_self" title="WAYBEL15:th.23">Th23</a></i>;<br/>
<a NAME="E9:29_1"/><b>then </b>
not <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c1">L</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E10:29_1"/>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="waybel_6.html#K4" title="WAYBEL_6:func.4">PRIME</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">L</font> <a href="yellow_7.html#NK1" title="YELLOW_7:NK.1">opp</a> </span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c1">L</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="txt" href="waybel15.html#E7:29_1"><i><font color="Green" title="E24">A4</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E2:29"/>
<span class="p1">(<span class="default"><a href="waybel_6.html#K4" title="WAYBEL_6:func.4">PRIME</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">L</font> <a href="yellow_7.html#NK1" title="YELLOW_7:NK.1">opp</a> </span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c1">L</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="waybel15.html#K1" title="WAYBEL15:func.1">ATOM</a> <font color="Maroon" title="c1">L</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="waybel_6.html#K4" title="WAYBEL_6:func.4">PRIME</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">L</font> <a href="yellow_7.html#NK1" title="YELLOW_7:NK.1">opp</a> </span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c1">L</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> or <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel15.html#K1" title="WAYBEL15:func.1">ATOM</a> <font color="Maroon" title="c1">L</font> )</span><br/>

<b>assume </b><a NAME="E1:29_2"/><i><font color="Green" title="E22">A5</font></i>: 
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="waybel_6.html#K4" title="WAYBEL_6:func.4">PRIME</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">L</font> <a href="yellow_7.html#NK1" title="YELLOW_7:NK.1">opp</a> </span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c1">L</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel15.html#K1" title="WAYBEL15:func.1">ATOM</a> <font color="Maroon" title="c1">L</font></span><br/>

<a NAME="E2:29_2"/><b>then </b><i><font color="Green" title="E23">A6</font></i>: 
( <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_6.html#K4" title="WAYBEL_6:func.4">PRIME</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">L</font> <a href="yellow_7.html#NK1" title="YELLOW_7:NK.1">opp</a> </span>)</span> &amp; not <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c1">L</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c3">x'</font> = <font color="Maroon" title="c2">x</font> as   <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">L</font> <a href="yellow_7.html#NK1" title="YELLOW_7:NK.1">opp</a> </span>)</span> <b>by </b><i><a class="txt" href="waybel15.html#E1:29_2"><i><font color="Green" title="E22">A5</font></i></a></i>;<br/>
<a NAME="E4:29_2"/>
<font color="Maroon" title="c3">x'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c1">L</font>
 
<b>by </b><i><a class="txt" href="waybel15.html#E2:29_2"><i><font color="Green" title="E23">A6</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<a NAME="E5:29_2"/><b>then </b><i><font color="Green" title="E24">A7</font></i>: 
 <a href="lattice3.html#K9" title="LATTICE3:func.9">~</a> <font color="Maroon" title="c3">x'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c1">L</font>
 
<b>by </b><i><a class="ref" href="lattice3.html#D7" title="LATTICE3:def.7">LATTICE3:def 7</a></i>;<br/>
<a NAME="E6:29_2"/><i><font color="Green" title="E25">A8</font></i>: 
<font color="Maroon" title="c3">x'</font> is <a href="waybel_6.html#V5" title="WAYBEL_6:attr.5">prime</a>
 
<b>by </b><i><a class="txt" href="waybel15.html#E2:29_2"><i><font color="Green" title="E23">A6</font></i></a>, <a class="ref" href="waybel_6.html#D7" title="WAYBEL_6:def.7">WAYBEL_6: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#K8" title="LATTICE3:func.8">~</a>  = 
 <a href="lattice3.html#K9" title="LATTICE3:func.9">~</a> <font color="Maroon" title="c3">x'</font>
<b>by </b><i><a class="ref" href="lattice3.html#D6" title="LATTICE3:def.6">LATTICE3:def 6</a></i>
<br/>.= 
<font color="Maroon" title="c3">x'</font>
<b>by </b><i><a class="ref" href="lattice3.html#D7" title="LATTICE3:def.7">LATTICE3:def 7</a></i>
;<br/>
<a NAME="E8:29_2"/><b>then </b>
 <a href="lattice3.html#K9" title="LATTICE3:func.9">~</a> <font color="Maroon" title="c3">x'</font> is <a href="waybel_6.html#V6" title="WAYBEL_6:attr.6">co-prime</a>
 
<b>by </b><i><a class="txt" href="waybel15.html#E6:29_2"><i><font color="Green" title="E25">A8</font></i></a>, <a class="ref" href="waybel_6.html#D8" title="WAYBEL_6:def.8">WAYBEL_6:def 8</a></i>;<br/>
<a NAME="E9:29_2"/><b>then </b>
 <a href="lattice3.html#K9" title="LATTICE3:func.9">~</a> <font color="Maroon" title="c3">x'</font> is <a href="waybel15.html#V1" title="WAYBEL15:attr.1">atom</a>
 
<b>by </b><i><a class="txt" href="waybel15.html#E5:29_2"><i><font color="Green" title="E24">A7</font></i></a>, <a class="ref" href="waybel15.html#T23" target="_self" title="WAYBEL15:th.23">Th23</a></i>;<br/>
<a NAME="E10:29_2"/><b>then </b>
 <a href="lattice3.html#K9" title="LATTICE3:func.9">~</a> <font color="Maroon" title="c3">x'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel15.html#K1" title="WAYBEL15:func.1">ATOM</a> <font color="Maroon" title="c1">L</font>
 
<b>by </b><i><a class="ref" href="waybel15.html#D2" target="_self" title="WAYBEL15:def.2">Def2</a></i>;<br/>
<b>hence </b><a NAME="E11:29_2"/>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel15.html#K1" title="WAYBEL15:func.1">ATOM</a> <font color="Maroon" title="c1">L</font>
 <b>by </b><i><a class="ref" href="lattice3.html#D7" title="LATTICE3:def.7">LATTICE3:def 7</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:29"/>
 <a href="waybel15.html#K1" title="WAYBEL15:func.1">ATOM</a> <font color="Maroon" title="c1">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="waybel_6.html#K4" title="WAYBEL_6:func.4">PRIME</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">L</font> <a href="yellow_7.html#NK1" title="YELLOW_7:NK.1">opp</a> </span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c1">L</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="txt" href="waybel15.html#E1:29"><i><font color="Green" title="E21">A1</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
