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<b>let </b><font color="Maroon" title="c1">L</font> be  <a href="yellow_0.html#V1" title="YELLOW_0:attr.1">lower-bounded</a> <a href="waybel_1.html#V2" title="WAYBEL_1:attr.2">distributive</a> <a href="waybel_3.html#V3" title="WAYBEL_3:attr.3">continuous</a> <a href="waybel_0.html#NM8" title="WAYBEL_0:NM.8">LATTICE</a>;<br/>

<b>assume </b><a NAME="E1:66"/><i><font color="Green" title="E38">A1</font></i>: 
for <font color="Olive" title="b1">p</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>  st <font color="Olive" title="b1">p</font> is <a href="waybel_7.html#V4" title="WAYBEL_7:attr.4">pseudoprime</a> holds <br/><font color="Olive" title="b1">p</font> is <a href="waybel_6.html#V5" title="WAYBEL_6:attr.5">prime</a>
 ;<br/>

<b>given </b><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">x</font>, <font color="Maroon" title="c4">y</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font><b> such that </b><a NAME="E2:66"/><i><font color="Green" title="E39">A2</font></i>: 
( <span class="p1"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">x</font></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">L</font> <a href="waybel_4.html#K1" title="WAYBEL_4:func.1">-waybelow</a>  &amp; <span class="p1"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">y</font></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">L</font> <a href="waybel_4.html#K1" title="WAYBEL_4:func.1">-waybelow</a>  &amp; not <span class="p1"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><font color="Maroon" title="c2">a</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c4">y</font></span>)</span></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">L</font> <a href="waybel_4.html#K1" title="WAYBEL_4:func.1">-waybelow</a>  )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_7.html#D7" target="_self" title="WAYBEL_7:def.7">WAYBEL_7:def 7</a><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:66_1"/><i><font color="Green" title="E40">A3</font></i>: 
( <font color="Maroon" title="c5">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_3.html#K1" title="WAYBEL_3:func.1">waybelow</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c4">y</font></span>)</span> &amp; <font color="Maroon" title="c5">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <font color="Maroon" title="c2">a</font> )
 ;<br/><b>then reconsider </b><font color="Maroon" title="c6">z</font> = <font color="Maroon" title="c5">z</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="E3:66_1"/>
( <font color="Maroon" title="c6">z</font> <a href="waybel_3.html#NR2" title="WAYBEL_3:NR.2">&lt;&lt;</a> <font color="Maroon" title="c3">x</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c4">y</font> &amp; <font color="Maroon" title="c6">z</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c2">a</font> )
 <b>by </b><i><a class="txt" href="waybel_7.html#E1:66_1"><i><font color="Green" title="E40">A3</font></i></a>, <a class="ref" href="waybel_0.html#T18" title="WAYBEL_0:th.18">WAYBEL_0:18</a>, <a class="ref" href="waybel_3.html#T7" title="WAYBEL_3:th.7">WAYBEL_3:7</a></i>;<br/><a NAME="E4:66_1"/><b>then </b>
<font color="Maroon" title="c2">a</font> <a href="waybel_3.html#NR2" title="WAYBEL_3:NR.2">&lt;&lt;</a> <font color="Maroon" title="c3">x</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c4">y</font>
 <b>by </b><i><a class="ref" href="waybel_3.html#T2" title="WAYBEL_3:th.2">WAYBEL_3:2</a></i>;<br/><b>hence </b><a NAME="E5:66_1"/>
contradiction
 <b>by </b><i><a class="txt" href="waybel_7.html#E2:66"><i><font color="Green" title="E39">A2</font></i></a>, <a class="ref" href="waybel_4.html#D2" title="WAYBEL_4:def.2">WAYBEL_4:def 2</a></i>;<br/></div><b>end;</b></div>
<a NAME="E4:66"/><b>then </b>
 <a href="waybel_3.html#K1" title="WAYBEL_3:func.1">waybelow</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c4">y</font></span>)</span> <a href="subset_1.html#R1" title="SUBSET_1:pred.1">misses</a>  <a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <font color="Maroon" title="c2">a</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c5">P</font> being   <a href="waybel_0.html#NM3" title="WAYBEL_0:NM.3">Ideal</a> of <font color="Maroon" title="c1">L</font><b> such that </b><br/><a NAME="E6:66"/><i><font color="Green" title="E40">A4</font></i>: 
( <font color="Maroon" title="c5">P</font> is <a href="waybel_7.html#V1" title="WAYBEL_7:attr.1">prime</a> &amp;  <a href="waybel_3.html#K1" title="WAYBEL_3:func.1">waybelow</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c4">y</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">P</font> &amp; <font color="Maroon" title="c5">P</font> <a href="subset_1.html#R1" title="SUBSET_1:pred.1">misses</a>  <a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <font color="Maroon" title="c2">a</font> )
 <b>by </b><i><a class="ref" href="waybel_7.html#T27" target="_self" title="WAYBEL_7:th.27">Th27</a></i>;<br/>
<b>set </b><font color="Maroon" title="c6">p</font> =  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c5">P</font>;<br/>
<a NAME="E7:66"/>
 <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c5">P</font> is <a href="waybel_7.html#V4" title="WAYBEL_7:attr.4">pseudoprime</a>
 
<b>by </b><i><a class="txt" href="waybel_7.html#E6:66"><i><font color="Green" title="E40">A4</font></i></a>, <a class="ref" href="waybel_7.html#D6" target="_self" title="WAYBEL_7:def.6">Def6</a></i>;<br/>
<a NAME="E8:66"/><b>then </b><i><font color="Green" title="E41">A5</font></i>: 
 <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c5">P</font> is <a href="waybel_6.html#V5" title="WAYBEL_6:attr.5">prime</a>
 
<b>by </b><i><a class="txt" href="waybel_7.html#E1:66"><i><font color="Green" title="E38">A1</font></i></a></i>;<br/>
<a NAME="E9:66"/>
( <font color="Maroon" title="c3">x</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1">(<span class="default"><a href="waybel_3.html#K1" title="WAYBEL_3:func.1">waybelow</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c4">y</font></span>)</span></span>)</span> &amp;  <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a>  <a href="waybel_3.html#K1" title="WAYBEL_3:func.1">waybelow</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c4">y</font></span>)</span>,<font color="Maroon" title="c1">L</font> &amp;  <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <font color="Maroon" title="c5">P</font>,<font color="Maroon" title="c1">L</font> )
 
<b>by </b><i><a class="ref" href="waybel_0.html#T75" title="WAYBEL_0:th.75">WAYBEL_0:75</a>, <a class="ref" href="waybel_3.html#D5" title="WAYBEL_3:def.5">WAYBEL_3:def 5</a></i>;<br/>
<a NAME="E10:66"/><b>then </b>
<font color="Maroon" title="c3">x</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c4">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a>  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c5">P</font>
 
<b>by </b><i><a class="txt" href="waybel_7.html#E6:66"><i><font color="Green" title="E40">A4</font></i></a>, <a class="ref" href="yellow_0.html#T34" title="YELLOW_0:th.34">YELLOW_0:34</a></i>;<br/>
<a NAME="E11:66"/><b>then </b>
( ( <font color="Maroon" title="c3">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a>  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c5">P</font> &amp; <font color="Maroon" title="c2">a</font> <a href="waybel_3.html#NR2" title="WAYBEL_3:NR.2">&lt;&lt;</a> <font color="Maroon" title="c3">x</font> ) or ( <font color="Maroon" title="c4">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a>  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c5">P</font> &amp; <font color="Maroon" title="c2">a</font> <a href="waybel_3.html#NR2" title="WAYBEL_3:NR.2">&lt;&lt;</a> <font color="Maroon" title="c4">y</font> ) )
 
<b>by </b><i><a class="txt" href="waybel_7.html#E2:66"><i><font color="Green" title="E39">A2</font></i></a>, <a class="txt" href="waybel_7.html#E8:66"><i><font color="Green" title="E41">A5</font></i></a>, <a class="ref" href="waybel_4.html#D2" title="WAYBEL_4:def.2">WAYBEL_4:def 2</a>, <a class="ref" href="waybel_6.html#D6" title="WAYBEL_6:def.6">WAYBEL_6:def 6</a></i>;<br/>
<a NAME="E12:66"/><b>then </b>
( <font color="Maroon" title="c2">a</font> <a href="waybel_3.html#NR2" title="WAYBEL_3:NR.2">&lt;&lt;</a>  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c5">P</font> &amp; <font color="Maroon" title="c2">a</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c2">a</font> )
 
<b>by </b><i><a class="ref" href="waybel_3.html#T2" title="WAYBEL_3:th.2">WAYBEL_3:2</a></i>;<br/>
<a NAME="E13:66"/><b>then </b>
( <font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P</font> &amp; <font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <font color="Maroon" title="c2">a</font> )
 
<b>by </b><i><a class="ref" href="waybel_0.html#T18" title="WAYBEL_0:th.18">WAYBEL_0:18</a>, <a class="ref" href="waybel_3.html#T20" title="WAYBEL_3:th.20">WAYBEL_3:20</a></i>;<br/>
<b>hence </b><a NAME="E14:66"/>
contradiction
 <b>by </b><i><a class="txt" href="waybel_7.html#E6:66"><i><font color="Green" title="E40">A4</font></i></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/>


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