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<b>let </b><font color="Maroon">c<sub>1</sub></font> be  <a href="yellow_0.html#V1">lower-bounded</a> <a href="waybel_1.html#V2">distributive</a> <a href="waybel_3.html#V3">continuous</a> <a href="waybel_0.html#NM8">LATTICE</a>;<br/>

<b>assume </b><a NAME="E1:66"/><i><font color="Green">E38</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>1</sub></font> st <font color="Olive">b<sub>1</sub></font> is <a href="waybel_7.html#V4" target="_self">pseudoprime</a> holds <br/><font color="Olive">b<sub>1</sub></font> is <a href="waybel_6.html#V5">prime</a>
 ;<br/>

<b>given </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><a NAME="E2:66"/><i><font color="Green">E39</font></i>: 
( <span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="yellow_3.html#K7">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> <a href="waybel_4.html#K1">-waybelow</a>  &amp; <span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="yellow_3.html#K7">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> <a href="waybel_4.html#K1">-waybelow</a>  &amp; not <span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<span class="p2">(<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></span><a href="yellow_3.html#K7">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> <a href="waybel_4.html#K1">-waybelow</a>  )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_7.html#D7" target="_self">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">c<sub>5</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:66_1"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="waybel_3.html#K1">waybelow</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> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">c<sub>2</sub></font> )
 ;<br/><b>then reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>5</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/><a NAME="E3:66_1"/>
( <font color="Maroon">c<sub>6</sub></font> <a href="waybel_3.html#NR2" target="_self">&lt;&lt;</a> <font color="Maroon">c<sub>3</sub></font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="waybel_7.html#E1:66_1"><i><font color="Green">E40</font></i></a>, <a class="ref" href="waybel_0.html#T18">WAYBEL_0:18</a>, <a class="ref" href="waybel_3.html#T7">WAYBEL_3:7</a></i>;<br/><a NAME="E4:66_1"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="waybel_3.html#NR2" target="_self">&lt;&lt;</a> <font color="Maroon">c<sub>3</sub></font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="waybel_3.html#T2">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">E39</font></i></a>, <a class="ref" href="waybel_4.html#D2">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">waybelow</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> <a href="subset_1.html#R1">misses</a>  <a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E6:66"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> is <a href="waybel_7.html#V1" target="_self">prime</a> &amp;  <a href="waybel_3.html#K1">waybelow</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> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#R1">misses</a>  <a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="ref" href="waybel_7.html#T27" target="_self">Th27</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>6</sub></font> =  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">c<sub>5</sub></font>;<br/>
<a NAME="E7:66"/>
 <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">c<sub>5</sub></font> is <a href="waybel_7.html#V4" target="_self">pseudoprime</a>
 
<b>by </b><i><a class="txt" href="waybel_7.html#E6:66"><i><font color="Green">E40</font></i></a>, <a class="ref" href="waybel_7.html#D6" target="_self">Def6</a></i>;<br/>
<a NAME="E8:66"/><b>then </b><i><font color="Green">E41</font></i>: 
 <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">c<sub>5</sub></font> is <a href="waybel_6.html#V5">prime</a>
 
<b>by </b><i><a class="txt" href="waybel_7.html#E1:66"><i><font color="Green">E38</font></i></a></i>;<br/>
<a NAME="E9:66"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><a href="waybel_3.html#K1">waybelow</a> <span class="p2">(<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></span>)</span> &amp;  <a href="yellow_0.html#R1">ex_sup_of</a>  <a href="waybel_3.html#K1">waybelow</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>,<font color="Maroon">c<sub>1</sub></font> &amp;  <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="ref" href="waybel_0.html#T75">WAYBEL_0:75</a>, <a class="ref" href="waybel_3.html#D5">WAYBEL_3:def 5</a></i>;<br/>
<a NAME="E10:66"/><b>then </b>
<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>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="waybel_7.html#E6:66"><i><font color="Green">E40</font></i></a>, <a class="ref" href="yellow_0.html#T34">YELLOW_0:34</a></i>;<br/>
<a NAME="E11:66"/><b>then </b>
( ( <font color="Maroon">c<sub>3</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>5</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="waybel_3.html#NR2" target="_self">&lt;&lt;</a> <font color="Maroon">c<sub>3</sub></font> ) or ( <font color="Maroon">c<sub>4</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>5</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="waybel_3.html#NR2" target="_self">&lt;&lt;</a> <font color="Maroon">c<sub>4</sub></font> ) )
 
<b>by </b><i><a class="txt" href="waybel_7.html#E2:66"><i><font color="Green">E39</font></i></a>, <a class="txt" href="waybel_7.html#E8:66"><i><font color="Green">E41</font></i></a>, <a class="ref" href="waybel_4.html#D2">WAYBEL_4:def 2</a>, <a class="ref" href="waybel_6.html#D6">WAYBEL_6:def 6</a></i>;<br/>
<a NAME="E12:66"/><b>then </b>
( <font color="Maroon">c<sub>2</sub></font> <a href="waybel_3.html#NR2" target="_self">&lt;&lt;</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="ref" href="waybel_3.html#T2">WAYBEL_3:2</a></i>;<br/>
<a NAME="E13:66"/><b>then </b>
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="ref" href="waybel_0.html#T18">WAYBEL_0:18</a>, <a class="ref" href="waybel_3.html#T20">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">E40</font></i></a>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>


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