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

<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be   <a href="waybel_0.html#NM8">LATTICE</a>;<br/>

<b>assume </b><a NAME="E1:25"/><i><font color="Green">E13</font></i>: 
 <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font> #) <a href="hidden.html#R1">=</a>  <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>2</sub></font> #)
 ;<br/>

<b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E2:25"/>
<font color="Maroon">c<sub>3</sub></font> is  <a href="waybel_7.html#V2" target="_self">prime</a> <a href="waybel_0.html#NM4">Filter</a> of <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>3</sub></font> as  <a href="waybel_7.html#V2" target="_self">prime</a> <a href="waybel_0.html#NM4">Filter</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</sub></font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="txt" href="waybel_7.html#E1:25"><i><font color="Green">E13</font></i></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="waybel_0.html#NM4">Filter</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="txt" href="waybel_7.html#E1:25"><i><font color="Green">E13</font></i></a>, <a class="ref" href="waybel_0.html#T4">WAYBEL_0:4</a>, <a class="ref" href="waybel_0.html#T25">WAYBEL_0:25</a></i>;<br/>
<a NAME="E6:25"/>
<font color="Maroon">c<sub>6</sub></font> is <a href="waybel_7.html#V2" target="_self">prime</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>8</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_7.html#D2" target="_self">WAYBEL_7:def 2</a><br/>

<b>reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>8</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="waybel_7.html#E1:25"><i><font color="Green">E13</font></i></a></i>;<br/>
<a NAME="E2:25_1"/>
(  <a href="yellow_0.html#R1">ex_sup_of</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="domain_1.html#K7">}</a></span>,<font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="domain_1.html#K7">}</a></span> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="domain_1.html#K7">}</a></span> )
 
<b>by </b><i><a class="ref" href="yellow_0.html#T20">YELLOW_0:20</a>, <a class="ref" href="yellow_0.html#T41">YELLOW_0:41</a></i>;<br/>
<a NAME="E3:25_1"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="txt" href="waybel_7.html#E1:25"><i><font color="Green">E13</font></i></a>, <a class="ref" href="yellow_0.html#T26">YELLOW_0:26</a></i>;<br/>
<b>hence </b><a NAME="E4:25_1"/>
(  not <font color="Maroon">c<sub>7</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> or <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> or <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="ref" href="waybel_7.html#D2" target="_self">Def2</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:25"/>
<font color="Maroon">c<sub>3</sub></font> is  <a href="waybel_7.html#V2" target="_self">prime</a> <a href="waybel_0.html#NM4">Filter</a> of <font color="Maroon">c<sub>2</sub></font>
 ;<br/>


</div>
