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

<b>set </b><font color="Maroon">c<sub>4</sub></font> = <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="tarski.html#K4">]</a></span> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>, B is    <a href="struct_0.html#M1">Element</a> of <font color="Maroon">c<sub>3</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>2</sub></font> </span>}</span> ;<br/>
<a NAME="E1:20_1_1"/>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow19.html#K3" target="_self">a_net</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><a href="yellow19.html#K3" target="_self">a_net</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><a href="yellow19.html#K3" target="_self">a_net</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> holds <br/>ex <font color="Olive">b<sub>3</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow19.html#K3" target="_self">a_net</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> st <br/>( <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><a href="yellow19.html#K3" target="_self">a_net</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> &amp; <font color="Olive">b<sub>1</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">b<sub>3</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">b<sub>3</sub></font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow19.html#K3" target="_self">a_net</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>;<br/>

<b>assume </b><a NAME="E1:20_1_1_1"/>
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><a href="yellow19.html#K3" target="_self">a_net</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><a href="yellow19.html#K3" target="_self">a_net</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> )
 ;<br/>

<a NAME="E2:20_1_1_1"/>
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow19.html#K3" target="_self">a_net</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow19.html#K3" target="_self">a_net</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> )
 
;<br/>
<a NAME="E3:20_1_1_1"/><b>then </b><i><font color="Green">E11</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="tarski.html#K4">]</a></span> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>, B is    <a href="struct_0.html#M1">Element</a> of <font color="Maroon">c<sub>3</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>2</sub></font> </span>}</span>  &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="tarski.html#K4">]</a></span> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>, B is    <a href="struct_0.html#M1">Element</a> of <font color="Maroon">c<sub>3</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>2</sub></font> </span>}</span>  )
 
<b>by </b><i><a class="ref" href="yellow19.html#D4" target="_self">Def4</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>8</sub></font> being    <a href="struct_0.html#M1">Element</a> of <font color="Maroon">c<sub>3</sub></font><b> such that </b><br/><a NAME="E5:20_1_1_1"/><i><font color="Green">E12</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> )
 ;<br/>
<b>consider </b><font color="Maroon">c<sub>9</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>10</sub></font> being    <a href="struct_0.html#M1">Element</a> of <font color="Maroon">c<sub>3</sub></font><b> such that </b><br/><a NAME="E7:20_1_1_1"/><i><font color="Green">E13</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font> )
 <b>by </b><i><a class="txt" href="yellow19.html#E3:20_1_1_1"><i><font color="Green">E11</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>8</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>10</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>11</sub></font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">c<sub>12</sub></font> as    <a href="struct_0.html#M1">Element</a> of <font color="Maroon">c<sub>3</sub></font> <b>by </b><i><a class="ref" href="waybel_0.html#T41">WAYBEL_0:41</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>14</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>13</sub></font>;<br/>
<a NAME="E12:20_1_1_1"/>
not  <a href="yellow_0.html#K3">Bottom</a> <span class="p1">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="waybel_7.html#T8">WAYBEL_7:8</a></i>;<br/>
<a NAME="E13:20_1_1_1"/><b>then </b>
not  <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="yellow_1.html#T18">YELLOW_1:18</a></i>;<br/>
<a NAME="E14:20_1_1_1"/><b>then </b><i><font color="Green">E14</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
;<br/>
<a NAME="E15:20_1_1_1"/><b>then </b><i><font color="Green">E15</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>13</sub></font>
 
;<br/>
<a NAME="E16:20_1_1_1"/>
<font color="Maroon">c<sub>13</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="waybel_8.html#T28">WAYBEL_8:28</a></i>;<br/>
<a NAME="E17:20_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="yellow19.html#E15:20_1_1_1"><i><font color="Green">E15</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>15</sub></font> = <font color="Maroon">c<sub>14</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<a NAME="E19:20_1_1_1"/><i><font color="Green">E16</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>13</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="tarski.html#K4">]</a></span> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>, B is    <a href="struct_0.html#M1">Element</a> of <font color="Maroon">c<sub>3</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>2</sub></font> </span>}</span> 
 
<b>by </b><i><a class="txt" href="yellow19.html#E14:20_1_1_1"><i><font color="Green">E14</font></i></a></i>;<br/>
<a NAME="E20:20_1_1_1"/><b>then </b><i><font color="Green">E17</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>13</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow19.html#K3" target="_self">a_net</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="yellow19.html#D4" target="_self">Def4</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>16</sub></font> = <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>13</sub></font></span><a href="tarski.html#K4">]</a></span> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow19.html#K3" target="_self">a_net</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <b>by </b><i><a class="txt" href="yellow19.html#E19:20_1_1_1"><i><font color="Green">E16</font></i></a>, <a class="ref" href="yellow19.html#D4" target="_self">Def4</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>17</sub></font> = <font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>18</sub></font> = <font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>19</sub></font> = <font color="Maroon">c<sub>16</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow19.html#K3" target="_self">a_net</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> ;<br/>
<a NAME="E23:20_1_1_1"/><i><font color="Green">E18</font></i>: 
( <font color="Maroon">c<sub>17</sub></font> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>18</sub></font> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>19</sub></font> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> )
 
<b>by </b><i><a class="txt" href="yellow19.html#E5:20_1_1_1"><i><font color="Green">E12</font></i></a>, <a class="txt" href="yellow19.html#E7:20_1_1_1"><i><font color="Green">E13</font></i></a>, <a class="ref" href="mcart_1.html#D2">MCART_1:def 2</a></i>;<br/>
<a NAME="E24:20_1_1_1"/>
( <font color="Maroon">c<sub>13</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>11</sub></font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">c<sub>12</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">c<sub>12</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>12</sub></font> )
 
<b>by </b><i><a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a>, <a class="ref" href="yellow_1.html#T17">YELLOW_1:17</a></i>;<br/>
<a NAME="E25:20_1_1_1"/><b>then </b><i><font color="Green">E19</font></i>: 
( <font color="Maroon">c<sub>13</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>12</sub></font> )
 
<b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>16</sub></font>
;<br/>

<b>thus </b><a NAME="E26:20_1_1_1"/>
( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><a href="yellow19.html#K3" target="_self">a_net</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>16</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>16</sub></font> )
 <b>by </b><i><a class="txt" href="yellow19.html#E20:20_1_1_1"><i><font color="Green">E17</font></i></a>, <a class="txt" href="yellow19.html#E23:20_1_1_1"><i><font color="Green">E18</font></i></a>, <a class="txt" href="yellow19.html#E25:20_1_1_1"><i><font color="Green">E19</font></i></a>, <a class="ref" href="yellow19.html#D4" target="_self">Def4</a>, <a class="ref" href="pre_topc.html#T12">PRE_TOPC:12</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:20_1_1"/><b>then </b>
 <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><a href="yellow19.html#K3" target="_self">a_net</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> is <a href="waybel_0.html#V1">directed</a>
 
<b>by </b><i><a class="ref" href="waybel_0.html#D1">WAYBEL_0:def 1</a></i>;<br/>
<b>hence </b><a NAME="E3:20_1_1"/>
 <a href="yellow19.html#K3" target="_self">a_net</a> <font color="Maroon">c<sub>3</sub></font> is <a href="waybel_0.html#V7">directed</a>
 <b>by </b><i><a class="ref" href="waybel_0.html#D6">WAYBEL_0:def 6</a></i>;<br/>


</div>
