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

<b>let </b><font color="Maroon">S</font> be  non <a href="struct_0.html#V3">empty</a> <a href="orders_2.html#NM1">Poset</a>;<br/><b>let </b><font color="Maroon">N</font> be  <a href="orders_2.html#V2">reflexive</a> <a href="waybel_0.html#V8">monotone</a> <a href="waybel_0.html#NM2">net</a> of <font color="Maroon">S</font>;<br/>



<b>set </b><font color="Maroon">X</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_0.html#K2">"/\"</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default"><font color="Maroon">N</font> <a href="waybel_0.html#K2">.</a> <font color="Olive">i</font></span>)</span> where <font color="Olive">i</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font> : <font color="Olive">i</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Olive">j</font> </span>}</span> ,<font color="Maroon">S</font></span>)</span> where <font color="Olive">j</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font> : verum </span>}</span> ;<br/>
<b>consider </b><font color="Maroon">jj</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font>;<br/>
<a NAME="E2:35"/><i><font color="Green">E26</font></i>: 
 <a href="yellow_0.html#K2">"/\"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">N</font> <a href="waybel_0.html#K2">.</a> <font color="Olive">i</font></span>)</span> where <font color="Olive">i</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font> : <font color="Olive">i</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">jj</font> </span>}</span> ,<font color="Maroon">S</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_0.html#K2">"/\"</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default"><font color="Maroon">N</font> <a href="waybel_0.html#K2">.</a> <font color="Olive">i</font></span>)</span> where <font color="Olive">i</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font> : <font color="Olive">i</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Olive">j</font> </span>}</span> ,<font color="Maroon">S</font></span>)</span> where <font color="Olive">j</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font> : verum </span>}</span> 
 
;<br/>
<a NAME="E3:35"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_0.html#K2">"/\"</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default"><font color="Maroon">N</font> <a href="waybel_0.html#K2">.</a> <font color="Olive">i</font></span>)</span> where <font color="Olive">i</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font> : <font color="Olive">i</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Olive">j</font> </span>}</span> ,<font color="Maroon">S</font></span>)</span> where <font color="Olive">j</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font> : verum </span>}</span>  <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:35_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_0.html#K2">"/\"</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default"><font color="Maroon">N</font> <a href="waybel_0.html#K2">.</a> <font color="Olive">i</font></span>)</span> where <font color="Olive">i</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font> : <font color="Olive">i</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Olive">j</font> </span>}</span> ,<font color="Maroon">S</font></span>)</span> where <font color="Olive">j</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font> : verum </span>}</span> 
 ;<br/>

<a NAME="E2:35_1"/><b>then </b>
 ex <font color="Olive">j</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font> st <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K2">"/\"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">N</font> <a href="waybel_0.html#K2">.</a> <font color="Olive">i</font></span>)</span> where <font color="Olive">i</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font> : <font color="Olive">i</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Olive">j</font> </span>}</span> ,<font color="Maroon">S</font>
 
;<br/>
<b>hence </b><a NAME="E3:35_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">X</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_0.html#K2">"/\"</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default"><font color="Maroon">N</font> <a href="waybel_0.html#K2">.</a> <font color="Olive">i</font></span>)</span> where <font color="Olive">i</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font> : <font color="Olive">i</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Olive">j</font> </span>}</span> ,<font color="Maroon">S</font></span>)</span> where <font color="Olive">j</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font> : verum </span>}</span>  as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">S</font> <b>by </b><i/>;<br/>
<a NAME="E5:35"/>
<font color="Maroon">X</font> is <a href="waybel_0.html#V1">directed</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font>, <font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_0.html#D1">WAYBEL_0:def 1</a><br/>

<b>assume </b><a NAME="E1:35_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/>

<b>then consider </b><font color="Maroon">j1</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font><b> such that </b><br/><a NAME="E3:35_2"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K2">"/\"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">N</font> <a href="waybel_0.html#K2">.</a> <font color="Olive">i</font></span>)</span> where <font color="Olive">i</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font> : <font color="Olive">i</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">j1</font> </span>}</span> ,<font color="Maroon">S</font>
 ;<br/>
<b>assume </b><a NAME="E4:35_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/>

<b>then consider </b><font color="Maroon">j2</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font><b> such that </b><br/><a NAME="E6:35_2"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K2">"/\"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">N</font> <a href="waybel_0.html#K2">.</a> <font color="Olive">i</font></span>)</span> where <font color="Olive">i</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font> : <font color="Olive">i</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">j2</font> </span>}</span> ,<font color="Maroon">S</font>
 ;<br/>
<b>reconsider </b><font color="Maroon">j1</font> = <font color="Maroon">j1</font>, <font color="Maroon">j2</font> = <font color="Maroon">j2</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font> ;<br/>
<a NAME="E8:35_2"/>
(  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">N</font> is <a href="waybel_0.html#V1">directed</a> &amp;  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">N</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">N</font> )
 
<b>by </b><i><a class="ref" href="pre_topc.html#T12">PRE_TOPC:12</a>, <a class="ref" href="waybel_0.html#D6">WAYBEL_0:def 6</a></i>;<br/>
<b>then consider </b><font color="Maroon">j</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font><b> such that </b><br/><a NAME="E10:35_2"/><i><font color="Green">E38</font></i>: 
( <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">N</font> &amp; <font color="Maroon">j</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">j1</font> &amp; <font color="Maroon">j</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">j2</font> )
 <b>by </b><i><a class="ref" href="waybel_0.html#D1">WAYBEL_0:def 1</a></i>;<br/>
<b>set </b><font color="Maroon">z</font> =  <a href="yellow_0.html#K2">"/\"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">N</font> <a href="waybel_0.html#K2">.</a> <font color="Olive">i</font></span>)</span> where <font color="Olive">i</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font> : <font color="Olive">i</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">j</font> </span>}</span> ,<font color="Maroon">S</font>;<br/>
<b>take </b>
 <a href="yellow_0.html#K2">"/\"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">N</font> <a href="waybel_0.html#K2">.</a> <font color="Olive">i</font></span>)</span> where <font color="Olive">i</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font> : <font color="Olive">i</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">j</font> </span>}</span> ,<font color="Maroon">S</font>
;<br/>

<b>thus </b><a NAME="E11:35_2"/>
 <a href="yellow_0.html#K2">"/\"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">N</font> <a href="waybel_0.html#K2">.</a> <font color="Olive">i</font></span>)</span> where <font color="Olive">i</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font> : <font color="Olive">i</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">j</font> </span>}</span> ,<font color="Maroon">S</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/>

<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font>) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">N</font> <a href="waybel_0.html#K2">.</a> <font color="Olive">i</font></span>)</span> where <font color="Olive">i</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font> : <font color="Olive">i</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">a<sub>1</sub></font> </span>}</span> ;<br/>
<a NAME="E12:35_2"/><i><font color="Green">E39</font></i>: 
for <font color="Olive">j</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font> holds   <a href="yellow_0.html#R2">ex_inf_of</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">j</font>),<font color="Maroon">S</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">j</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font>;<br/>

<b>reconsider </b><font color="Maroon">j'</font> = <font color="Maroon">j</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font> ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_2_1_1"><b>now </b></a><div class="add"><b>take </b><font color="Maroon">x</font> = <font color="Maroon">N</font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">j</font>;<br/><a NAME="E1:35_2_1_1"/>
<font color="Maroon">j'</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">j'</font>
 ;<br/><a NAME="E2:35_2_1_1"/><b>then </b><i><font color="Green">E42</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">j</font>)
 ;<br/><b>thus </b><a NAME="E3:35_2_1_1"/>
<font color="Maroon">x</font> <a href="lattice3.html#R1">is_&lt;=_than</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">j</font>)
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_2_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D8">LATTICE3:def 8</a><br/>

<b>assume </b><a NAME="E1:35_2_1_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">j</font>)
 ;<br/>

<a NAME="E2:35_2_1_1_1"/><b>then </b>
 ex <font color="Olive">i</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font> st <br/>( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">N</font> <a href="waybel_0.html#K2">.</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">j</font> )
 
;<br/>
<b>hence </b><a NAME="E3:35_2_1_1_1"/>
<font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">y</font>
 <b>by </b><i><a class="ref" href="waybel11.html#D9">WAYBEL11:def 9</a></i>;<br/>


</div><b>end;</b></div><b>let </b><font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font>;<br/><b>assume </b><a NAME="E4:35_2_1_1"/>
<font color="Maroon">y</font> <a href="lattice3.html#R1">is_&lt;=_than</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">j</font>)
 ;<br/><b>hence </b><a NAME="E5:35_2_1_1"/>
<font color="Maroon">y</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">x</font>
 <b>by </b><i>, <a class="ref" href="lattice3.html#D8">LATTICE3:def 8</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:35_2_1"/>
 <a href="yellow_0.html#R2">ex_inf_of</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">j</font>),<font color="Maroon">S</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T16">YELLOW_0:16</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E13:35_2"/><b>then </b><i><font color="Green">E43</font></i>: 
 <a href="yellow_0.html#R2">ex_inf_of</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">j1</font>),<font color="Maroon">S</font>
 
;<br/>
<a NAME="E14:35_2"/><i><font color="Green">E44</font></i>: 
 <a href="yellow_0.html#R2">ex_inf_of</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">j2</font>),<font color="Maroon">S</font>
 
<b>by </b><i/>;<br/>
<a NAME="E15:35_2"/><i><font color="Green">E45</font></i>: 
 <a href="yellow_0.html#R2">ex_inf_of</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">j</font>),<font color="Maroon">S</font>
 
<b>by </b><i/>;<br/>
<b>set </b><font color="Maroon">A</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">N</font> <a href="waybel_0.html#K2">.</a> <font color="Olive">i</font></span>)</span> where <font color="Olive">i</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font> : <font color="Olive">i</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">j</font> </span>}</span> ;<br/>
<a NAME="E16:35_2"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">N</font> <a href="waybel_0.html#K2">.</a> <font color="Olive">i</font></span>)</span> where <font color="Olive">i</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font> : <font color="Olive">i</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">j</font> </span>}</span>  <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">N</font> <a href="waybel_0.html#K2">.</a> <font color="Olive">k</font></span>)</span> where <font color="Olive">k</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font> : <font color="Olive">k</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">j1</font> </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:35_2_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">N</font> <a href="waybel_0.html#K2">.</a> <font color="Olive">i</font></span>)</span> where <font color="Olive">i</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font> : <font color="Olive">i</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">j</font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">k</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font><b> such that </b><br/><a NAME="E3:35_2_2"/><i><font color="Green">E51</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">N</font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">j</font> )
 ;<br/>
<b>reconsider </b><font color="Maroon">k</font> = <font color="Maroon">k</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font> ;<br/>
<a NAME="E5:35_2_2"/>
<font color="Maroon">k</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">j1</font>
 
<b>by </b><i>, , <a class="ref" href="orders_2.html#T26">ORDERS_2:26</a></i>;<br/>
<b>hence </b><a NAME="E6:35_2_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">N</font> <a href="waybel_0.html#K2">.</a> <font color="Olive">k</font></span>)</span> where <font color="Olive">k</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font> : <font color="Olive">k</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">j1</font> </span>}</span> 
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E17:35_2"/>
 <a href="yellow_0.html#K2">"/\"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">N</font> <a href="waybel_0.html#K2">.</a> <font color="Olive">i</font></span>)</span> where <font color="Olive">i</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font> : <font color="Olive">i</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">j</font> </span>}</span> ,<font color="Maroon">S</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">x</font>
 <b>by </b><i>, , , <a class="ref" href="yellow_0.html#T35">YELLOW_0:35</a></i>;<br/>

<a NAME="E18:35_2"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">N</font> <a href="waybel_0.html#K2">.</a> <font color="Olive">i</font></span>)</span> where <font color="Olive">i</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font> : <font color="Olive">i</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">j</font> </span>}</span>  <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">N</font> <a href="waybel_0.html#K2">.</a> <font color="Olive">k</font></span>)</span> where <font color="Olive">k</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font> : <font color="Olive">k</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">j2</font> </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_2_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:35_2_3"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">N</font> <a href="waybel_0.html#K2">.</a> <font color="Olive">i</font></span>)</span> where <font color="Olive">i</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font> : <font color="Olive">i</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">j</font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">k</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font><b> such that </b><br/><a NAME="E3:35_2_3"/><i><font color="Green">E52</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">N</font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">j</font> )
 ;<br/>
<b>reconsider </b><font color="Maroon">k</font> = <font color="Maroon">k</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font> ;<br/>
<a NAME="E5:35_2_3"/>
<font color="Maroon">k</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">j2</font>
 
<b>by </b><i>, , <a class="ref" href="orders_2.html#T26">ORDERS_2:26</a></i>;<br/>
<b>hence </b><a NAME="E6:35_2_3"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">N</font> <a href="waybel_0.html#K2">.</a> <font color="Olive">k</font></span>)</span> where <font color="Olive">k</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font> : <font color="Olive">k</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">j2</font> </span>}</span> 
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E19:35_2"/>
 <a href="yellow_0.html#K2">"/\"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">N</font> <a href="waybel_0.html#K2">.</a> <font color="Olive">i</font></span>)</span> where <font color="Olive">i</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font> : <font color="Olive">i</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">j</font> </span>}</span> ,<font color="Maroon">S</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">y</font>
 <b>by </b><i>, , , <a class="ref" href="yellow_0.html#T35">YELLOW_0:35</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:35"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_0.html#K2">"/\"</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default"><font color="Maroon">N</font> <a href="waybel_0.html#K2">.</a> <font color="Olive">i</font></span>)</span> where <font color="Olive">i</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font> : <font color="Olive">i</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Olive">j</font> </span>}</span> ,<font color="Maroon">S</font></span>)</span> where <font color="Olive">j</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font> : verum </span>}</span>  is  non <a href="xboole_0.html#V1">empty</a> <a href="waybel_0.html#V1">directed</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">S</font>
 ;<br/>


</div>
