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

<b>let </b><font color="Maroon">L</font> be  non <a href="struct_0.html#V3">empty</a> <a href="orders_2.html#NM1">Poset</a>;<br/>

<b>assume </b><a NAME="E1:7"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">L</font> is <a href="lattice3.html#V1">with_suprema</a> or <font color="Maroon">L</font> is <a href="waybel_0.html#V25">/\-complete</a> )
 ;<br/>

<b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>, <font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>, <font color="Maroon">z</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/>

<b>assume </b><a NAME="E2:7"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">z</font> <a href="waybel_3.html#NR3" target="_self">&gt;&gt;</a> <font color="Maroon">x</font>
 ;<br/>

<a NAME="E3:7"/><b>then </b><i><font color="Green">E39</font></i>: 
( <font color="Maroon">z</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">x</font> &amp; ( for <font color="Olive">D</font> being 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">L</font>  st <font color="Maroon">z</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Olive">D</font> holds <br/> ex <font color="Olive">d</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> st <br/>( <font color="Olive">d</font> <a href="hidden.html#R2">in</a> <font color="Olive">D</font> &amp; <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">d</font> ) ) )
 
<b>by </b><i>, </i>;<br/>
<b>assume </b><a NAME="E4:7"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">z</font> <a href="waybel_3.html#NR3" target="_self">&gt;&gt;</a> <font color="Maroon">y</font>
 ;<br/>

<a NAME="E5:7"/><b>then </b><i><font color="Green">E42</font></i>: 
( <font color="Maroon">z</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">y</font> &amp; ( for <font color="Olive">D</font> being 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">L</font>  st <font color="Maroon">z</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Olive">D</font> holds <br/> ex <font color="Olive">d</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> st <br/>( <font color="Olive">d</font> <a href="hidden.html#R2">in</a> <font color="Olive">D</font> &amp; <font color="Maroon">y</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">d</font> ) ) )
 
<b>by </b><i>, </i>;<br/>
<div><b>hereby </b>
<div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:7_1_1"/>
( <font color="Maroon">L</font> is <a href="lattice3.html#V1">with_suprema</a> or <font color="Maroon">L</font> is <a href="waybel_0.html#V25">/\-complete</a> )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_1_1_1"><b>suppose </b></a><a NAME="E1:7_1_1_1"/>
<font color="Maroon">L</font> is <a href="lattice3.html#V1">with_suprema</a>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:7_1_1_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">x</font>,<font color="Maroon">y</font></span><a href="domain_1.html#K7">}</a></span>,<font color="Maroon">L</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T20">YELLOW_0:20</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_1_1_2"><b>suppose </b></a><a NAME="E1:7_1_1_2"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">L</font> is <a href="waybel_0.html#V25">/\-complete</a>
 ;<br/><div class="add"><b>set </b><font color="Maroon">X</font> = <span class="p1">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> : ( <font color="Olive">a</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">x</font> &amp; <font color="Olive">a</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">y</font> ) </span>}</span> ;<br/><a NAME="E2:7_1_1_2"/>
<span class="p1">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> : ( <font color="Olive">a</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">x</font> &amp; <font color="Olive">a</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">y</font> ) </span>}</span>  <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_1_2_1"><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="E1:7_1_1_2_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> : ( <font color="Olive">a</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">x</font> &amp; <font color="Olive">a</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">y</font> ) </span>}</span> 
 ;<br/>

<a NAME="E2:7_1_1_2_1"/><b>then </b>
 ex <font color="Olive">z</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> st <br/>( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">z</font> &amp; <font color="Olive">z</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">x</font> &amp; <font color="Olive">z</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">y</font> )
 
;<br/>
<b>hence </b><a NAME="E3:7_1_1_2_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>
 ;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">X</font> = <span class="p1">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> : ( <font color="Olive">a</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">x</font> &amp; <font color="Olive">a</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">y</font> ) </span>}</span>  as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font> ;<br/><a NAME="E4:7_1_1_2"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i>, </i>;<br/><a NAME="E5:7_1_1_2"/><b>then </b>
 <a href="yellow_0.html#R2">ex_inf_of</a> <font color="Maroon">X</font>,<font color="Maroon">L</font>
 <b>by </b><i>, <a class="ref" href="waybel_0.html#T76">WAYBEL_0:76</a></i>;<br/><b>then consider </b><font color="Maroon">c</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E7:7_1_1_2"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">c</font> <a href="lattice3.html#R1">is_&lt;=_than</a> <font color="Maroon">X</font>
 <b>and </b><br/><a NAME="E8:7_1_1_2"/><i><font color="Green">E49</font></i>: 
for <font color="Olive">d</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>  st <font color="Olive">d</font> <a href="lattice3.html#R1">is_&lt;=_than</a> <font color="Maroon">X</font> holds <br/><font color="Olive">d</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T16">YELLOW_0:16</a></i>;<br/><a NAME="E9:7_1_1_2"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">c</font> <a href="lattice3.html#NR4" target="_self">is_&gt;=_than</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="domain_1.html#K7">}</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D9">LATTICE3:def 9</a><br/>

<b>assume </b><a NAME="E1:7_1_1_2_2"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="domain_1.html#K7">}</a></span>
 ;<br/>

<a NAME="E2:7_1_1_2_2"/>
<font color="Maroon">a</font> <a href="lattice3.html#R1">is_&lt;=_than</a> <font color="Maroon">X</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_1_2_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</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="E1:7_1_1_2_2_1"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/>

<a NAME="E2:7_1_1_2_2_1"/><b>then </b>
 ex <font color="Olive">z</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> st <br/>( <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Olive">z</font> &amp; <font color="Olive">z</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">x</font> &amp; <font color="Olive">z</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">y</font> )
 
;<br/>
<b>hence </b><a NAME="E3:7_1_1_2_2_1"/>
<font color="Maroon">a</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">b</font>
 <b>by </b><i>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:7_1_1_2_2"/>
<font color="Maroon">a</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_1_2_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/><b>assume </b><a NAME="E1:7_1_1_2_3"/>
<font color="Maroon">a</font> <a href="lattice3.html#NR4" target="_self">is_&gt;=_than</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="domain_1.html#K7">}</a></span>
 ;<br/><a NAME="E2:7_1_1_2_3"/><b>then </b>
( <font color="Maroon">a</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">a</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#T8">YELLOW_0:8</a></i>;<br/><a NAME="E3:7_1_1_2_3"/><b>then </b>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/><b>hence </b><a NAME="E4:7_1_1_2_3"/>
<font color="Maroon">c</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">a</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="E11:7_1_1_2"/>
 <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">x</font>,<font color="Maroon">y</font></span><a href="domain_1.html#K7">}</a></span>,<font color="Maroon">L</font>
 <b>by </b><i>, <a class="ref" href="yellow_0.html#T15">YELLOW_0:15</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div>
<b>end;</b></div>

<b>let </b><font color="Maroon">D</font> be  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">L</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_3.html#D1" target="_self">WAYBEL_3:def 1</a><br/>

<b>assume </b><a NAME="E7:7"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">z</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">D</font>
 ;<br/>

<b>then consider </b><font color="Maroon">d1</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E9:7"/><i><font color="Green">E55</font></i>: 
( <font color="Maroon">d1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> &amp; <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">d1</font> )
 <b>by </b><i>, </i>;<br/>
<b>consider </b><font color="Maroon">d2</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E11:7"/><i><font color="Green">E57</font></i>: 
( <font color="Maroon">d2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> &amp; <font color="Maroon">y</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">d2</font> )
 <b>by </b><i>, , </i>;<br/>
<b>consider </b><font color="Maroon">d</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E13:7"/><i><font color="Green">E58</font></i>: 
( <font color="Maroon">d</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> &amp; <font color="Maroon">d1</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">d</font> &amp; <font color="Maroon">d2</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">d</font> )
 <b>by </b><i>, , <a class="ref" href="waybel_0.html#D1">WAYBEL_0:def 1</a></i>;<br/>
<a NAME="E14:7"/><i><font color="Green">E59</font></i>: 
( <font color="Maroon">d</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> &amp; <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">d</font> &amp; <font color="Maroon">y</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">d</font> )
 
<b>by </b><i>, , , <a class="ref" href="orders_2.html#T26">ORDERS_2:26</a></i>;<br/>
<b>take </b>
<font color="Maroon">d</font>
;<br/>

<b>thus </b><a NAME="E15:7"/>
( <font color="Maroon">d</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> &amp; <font color="Maroon">x</font> <a href="lattice3.html#K10">"\/"</a> <font color="Maroon">y</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">d</font> )
 <b>by </b><i>, , <a class="ref" href="yellow_0.html#T18">YELLOW_0:18</a></i>;<br/>


</div>
