<?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/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:70_1"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">L</font> is <a href="lattice3.html#V2">with_infima</a>
 ;<br/><b>let </b><font color="Maroon">X</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font>;<br/><b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ] <b>means </b>( not <font color="Maroon">a<sub>1</sub></font> is <a href="xboole_0.html#V1">empty</a> implies  <a href="yellow_0.html#R2" target="_self">ex_inf_of</a> <font color="Maroon">a<sub>1</sub></font>,<font color="Maroon">L</font> );<br/><a NAME="E2:70_1"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">X</font> is <a href="finset_1.html#V1">finite</a>
 ;<br/><a NAME="E3:70_1"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="xboole_0.html#K1">{}</a> ]
 ;<br/><a NAME="E4:70_1"/><i><font color="Green">E31</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">Y</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Olive">Y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">Y</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">Y</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">x</font></span><a href="tarski.html#K1">}</a></span>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">Y</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:70_1_1"/><i><font color="Green">E32</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">Y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font> )
 <b>and </b><br/><a NAME="E2:70_1_1"/><i><font color="Green">E33</font></i>: 
( not <font color="Maroon">Y</font> is <a href="xboole_0.html#V1">empty</a> implies  <a href="yellow_0.html#R2" target="_self">ex_inf_of</a> <font color="Maroon">Y</font>,<font color="Maroon">L</font> )
 ;<br/>

<b>assume </b><a NAME="E3:70_1_1"/>
not <font color="Maroon">Y</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> is <a href="xboole_0.html#V1">empty</a>
 ;<br/>

<b>reconsider </b><font color="Maroon">z</font> = <font color="Maroon">x</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> <b>by </b><i/>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="70_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:70_1_1_1"/>
( <font color="Maroon">Y</font> is <a href="xboole_0.html#V1">empty</a> or  not <font color="Maroon">Y</font> is <a href="xboole_0.html#V1">empty</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="70_1_1_1_1"><b>suppose </b></a><a NAME="E1:70_1_1_1_1"/>
<font color="Maroon">Y</font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/><div class="add"><a NAME="E2:70_1_1_1_1"/><b>then </b>
<font color="Maroon">Y</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">z</font></span><a href="domain_1.html#K6">}</a></span>
 ;<br/><b>hence </b><a NAME="E3:70_1_1_1_1"/>
 <a href="yellow_0.html#R2" target="_self">ex_inf_of</a> <font color="Maroon">Y</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>,<font color="Maroon">L</font>
 <b>by </b><i><a class="txt" href="yellow_0.html#E1:70_1_1"><i><font color="Green">E32</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="70_1_1_1_2"><b>suppose </b></a><a NAME="E1:70_1_1_1_2"/><i><font color="Green">E67</font></i>: 
not <font color="Maroon">Y</font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/><div class="add"><b>thus </b><a NAME="E2:70_1_1_1_2"/>
 <a href="yellow_0.html#R2" target="_self">ex_inf_of</a> <font color="Maroon">Y</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>,<font color="Maroon">L</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_1_1_1_2_1"><b>proof </b></a><div class="add">

<b>take </b><font color="Maroon">a</font> = <span class="p1">(<span class="default"><a href="yellow_0.html#K2" target="_self">"/\"</a> <font color="Maroon">Y</font>,<font color="Maroon">L</font></span>)</span> <a href="lattice3.html#K11">"/\"</a> <font color="Maroon">z</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow_0.html#D8" target="_self">YELLOW_0:def 8</a><br/>

<a NAME="E1:70_1_1_1_2_1"/><i><font color="Green">E68</font></i>: 
 <a href="yellow_0.html#R2" target="_self">ex_inf_of</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#K2" target="_self">"/\"</a> <font color="Maroon">Y</font>,<font color="Maroon">L</font></span>)</span>,<font color="Maroon">z</font></span><a href="domain_1.html#K7">}</a></span>,<font color="Maroon">L</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E2:70_1_1_1_2_1"/><b>then </b>
( <font color="Maroon">Y</font> <a href="lattice3.html#NR3" target="_self">is_&gt;=_than</a>  <a href="yellow_0.html#K2" target="_self">"/\"</a> <font color="Maroon">Y</font>,<font color="Maroon">L</font> &amp;  <a href="yellow_0.html#K2" target="_self">"/\"</a> <font color="Maroon">Y</font>,<font color="Maroon">L</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">a</font> &amp; <font color="Maroon">z</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">a</font> )
 
<b>by </b><i><a class="ref" href="yellow_0.html#T4" target="_self">Th4</a>, <a class="ref" href="yellow_0.html#T5" target="_self">Th5</a>, , </i>;<br/>
<a NAME="E3:70_1_1_1_2_1"/><b>then </b>
( <font color="Maroon">Y</font> <a href="lattice3.html#NR3" target="_self">is_&gt;=_than</a> <font color="Maroon">a</font> &amp; <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> <a href="lattice3.html#NR3" target="_self">is_&gt;=_than</a> <font color="Maroon">a</font> )
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E4:70_1_1_1_2_1"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">Y</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> <a href="lattice3.html#NR3" target="_self">is_&gt;=_than</a> <font color="Maroon">a</font>
 <b>by </b><i/>;<br/>

<div><b>hereby </b>
<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>;<br/><b>assume </b><a NAME="E1:70_1_1_1_2_1_1"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">Y</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> <a href="lattice3.html#NR3" target="_self">is_&gt;=_than</a> <font color="Maroon">b</font>
 ;<br/><a NAME="E2:70_1_1_1_2_1_1"/>
( <font color="Maroon">Y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> )
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a>, <a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/><a NAME="E3:70_1_1_1_2_1_1"/><b>then </b>
( <font color="Maroon">Y</font> <a href="lattice3.html#NR3" target="_self">is_&gt;=_than</a> <font color="Maroon">b</font> &amp; <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> )
 <b>by </b><i><a class="ref" href="yellow_0.html#T7" target="_self">Th7</a>, , <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E4:70_1_1_1_2_1_1"/><b>then </b>
(  <a href="yellow_0.html#K2" target="_self">"/\"</a> <font color="Maroon">Y</font>,<font color="Maroon">L</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">b</font> &amp; <font color="Maroon">z</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">b</font> )
 <b>by </b><i><a class="ref" href="yellow_0.html#T4" target="_self">Th4</a>, <a class="ref" href="yellow_0.html#T5" target="_self">Th5</a>, <a class="ref" href="yellow_0.html#T7" target="_self">Th7</a>, , <a class="ref" href="lattice3.html#D8">LATTICE3:def 8</a></i>;<br/><b>hence </b><a NAME="E5:70_1_1_1_2_1_1"/>
<font color="Maroon">b</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">a</font>
 <b>by </b><i>, </i>;<br/></div>
<b>end;</b></div>

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

<b>assume </b><b>that </b><br/><a NAME="E6:70_1_1_1_2_1"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">Y</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> <a href="lattice3.html#NR3" target="_self">is_&gt;=_than</a> <font color="Maroon">c</font>
 <b>and </b><br/><a NAME="E7:70_1_1_1_2_1"/><i><font color="Green">E101</font></i>: 
for <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>  st <font color="Maroon">Y</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> <a href="lattice3.html#NR3" target="_self">is_&gt;=_than</a> <font color="Olive">b</font> holds <br/><font color="Olive">b</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c</font>
 ;<br/>

<a NAME="E8:70_1_1_1_2_1"/><i><font color="Green">E102</font></i>: 
<font color="Maroon">a</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#T6" target="_self">Th6</a>, </i>;<br/>
<a NAME="E9:70_1_1_1_2_1"/>
( <font color="Maroon">Y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> )
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a>, <a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E10:70_1_1_1_2_1"/><b>then </b>
( <font color="Maroon">Y</font> <a href="lattice3.html#NR3" target="_self">is_&gt;=_than</a> <font color="Maroon">c</font> &amp; <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> )
 
<b>by </b><i><a class="ref" href="yellow_0.html#T7" target="_self">Th7</a>, , <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E11:70_1_1_1_2_1"/><b>then </b>
(  <a href="yellow_0.html#K2" target="_self">"/\"</a> <font color="Maroon">Y</font>,<font color="Maroon">L</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c</font> &amp; <font color="Maroon">z</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c</font> )
 
<b>by </b><i><a class="ref" href="yellow_0.html#T4" target="_self">Th4</a>, <a class="ref" href="yellow_0.html#T5" target="_self">Th5</a>, <a class="ref" href="yellow_0.html#T7" target="_self">Th7</a>, , <a class="ref" href="lattice3.html#D8">LATTICE3:def 8</a></i>;<br/>
<a NAME="E12:70_1_1_1_2_1"/><b>then </b>
<font color="Maroon">c</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">a</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E13:70_1_1_1_2_1"/>
<font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T8" target="_self">Th8</a>, <a class="ref" href="orders_2.html#T25">ORDERS_2:25</a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E5:70_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">X</font>]
 <b>from </b><i><a class="ref" href="finset_1.html#S2">FINSET_1:sch 2</a>(, <a class="ref" href="yellow_0.html#T2" target="_self">Th2</a>, );<br/></i><b>hence </b><a NAME="E6:70_1"/>
 <a href="yellow_0.html#R2" target="_self">ex_inf_of</a> <font color="Maroon">X</font>,<font color="Maroon">L</font>
 ;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:70"/>
for <font color="Olive">X</font> being non <a href="xboole_0.html#V1">empty</a> <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font> holds   <a href="yellow_0.html#R2" target="_self">ex_inf_of</a> <font color="Olive">X</font>,<font color="Maroon">L</font>
 ;<br/>

<a NAME="E3:70"/><b>then </b>
for <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> holds   <a href="yellow_0.html#R2" target="_self">ex_inf_of</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Olive">a</font>,<font color="Olive">b</font></span><a href="domain_1.html#K7">}</a></span>,<font color="Maroon">L</font>
 
;<br/>
<b>hence </b><a NAME="E4:70"/>
<font color="Maroon">L</font> is <a href="lattice3.html#V2">with_infima</a>
 <b>by </b><i/>;<br/>


</div>
