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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="struct_0.html#V3">empty</a>  <a href="orders_2.html#L1">RelStr</a> ;<br/>

<b>assume </b><a NAME="E1:31"/><i><font color="Green">E13</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> holds  <a href="yellow_0.html#R2">ex_inf_of</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>2</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>set </b><font color="Maroon">c<sub>3</sub></font> = <font color="Maroon">c<sub>2</sub></font> <a href="xboole_0.html#K3">/\</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>4</sub></font> =  <a href="yellow_0.html#K2">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="xboole_0.html#K3">/\</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span>,<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>2</sub></font> <a href="xboole_0.html#K3">/\</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/>
<a NAME="E3:31"/><i><font color="Green">E14</font></i>: 
 <a href="yellow_0.html#R2">ex_inf_of</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="yellow_2.html#E1:31"><i><font color="Green">E13</font></i></a></i>;<br/>
<a NAME="E4:31"/><b>then </b><i><font color="Green">E15</font></i>: 
 <a href="yellow_0.html#R2">ex_inf_of</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#T50">YELLOW_0:50</a></i>;<br/>
<a NAME="E5:31"/><i><font color="Green">E16</font></i>: 
 <a href="yellow_0.html#K2">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="xboole_0.html#K3">/\</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font> <a href="lattice3.html#R1">is_&lt;=_than</a> <font color="Maroon">c<sub>2</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></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="E1:31_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<a NAME="E2:31_1"/><b>then </b><i><font color="Green">E17</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E3:31_1"/>
 <a href="yellow_0.html#K2">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="xboole_0.html#K3">/\</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font> <a href="lattice3.html#R1">is_&lt;=_than</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="yellow_2.html#E3:31"><i><font color="Green">E14</font></i></a>, <a class="ref" href="yellow_0.html#D10">YELLOW_0:def 10</a></i>;<br/>
<b>hence </b><a NAME="E4:31_1"/>
 <a href="yellow_0.html#K2">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="xboole_0.html#K3">/\</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="yellow_2.html#E2:31_1"><i><font color="Green">E17</font></i></a>, <a class="ref" href="lattice3.html#D8">LATTICE3:def 8</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:31"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="lattice3.html#R1">is_&lt;=_than</a> <font color="Maroon">c<sub>2</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="orders_2.html#R1">&lt;=</a>  <a href="yellow_0.html#K2">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="xboole_0.html#K3">/\</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>assume </b><a NAME="E1:31_2"/><i><font color="Green">E17</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="lattice3.html#R1">is_&lt;=_than</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<a NAME="E2:31_2"/>
<font color="Maroon">c<sub>5</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="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/>
<a NAME="E3:31_2"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="lattice3.html#R1">is_&lt;=_than</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="yellow_2.html#E1:31_2"><i><font color="Green">E17</font></i></a>, <a class="ref" href="yellow_0.html#T9">YELLOW_0:9</a></i>;<br/>
<b>hence </b><a NAME="E4:31_2"/>
<font color="Maroon">c<sub>6</sub></font> <a href="orders_2.html#R1">&lt;=</a>  <a href="yellow_0.html#K2">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="xboole_0.html#K3">/\</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="yellow_2.html#E3:31"><i><font color="Green">E14</font></i></a>, <a class="ref" href="yellow_0.html#D10">YELLOW_0:def 10</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:31"/>
(  <a href="yellow_0.html#R2">ex_inf_of</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font> &amp;  <a href="yellow_0.html#K2">"/\"</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K2">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="xboole_0.html#K3">/\</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="yellow_2.html#E4:31"><i><font color="Green">E15</font></i></a>, <a class="txt" href="yellow_2.html#E5:31"><i><font color="Green">E16</font></i></a>, <a class="ref" href="yellow_0.html#D10">YELLOW_0:def 10</a></i>;<br/>


</div>
