<?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#NM1">Poset</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font>;<br/>



<b>assume </b><a NAME="E1:90"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is <a href="orders_3.html#V5">monotone</a>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>assume </b><a NAME="E2:90"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> </span>}</span> 
 ;<br/>

<b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow_0.html#D19">YELLOW_0:def 19</a><br/>

<b>assume </b><a NAME="E3:90"/><i><font color="Green">E59</font></i>: 
 <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<a NAME="E4:90"/>
<font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K1">"\/"</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="lattice3.html#NR4" target="_self">is_&gt;=_than</a> <font color="Maroon">c<sub>4</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="90_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</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#D9">LATTICE3:def 9</a><br/>

<b>assume </b><a NAME="E1:90_1"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<a NAME="E2:90_1"/><b>then </b>
<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="yellow_0.html#K5">subrelstr</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 
;<br/>
<a NAME="E3:90_1"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#D15">YELLOW_0:def 15</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>6</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E5:90_1"/><i><font color="Green">E61</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="txt" href="waybel_1.html#E2:90"><i><font color="Green">E58</font></i></a></i>;<br/>
<a NAME="E6:90_1"/>
 <a href="yellow_0.html#K1">"\/"</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font> <a href="lattice3.html#NR4" target="_self">is_&gt;=_than</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="waybel_1.html#E3:90"><i><font color="Green">E59</font></i></a>, <a class="ref" href="yellow_0.html#T30">YELLOW_0:30</a></i>;<br/>
<a NAME="E7:90_1"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#K1">"\/"</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="waybel_1.html#E1:90_1"><i><font color="Green">E60</font></i></a>, <a class="ref" href="lattice3.html#D9">LATTICE3:def 9</a></i>;<br/>
<a NAME="E8:90_1"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K1">"\/"</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel_1.html#E1:90"><i><font color="Green">E57</font></i></a>, <a class="ref" href="waybel_1.html#D2" target="_self">Def2</a></i>;<br/>
<b>hence </b><a NAME="E9:90_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K1">"\/"</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="waybel_1.html#E5:90_1"><i><font color="Green">E61</font></i></a>, <a class="ref" href="orders_2.html#T26">ORDERS_2:26</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:90"/><b>then </b>
 <a href="yellow_0.html#K1">"\/"</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K1">"\/"</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel_1.html#E3:90"><i><font color="Green">E59</font></i></a>, <a class="ref" href="yellow_0.html#T30">YELLOW_0:30</a></i>;<br/>
<a NAME="E6:90"/><b>then </b>
 <a href="yellow_0.html#K1">"\/"</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="waybel_1.html#E2:90"><i><font color="Green">E58</font></i></a></i>;<br/>
<b>hence </b><a NAME="E7:90"/>
 <a href="yellow_0.html#K1">"\/"</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</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="yellow_0.html#K5">subrelstr</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="yellow_0.html#D15">YELLOW_0:def 15</a></i>;<br/>


</div>
