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<div class="add">

<b>let </b><font color="Maroon" title="c1">L</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="orders_2.html#NM1" title="ORDERS_2:NM.1">Poset</a>;<br/><b>let </b><font color="Maroon" title="c2">p</font> be   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">L</font>,<font color="Maroon" title="c1">L</font>;<br/>



<b>assume </b><a NAME="E1:91"/><i><font color="Green" title="E57">A1</font></i>: 
<font color="Maroon" title="c2">p</font> is <a href="orders_3.html#V5" title="ORDERS_3:attr.5">monotone</a>
 ;<br/>

<b>let </b><font color="Maroon" title="c3">Lk</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font>;<br/>

<b>assume </b><a NAME="E2:91"/><i><font color="Green" title="E58">A2</font></i>: 
<font color="Maroon" title="c3">Lk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">k</font> where <font color="Olive" title="b1">k</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> : <font color="Maroon" title="c2">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">k</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b1">k</font> </span>}</span> 
 ;<br/>

<b>let </b><font color="Maroon" title="c4">X</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c3">Lk</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow_0.html#D18" title="YELLOW_0:def.18">YELLOW_0:def 18</a><br/>

<b>assume </b><a NAME="E3:91"/><i><font color="Green" title="E59">A3</font></i>: 
 <a href="yellow_0.html#R2" title="YELLOW_0:pred.2">ex_inf_of</a> <font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c1">L</font>
 ;<br/>

<a NAME="E4:91"/>
<font color="Maroon" title="c2">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K2" title="YELLOW_0:func.2">"/\"</a> <font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c1">L</font></span>)</span> <a href="lattice3.html#R1" title="LATTICE3:pred.1">is_&lt;=_than</a> <font color="Maroon" title="c4">X</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="91_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D8" title="LATTICE3:def.8">LATTICE3:def 8</a><br/>

<b>assume </b><a NAME="E1:91_1"/><i><font color="Green" title="E60">A4</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font>
 ;<br/>

<a NAME="E2:91_1"/><b>then </b>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c3">Lk</font></span>)</span>
 
;<br/>
<a NAME="E3:91_1"/><b>then </b>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Lk</font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#D15" title="YELLOW_0:def.15">YELLOW_0:def 15</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c6">l</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font><b> such that </b><br/><a NAME="E5:91_1"/><i><font color="Green" title="E61">A5</font></i>: 
( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">l</font> &amp; <font color="Maroon" title="c6">l</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c2">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">l</font> )
 <b>by </b><i><a class="txt" href="waybel_1.html#E2:91"><i><font color="Green" title="E58">A2</font></i></a></i>;<br/>
<a NAME="E6:91_1"/>
 <a href="yellow_0.html#K2" title="YELLOW_0:func.2">"/\"</a> <font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c1">L</font> <a href="lattice3.html#R1" title="LATTICE3:pred.1">is_&lt;=_than</a> <font color="Maroon" title="c4">X</font>
 
<b>by </b><i><a class="txt" href="waybel_1.html#E3:91"><i><font color="Green" title="E59">A3</font></i></a>, <a class="ref" href="yellow_0.html#T31" title="YELLOW_0:th.31">YELLOW_0:31</a></i>;<br/>
<a NAME="E7:91_1"/><b>then </b>
<font color="Maroon" title="c5">x</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a>  <a href="yellow_0.html#K2" title="YELLOW_0:func.2">"/\"</a> <font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c1">L</font>
 
<b>by </b><i><a class="txt" href="waybel_1.html#E1:91_1"><i><font color="Green" title="E60">A4</font></i></a>, <a class="ref" href="lattice3.html#D8" title="LATTICE3:def.8">LATTICE3:def 8</a></i>;<br/>
<a NAME="E8:91_1"/><b>then </b>
<font color="Maroon" title="c2">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">x</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c2">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K2" title="YELLOW_0:func.2">"/\"</a> <font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c1">L</font></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel_1.html#E1:91"><i><font color="Green" title="E57">A1</font></i></a>, <a class="ref" href="waybel_1.html#D2" target="_self" title="WAYBEL_1:def.2">Def2</a></i>;<br/>
<b>hence </b><a NAME="E9:91_1"/>
<font color="Maroon" title="c5">x</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c2">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K2" title="YELLOW_0:func.2">"/\"</a> <font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c1">L</font></span>)</span>
 <b>by </b><i><a class="txt" href="waybel_1.html#E5:91_1"><i><font color="Green" title="E61">A5</font></i></a>, <a class="ref" href="orders_2.html#T26" title="ORDERS_2:th.26">ORDERS_2:26</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:91"/><b>then </b>
 <a href="yellow_0.html#K2" title="YELLOW_0:func.2">"/\"</a> <font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c1">L</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c2">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K2" title="YELLOW_0:func.2">"/\"</a> <font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c1">L</font></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel_1.html#E3:91"><i><font color="Green" title="E59">A3</font></i></a>, <a class="ref" href="yellow_0.html#T31" title="YELLOW_0:th.31">YELLOW_0:31</a></i>;<br/>
<a NAME="E6:91"/><b>then </b>
 <a href="yellow_0.html#K2" title="YELLOW_0:func.2">"/\"</a> <font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c1">L</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Lk</font>
 
<b>by </b><i><a class="txt" href="waybel_1.html#E2:91"><i><font color="Green" title="E58">A2</font></i></a></i>;<br/>
<b>hence </b><a NAME="E7:91"/>
 <a href="yellow_0.html#K2" title="YELLOW_0:func.2">"/\"</a> <font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c1">L</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c3">Lk</font></span>)</span>
 <b>by </b><i><a class="ref" href="yellow_0.html#D15" title="YELLOW_0:def.15">YELLOW_0:def 15</a></i>;<br/>


</div>
