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<b>let </b><font color="Maroon" title="c1">L</font> be  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> <a href="orders_2.html#V2" title="ORDERS_2:attr.2">reflexive</a> <a href="orders_2.html#V4" title="ORDERS_2:attr.4">antisymmetric</a>  <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a> ;<br/><b>let </b><font color="Maroon" title="c2">J</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c3">f</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c2">J</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:39"/><i><font color="Green" title="E25">A1</font></i>: 
(  <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">L</font> &amp;  <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K1" title="WAYBEL_0:func.1">netmap</a> <span class="p2">(<span class="default"><a href="waybel_2.html#K2" title="WAYBEL_2:func.2">FinSups</a> <font color="Maroon" title="c3">f</font></span>)</span>,<font color="Maroon" title="c1">L</font></span>)</span>,<font color="Maroon" title="c1">L</font> )
 <b>and </b><br/><a NAME="E2:39"/><i><font color="Green" title="E26">A2</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> <font color="Maroon" title="c2">J</font> holds   <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Olive" title="b1">x</font>,<font color="Maroon" title="c1">L</font>
 ;<br/>

<b>set </b><font color="Maroon" title="c4">h</font> =  <a href="waybel_0.html#K1" title="WAYBEL_0:func.1">netmap</a> <span class="p1">(<span class="default"><a href="waybel_2.html#K2" title="WAYBEL_2:func.2">FinSups</a> <font color="Maroon" title="c3">f</font></span>)</span>,<font color="Maroon" title="c1">L</font>;<br/>
<a NAME="E3:39"/>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c3">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K1" title="WAYBEL_0:func.1">netmap</a> <span class="p2">(<span class="default"><a href="waybel_2.html#K2" title="WAYBEL_2:func.2">FinSups</a> <font color="Maroon" title="c3">f</font></span>)</span>,<font color="Maroon" title="c1">L</font></span>)</span>
 
<b>by </b><i><a class="ref" href="waybel_2.html#T25" target="_self" title="WAYBEL_2:th.25">Th25</a></i>;<br/>
<a NAME="E4:39"/><b>then </b><i><font color="Green" title="E27">A3</font></i>: 
 <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c3">f</font></span>)</span>,<font color="Maroon" title="c1">L</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a>  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K1" title="WAYBEL_0:func.1">netmap</a> <span class="p3">(<span class="default"><a href="waybel_2.html#K2" title="WAYBEL_2:func.2">FinSups</a> <font color="Maroon" title="c3">f</font></span>)</span>,<font color="Maroon" title="c1">L</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel_2.html#E1:39"><i><font color="Green" title="E25">A1</font></i></a>, <a class="ref" href="yellow_0.html#T34" title="YELLOW_0:th.34">YELLOW_0:34</a></i>;<br/>
<a NAME="E5:39"/>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K1" title="WAYBEL_0:func.1">netmap</a> <span class="p2">(<span class="default"><a href="waybel_2.html#K2" title="WAYBEL_2:func.2">FinSups</a> <font color="Maroon" title="c3">f</font></span>)</span>,<font color="Maroon" title="c1">L</font></span>)</span> <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a>  <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c3">f</font></span>)</span>,<font color="Maroon" title="c1">L</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">a</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#D9" title="LATTICE3:def.9">LATTICE3:def 9</a><br/>

<b>assume </b><a NAME="E1:39_1"/>
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K1" title="WAYBEL_0:func.1">netmap</a> <span class="p2">(<span class="default"><a href="waybel_2.html#K2" title="WAYBEL_2:func.2">FinSups</a> <font color="Maroon" title="c3">f</font></span>)</span>,<font color="Maroon" title="c1">L</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c6">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:39_1"/><i><font color="Green" title="E28">A4</font></i>: 
( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K1" title="WAYBEL_0:func.1">netmap</a> <span class="p2">(<span class="default"><a href="waybel_2.html#K2" title="WAYBEL_2:func.2">FinSups</a> <font color="Maroon" title="c3">f</font></span>)</span>,<font color="Maroon" title="c1">L</font></span>)</span> &amp; <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K1" title="WAYBEL_0:func.1">netmap</a> <span class="p2">(<span class="default"><a href="waybel_2.html#K2" title="WAYBEL_2:func.2">FinSups</a> <font color="Maroon" title="c3">f</font></span>)</span>,<font color="Maroon" title="c1">L</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c7">g</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> <font color="Maroon" title="c2">J</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font><b> such that </b><br/><a NAME="E5:39_1"/><i><font color="Green" title="E29">A5</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> <font color="Maroon" title="c2">J</font> holds <br/> ( <font color="Maroon" title="c7">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Olive" title="b1">x</font></span>)</span> &amp;  <a href="waybel_2.html#K2" title="WAYBEL_2:func.2">FinSups</a> <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="waybel_0.html#G1" title="WAYBEL_0:aggr.1">NetStr</a>(# <span class="p1">(<span class="default"><a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> <font color="Maroon" title="c2">J</font></span>)</span>,<span class="p1">(<span class="default"><a href="yellow_1.html#K1" title="YELLOW_1:func.1">RelIncl</a> <span class="p2">(<span class="default"><a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> <font color="Maroon" title="c2">J</font></span>)</span></span>)</span>,<font color="Maroon" title="c7">g</font> #) )
 <b>by </b><i><a class="ref" href="waybel_2.html#D2" target="_self" title="WAYBEL_2:def.2">Def2</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c8">x</font> = <font color="Maroon" title="c6">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> <font color="Maroon" title="c2">J</font> <b>by </b><i><a class="txt" href="waybel_2.html#E3:39_1"><i><font color="Green" title="E28">A4</font></i></a>, <a class="txt" href="waybel_2.html#E5:39_1"><i><font color="Green" title="E29">A5</font></i></a></i>;<br/>
<a NAME="E7:39_1"/><i><font color="Green" title="E30">A6</font></i>: 
 <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c1">L</font>
 
<b>by </b><i><a class="txt" href="waybel_2.html#E2:39"><i><font color="Green" title="E26">A2</font></i></a></i>;<br/>
<a NAME="E8:39_1"/>
<font color="Maroon" title="c3">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c8">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c3">f</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T144" title="RELAT_1:th.144">RELAT_1:144</a></i>;<br/>
<a NAME="E9:39_1"/><b>then </b>
 <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a>  <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c3">f</font></span>)</span>,<font color="Maroon" title="c1">L</font>
 
<b>by </b><i><a class="txt" href="waybel_2.html#E1:39"><i><font color="Green" title="E25">A1</font></i></a>, <a class="txt" href="waybel_2.html#E7:39_1"><i><font color="Green" title="E30">A6</font></i></a>, <a class="ref" href="yellow_0.html#T34" title="YELLOW_0:th.34">YELLOW_0:34</a></i>;<br/>
<b>hence </b><a NAME="E10:39_1"/>
<font color="Maroon" title="c5">a</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a>  <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c3">f</font></span>)</span>,<font color="Maroon" title="c1">L</font>
 <b>by </b><i><a class="txt" href="waybel_2.html#E3:39_1"><i><font color="Green" title="E28">A4</font></i></a>, <a class="txt" href="waybel_2.html#E5:39_1"><i><font color="Green" title="E29">A5</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:39"/><b>then </b><i><font color="Green" title="E28">A7</font></i>: 
 <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K1" title="WAYBEL_0:func.1">netmap</a> <span class="p3">(<span class="default"><a href="waybel_2.html#K2" title="WAYBEL_2:func.2">FinSups</a> <font color="Maroon" title="c3">f</font></span>)</span>,<font color="Maroon" title="c1">L</font></span>)</span></span>)</span> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a>  <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c3">f</font></span>)</span>,<font color="Maroon" title="c1">L</font>
 
<b>by </b><i><a class="txt" href="waybel_2.html#E1:39"><i><font color="Green" title="E25">A1</font></i></a>, <a class="ref" href="yellow_0.html#D9" title="YELLOW_0:def.9">YELLOW_0:def 9</a></i>;<br/>
<b>thus </b> <a href="yellow_2.html#NK6" title="YELLOW_2:NK.6">Sup</a>  = 
 <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c3">f</font></span>)</span>,<font color="Maroon" title="c1">L</font>
<b>by </b><i><a class="ref" href="yellow_2.html#D5" title="YELLOW_2:def.5">YELLOW_2:def 5</a></i>
<br/>.= 
 <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K1" title="WAYBEL_0:func.1">netmap</a> <span class="p3">(<span class="default"><a href="waybel_2.html#K2" title="WAYBEL_2:func.2">FinSups</a> <font color="Maroon" title="c3">f</font></span>)</span>,<font color="Maroon" title="c1">L</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="waybel_2.html#E4:39"><i><font color="Green" title="E27">A3</font></i></a>, <a class="txt" href="waybel_2.html#E6:39"><i><font color="Green" title="E28">A7</font></i></a>, <a class="ref" href="orders_2.html#T25" title="ORDERS_2:th.25">ORDERS_2:25</a></i>
<br/>.= 
 <a href="waybel_2.html#K1" title="WAYBEL_2:func.1">sup</a> <span class="p1">(<span class="default"><a href="waybel_2.html#K2" title="WAYBEL_2:func.2">FinSups</a> <font color="Maroon" title="c3">f</font></span>)</span>
<b>by </b><i><a class="ref" href="yellow_2.html#D5" title="YELLOW_2:def.5">YELLOW_2:def 5</a></i>
;<br/>


</div>
