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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#V3" title="ORDERS_2:attr.3">transitive</a>  <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a> ;<br/>

<b>assume </b><a NAME="E1:16"/><i><font color="Green" title="E13">A1</font></i>: 
for <font color="Olive" title="b1">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font><br/> for <font color="Olive" title="b2">N</font> being  <a href="waybel_0.html#NM2" title="WAYBEL_0:NM.2">net</a> of <font color="Maroon" title="c1">L</font>  st <font color="Olive" title="b2">N</font> is <a href="waybel_0.html#V10" title="WAYBEL_0:attr.10">eventually-directed</a> holds <br/><font color="Olive" title="b1">x</font> <a href="lattice3.html#K11" title="LATTICE3:func.11">"/\"</a> <span class="p1">(<span class="default"><a href="waybel_2.html#K1" title="WAYBEL_2:func.1">sup</a> <font color="Olive" title="b2">N</font></span>)</span> <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"><span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Olive" title="b1">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="yellow_4.html#K3" title="YELLOW_4:func.3">"/\"</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p3">(<span class="default"><a href="waybel_0.html#K1" title="WAYBEL_0:func.1">netmap</a> <font color="Olive" title="b2">N</font>,<font color="Maroon" title="c1">L</font></span>)</span></span>)</span></span>)</span>
 ;<br/>

<b>let </b><font color="Maroon" title="c2">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="waybel_2.html#D6" title="WAYBEL_2:def.6">WAYBEL_2:def 6</a><br/><b>let </b><font color="Maroon" title="c3">D</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="waybel_0.html#V1" title="WAYBEL_0:attr.1">directed</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font>;<br/>



<b>reconsider </b><font color="Maroon" title="c4">n</font> =  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <font color="Maroon" title="c3">D</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c3">D</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font> <b>by </b><i><a class="ref" href="funct_2.html#T9" title="FUNCT_2:th.9">FUNCT_2:9</a></i>;<br/>
<b>set </b><font color="Maroon" title="c5">N</font> =  <a href="waybel_0.html#G1" title="WAYBEL_0:aggr.1">NetStr</a>(# <font color="Maroon" title="c3">D</font>,<span class="p1">(<span class="default">the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">L</font> <a href="toler_1.html#K1" title="TOLER_1:func.1">|_2</a> <font color="Maroon" title="c3">D</font></span>)</span>,<font color="Maroon" title="c4">n</font> #);<br/>
<a NAME="E3:16"/>
<font color="Maroon" title="c3">D</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">D</font>
 
;<br/>
<b>then </b><i><font color="Green" title="E14">A2</font></i>: <font color="Maroon" title="c3">D</font> = 
<font color="Maroon" title="c4">n</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c3">D</font>
<b>by </b><i><a class="ref" href="funct_1.html#T162" title="FUNCT_1:th.162">FUNCT_1:162</a></i>
<br/>.= 
 <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> <a href="waybel_0.html#G1" title="WAYBEL_0:aggr.1">NetStr</a>(# <font color="Maroon" title="c3">D</font>,<span class="p1">(<span class="default">the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">L</font> <a href="toler_1.html#K1" title="TOLER_1:func.1">|_2</a> <font color="Maroon" title="c3">D</font></span>)</span>,<font color="Maroon" title="c4">n</font> #),<font color="Maroon" title="c1">L</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#T45" title="FUNCT_2:th.45">FUNCT_2:45</a></i>
;<br/>
<a NAME="E5:16"/><i><font color="Green" title="E15">A3</font></i>: 
 <a href="waybel_0.html#G1" title="WAYBEL_0:aggr.1">NetStr</a>(# <font color="Maroon" title="c3">D</font>,<span class="p1">(<span class="default">the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">L</font> <a href="toler_1.html#K1" title="TOLER_1:func.1">|_2</a> <font color="Maroon" title="c3">D</font></span>)</span>,<font color="Maroon" title="c4">n</font> #) is <a href="waybel_0.html#V10" title="WAYBEL_0:attr.10">eventually-directed</a>
 
<b>by </b><i><a class="ref" href="waybel_2.html#T20" title="WAYBEL_2:th.20">WAYBEL_2:20</a></i>;<br/>
<a NAME="E6:16"/><i><font color="Green" title="E16">A4</font></i>: 
 <a href="yellow_2.html#NK6" title="YELLOW_2:NK.6">Sup</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="waybel_2.html#K1" title="WAYBEL_2:func.1">sup</a> <a href="waybel_0.html#G1" title="WAYBEL_0:aggr.1">NetStr</a>(# <font color="Maroon" title="c3">D</font>,<span class="p1">(<span class="default">the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">L</font> <a href="toler_1.html#K1" title="TOLER_1:func.1">|_2</a> <font color="Maroon" title="c3">D</font></span>)</span>,<font color="Maroon" title="c4">n</font> #)
 
;<br/>
<b>thus </b><font color="Maroon" title="c2">x</font> <a href="lattice3.html#K11" title="LATTICE3:func.11">"/\"</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c3">D</font></span>)</span> = 
<font color="Maroon" title="c2">x</font> <a href="lattice3.html#K11" title="LATTICE3:func.11">"/\"</a> <span class="p1">(<span class="default"><a href="yellow_2.html#NK6" title="YELLOW_2:NK.6">Sup</a> </span>)</span>
<b>by </b><i><a class="txt" href="waybel_9.html#E4:16"><i><font color="Green" title="E14">A2</font></i></a>, <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"><span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="yellow_4.html#K3" title="YELLOW_4:func.3">"/\"</a> <font color="Maroon" title="c3">D</font></span>)</span>
<b>by </b><i><a class="txt" href="waybel_9.html#E1:16"><i><font color="Green" title="E13">A1</font></i></a>, <a class="txt" href="waybel_9.html#E4:16"><i><font color="Green" title="E14">A2</font></i></a>, <a class="txt" href="waybel_9.html#E5:16"><i><font color="Green" title="E15">A3</font></i></a>, <a class="txt" href="waybel_9.html#E6:16"><i><font color="Green" title="E16">A4</font></i></a></i>
;<br/>


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