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<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="waybel_0.html#V24" title="WAYBEL_0:attr.24">up-complete</a> <a href="orders_2.html#NM1" title="ORDERS_2:NM.1">Poset</a>;<br/><b>let </b><font color="Maroon" title="c2">S</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="yellow_0.html#V4" title="YELLOW_0:attr.4">full</a>  <a href="yellow_0.html#M1" title="YELLOW_0:mode.1">SubRelStr</a> of <font color="Maroon" title="c1">L</font>;<br/>



<b>set </b><font color="Maroon" title="c3">f</font> =  <a href="waybel23.html#K3" title="WAYBEL23:func.3">supMap</a> <font color="Maroon" title="c2">S</font>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="94_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">x</font>, <font color="Maroon" title="c5">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K7" title="WAYBEL_0:func.7">Ids</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span>;<br/><b>assume </b><a NAME="E1:94_1"/><i><font color="Green" title="E52">A1</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c5">y</font>
 ;<br/><b>reconsider </b><font color="Maroon" title="c6">I</font> = <font color="Maroon" title="c4">x</font>, <font color="Maroon" title="c7">J</font> = <font color="Maroon" title="c5">y</font> as   <a href="waybel_0.html#NM3" title="WAYBEL_0:NM.3">Ideal</a> of <font color="Maroon" title="c2">S</font> <b>by </b><i><a class="ref" href="yellow_2.html#T43" title="YELLOW_2:th.43">YELLOW_2:43</a></i>;<br/><a NAME="E3:94_1"/>
( <font color="Maroon" title="c6">I</font> is  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> &amp; <font color="Maroon" title="c7">J</font> is  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> )
 <b>by </b><i><a class="ref" href="yellow_2.html#T7" title="YELLOW_2:th.7">YELLOW_2:7</a></i>;<br/><a NAME="E4:94_1"/><b>then </b><i><font color="Green" title="E53">A2</font></i>: 
(  <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <font color="Maroon" title="c6">I</font>,<font color="Maroon" title="c1">L</font> &amp;  <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <font color="Maroon" title="c7">J</font>,<font color="Maroon" title="c1">L</font> )
 <b>by </b><i><a class="ref" href="waybel_0.html#T75" title="WAYBEL_0:th.75">WAYBEL_0:75</a></i>;<br/><a NAME="E5:94_1"/><i><font color="Green" title="E54">A3</font></i>: 
<font color="Maroon" title="c6">I</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">J</font>
 <b>by </b><i><a class="txt" href="waybel23.html#E1:94_1"><i><font color="Green" title="E52">A1</font></i></a>, <a class="ref" href="yellow_1.html#T3" title="YELLOW_1:th.3">YELLOW_1:3</a></i>;<br/><a NAME="E6:94_1"/>
( <span class="p1">(<span class="default"><a href="waybel23.html#K3" title="WAYBEL23:func.3">supMap</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Maroon" title="c6">I</font>,<font color="Maroon" title="c1">L</font> &amp; <span class="p1">(<span class="default"><a href="waybel23.html#K3" title="WAYBEL23:func.3">supMap</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Maroon" title="c7">J</font>,<font color="Maroon" title="c1">L</font> )
 <b>by </b><i><a class="ref" href="waybel23.html#D10" target="_self" title="WAYBEL23:def.10">Def10</a></i>;<br/><b>hence </b><a NAME="E7:94_1"/>
<span class="p1">(<span class="default"><a href="waybel23.html#K3" title="WAYBEL23:func.3">supMap</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><a href="waybel23.html#K3" title="WAYBEL23:func.3">supMap</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">y</font>
 <b>by </b><i><a class="txt" href="waybel23.html#E4:94_1"><i><font color="Green" title="E53">A2</font></i></a>, <a class="txt" href="waybel23.html#E5:94_1"><i><font color="Green" title="E54">A3</font></i></a>, <a class="ref" href="yellow_0.html#T34" title="YELLOW_0:th.34">YELLOW_0:34</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:94"/>
 <a href="waybel23.html#K3" title="WAYBEL23:func.3">supMap</a> <font color="Maroon" title="c2">S</font> is <a href="orders_3.html#V5" title="ORDERS_3:attr.5">monotone</a>
 <b>by </b><i><a class="ref" href="waybel_1.html#D2" title="WAYBEL_1:def.2">WAYBEL_1:def 2</a></i>;<br/>


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