<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon" title="c1">I</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c2">J</font> be  <a href="waybel_3.html#V4" title="WAYBEL_3:attr.4">non-Empty</a> <a href="yellow16.html#V1" title="YELLOW16:attr.1">Poset-yielding</a>  <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c1">I</font>;<br/>



<b>assume </b><a NAME="E1:14"/><i><font color="Green" title="E10">A1</font></i>: 
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">I</font> holds  <font color="Maroon" title="c2">J</font> <a href="yellow16.html#K2" title="YELLOW16:func.2">.</a> <font color="Olive" title="b1">i</font> is <a href="waybel_0.html#V24" title="WAYBEL_0:attr.24">up-complete</a>
 ;<br/>

<b>set </b><font color="Maroon" title="c3">L</font> = <font color="Maroon" title="c1">I</font> <a href="waybel26.html#NK1" title="WAYBEL26:NK.1">-POS_prod</a> <font color="Maroon" title="c2">J</font>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">A</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 <span class="p1">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="waybel26.html#NK1" title="WAYBEL26:NK.1">-POS_prod</a> <font color="Maroon" title="c2">J</font></span>)</span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">I</font>;<br/><a NAME="E1:14_1_1"/>
( <font color="Maroon" title="c2">J</font> <a href="yellow16.html#K2" title="YELLOW16:func.2">.</a> <font color="Maroon" title="c5">x</font> is  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> &amp;  <a href="waybel_3.html#K5" title="WAYBEL_3:func.5">pi</a> <font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c5">x</font> is <a href="waybel_0.html#V1" title="WAYBEL_0:attr.1">directed</a> &amp; not  <a href="waybel_3.html#K5" title="WAYBEL_3:func.5">pi</a> <font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c5">x</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> )
 <b>by </b><i><a class="txt" href="waybel29.html#E1:14"><i><font color="Green" title="E10">A1</font></i></a>, <a class="ref" href="yellow16.html#T37" title="YELLOW16:th.37">YELLOW16:37</a></i>;<br/><b>hence </b><a NAME="E2:14_1_1"/>
 <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a>  <a href="waybel_3.html#K5" title="WAYBEL_3:func.5">pi</a> <font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c2">J</font> <a href="yellow16.html#K2" title="YELLOW16:func.2">.</a> <font color="Maroon" title="c5">x</font>
 <b>by </b><i><a class="ref" href="waybel_0.html#T75" title="WAYBEL_0:th.75">WAYBEL_0:75</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E2:14_1"/>
 <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c1">I</font> <a href="waybel26.html#NK1" title="WAYBEL26:NK.1">-POS_prod</a> <font color="Maroon" title="c2">J</font>
 <b>by </b><i><a class="ref" href="yellow16.html#T33" title="YELLOW16:th.33">YELLOW16:33</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:14"/>
<font color="Maroon" title="c1">I</font> <a href="waybel26.html#NK1" title="WAYBEL26:NK.1">-POS_prod</a> <font color="Maroon" title="c2">J</font> is <a href="waybel_0.html#V24" title="WAYBEL_0:attr.24">up-complete</a>
 <b>by </b><i><a class="ref" href="waybel_0.html#T75" title="WAYBEL_0:th.75">WAYBEL_0:75</a></i>;<br/>


</div>
