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<b>let </b><font color="Maroon">I</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">J</font> be  <a href="waybel_3.html#V4">non-Empty</a> <a href="yellow16.html#V1">Poset-yielding</a>  <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">I</font>;<br/>



<b>assume </b><a NAME="E1:14"/><i><font color="Green">E46</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font> holds  <font color="Maroon">J</font> <a href="yellow16.html#K2">.</a> <font color="Olive">i</font> is <a href="waybel_0.html#V24">up-complete</a>
 ;<br/>

<b>set </b><font color="Maroon">L</font> = <font color="Maroon">I</font> <a href="waybel26.html#NK1" target="_self">-POS_prod</a> <font color="Maroon">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">A</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="waybel_0.html#V1">directed</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon">I</font> <a href="waybel26.html#NK1" target="_self">-POS_prod</a> <font color="Maroon">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">x</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font>;<br/><a NAME="E1:14_1_1"/>
( <font color="Maroon">J</font> <a href="yellow16.html#K2">.</a> <font color="Maroon">x</font> is  non <a href="struct_0.html#V3">empty</a> <a href="waybel_0.html#V24">up-complete</a> <a href="orders_2.html#NM1">Poset</a> &amp;  <a href="waybel_3.html#K5">pi</a> <font color="Maroon">A</font>,<font color="Maroon">x</font> is <a href="waybel_0.html#V1">directed</a> &amp; not  <a href="waybel_3.html#K5">pi</a> <font color="Maroon">A</font>,<font color="Maroon">x</font> is <a href="xboole_0.html#V1">empty</a> )
 <b>by </b><i>, <a class="ref" href="yellow16.html#T37">YELLOW16:37</a></i>;<br/><b>hence </b><a NAME="E2:14_1_1"/>
 <a href="yellow_0.html#R1">ex_sup_of</a>  <a href="waybel_3.html#K5">pi</a> <font color="Maroon">A</font>,<font color="Maroon">x</font>,<font color="Maroon">J</font> <a href="yellow16.html#K2">.</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="waybel_0.html#T75">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">ex_sup_of</a> <font color="Maroon">A</font>,<font color="Maroon">I</font> <a href="waybel26.html#NK1" target="_self">-POS_prod</a> <font color="Maroon">J</font>
 <b>by </b><i><a class="ref" href="yellow16.html#T33">YELLOW16:33</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:14"/>
<font color="Maroon">I</font> <a href="waybel26.html#NK1" target="_self">-POS_prod</a> <font color="Maroon">J</font> is <a href="waybel_0.html#V24">up-complete</a>
 <b>by </b><i><a class="ref" href="waybel_0.html#T75">WAYBEL_0:75</a></i>;<br/>


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