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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" target="_self">Poset-yielding</a>  <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">I</font>;<br/><b>let </b><font color="Maroon">f</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5">product</a> <font color="Maroon">J</font></span>)</span>;<br/><b>let </b><font color="Maroon">X</font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5">product</a> <font color="Maroon">J</font></span>)</span>;<br/>







<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:43_1"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">f</font> <a href="lattice3.html#R1">is_&lt;=_than</a> <font color="Maroon">X</font>
 ;<br/><b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font>;<br/><b>thus </b><a NAME="E2:43_1"/>
<font color="Maroon">f</font> <a href="waybel_3.html#K7">.</a> <font color="Maroon">i</font> <a href="lattice3.html#R1">is_&lt;=_than</a>  <a href="waybel_3.html#K5">pi</a> <font color="Maroon">X</font>,<font color="Maroon">i</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">J</font> <a href="yellow16.html#K2" target="_self">.</a> <font color="Maroon">i</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D8">LATTICE3:def 8</a><br/>

<b>assume </b><a NAME="E1:43_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="waybel_3.html#K5">pi</a> <font color="Maroon">X</font>,<font color="Maroon">i</font>
 ;<br/>

<b>then consider </b><font color="Maroon">g</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E3:43_1_1"/><i><font color="Green">E32</font></i>: 
( <font color="Maroon">g</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> )
 <b>by </b><i><a class="ref" href="card_3.html#D6">CARD_3:def 6</a></i>;<br/>
<b>reconsider </b><font color="Maroon">g</font> = <font color="Maroon">g</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5">product</a> <font color="Maroon">J</font></span>)</span> <b>by </b><i/>;<br/>
<a NAME="E5:43_1_1"/>
<font color="Maroon">g</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">f</font>
 
<b>by </b><i><a class="txt" href="yellow16.html#E1:43_1"><i><font color="Green">E29</font></i></a>, , <a class="ref" href="lattice3.html#D8">LATTICE3:def 8</a></i>;<br/>
<b>hence </b><a NAME="E6:43_1_1"/>
<font color="Maroon">f</font> <a href="waybel_3.html#K7">.</a> <font color="Maroon">i</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">x</font>
 <b>by </b><i>, <a class="ref" href="waybel_3.html#T28">WAYBEL_3:28</a></i>;<br/>


</div><b>end;</b></div></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:43"/><i><font color="Green">E47</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">f</font> <a href="waybel_3.html#K7">.</a> <font color="Olive">i</font> <a href="lattice3.html#R1">is_&lt;=_than</a>  <a href="waybel_3.html#K5">pi</a> <font color="Maroon">X</font>,<font color="Olive">i</font>
 ;<br/>

<b>let </b><font color="Maroon">g</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5">product</a> <font color="Maroon">J</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D8">LATTICE3:def 8</a><br/>

<b>assume </b><a NAME="E3:43"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">g</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font>;<br/><a NAME="E1:43_2"/>
( <font color="Maroon">g</font> <a href="waybel_3.html#K7">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="waybel_3.html#K5">pi</a> <font color="Maroon">X</font>,<font color="Maroon">i</font> &amp; <font color="Maroon">f</font> <a href="waybel_3.html#K7">.</a> <font color="Maroon">i</font> <a href="lattice3.html#R1">is_&lt;=_than</a>  <a href="waybel_3.html#K5">pi</a> <font color="Maroon">X</font>,<font color="Maroon">i</font> )
 <b>by </b><i>, , <a class="ref" href="card_3.html#D6">CARD_3:def 6</a></i>;<br/><b>hence </b><a NAME="E2:43_2"/>
<font color="Maroon">g</font> <a href="waybel_3.html#K7">.</a> <font color="Maroon">i</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">f</font> <a href="waybel_3.html#K7">.</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="ref" href="lattice3.html#D8">LATTICE3:def 8</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:43"/>
<font color="Maroon">f</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">g</font>
 <b>by </b><i><a class="ref" href="waybel_3.html#T28">WAYBEL_3:28</a></i>;<br/>


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