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<b>let </b><font color="Maroon">S</font> be  non <a href="struct_0.html#V3">empty</a> <a href="orders_2.html#NM1">Poset</a>, <font color="Maroon">T</font> be  non <a href="struct_0.html#V3">empty</a> <a href="orders_2.html#NM1">Poset</a>;<br/><b>let </b><font color="Maroon">d</font> be   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">T</font>,<font color="Maroon">S</font>;<br/>



<b>given </b><font color="Maroon">g</font> being   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">S</font>,<font color="Maroon">T</font><b> such that </b><a NAME="E1:30"/><i><font color="Green">E32</font></i>: 
<span class="p1"><a href="waybel_1.html#K1" target="_self">[</a><span class="default"><font color="Maroon">g</font>,<font color="Maroon">d</font></span><a href="waybel_1.html#K1" target="_self">]</a></span> is <a href="waybel_1.html#V3" target="_self">Galois</a>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_1.html#D12" target="_self">WAYBEL_1:def 12</a><br/>

<b>let </b><font color="Maroon">X</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_0.html#D33">WAYBEL_0:def 33</a><br/>

<b>assume </b><a NAME="E2:30"/><i><font color="Green">E35</font></i>: 
 <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Maroon">X</font>,<font color="Maroon">T</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_0.html#D31">WAYBEL_0:def 31</a><br/>

<b>set </b><font color="Maroon">t</font> =  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">X</font>;<br/>
<a NAME="E3:30"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">X</font></span>)</span> <a href="lattice3.html#NR4" target="_self">is_&gt;=_than</a> <font color="Maroon">d</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">X</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">s</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D9">LATTICE3:def 9</a><br/>

<a NAME="E1:30_1"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">d</font> is <a href="orders_3.html#V5">monotone</a>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E2:30_1"/><i><font color="Green">E49</font></i>: 
 <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">X</font> <a href="lattice3.html#NR4" target="_self">is_&gt;=_than</a> <font color="Maroon">X</font>
 
<b>by </b><i>, <a class="ref" href="yellow_0.html#T30">YELLOW_0:30</a></i>;<br/>
<b>assume </b><a NAME="E3:30_1"/>
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">d</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">X</font>
 ;<br/>

<b>then consider </b><font color="Maroon">ti</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E5:30_1"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">ti</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>and </b><br/><a NAME="E6:30_1"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">ti</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T116">FUNCT_2:116</a></i>;<br/>
<b>reconsider </b><font color="Maroon">ti</font> = <font color="Maroon">ti</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font> ;<br/>
<a NAME="E8:30_1"/>
 <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">X</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">ti</font>
 
<b>by </b><i><a class="ref" href="waybel_1.html#D2" target="_self">Def2</a>, , <a class="ref" href="lattice3.html#D9">LATTICE3:def 9</a></i>;<br/>
<b>hence </b><a NAME="E9:30_1"/>
<font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">X</font></span>)</span> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">s</font>
 <b>by </b><i>, , </i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:30"/>
for <font color="Olive">s</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font>  st <font color="Olive">s</font> <a href="lattice3.html#NR4" target="_self">is_&gt;=_than</a> <font color="Maroon">d</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">X</font> holds <br/><font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">X</font></span>)</span> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">s</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">s</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font>;<br/>

<b>assume </b><a NAME="E1:30_2"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">s</font> <a href="lattice3.html#NR4" target="_self">is_&gt;=_than</a> <font color="Maroon">d</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">X</font>
 ;<br/>

<a NAME="E2:30_2"/>
<font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">s</font> <a href="lattice3.html#NR4" target="_self">is_&gt;=_than</a> <font color="Maroon">X</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">ti</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D9">LATTICE3:def 9</a><br/>

<b>assume </b><a NAME="E1:30_2_1"/>
<font color="Maroon">ti</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/>

<a NAME="E2:30_2_1"/><b>then </b>
<font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">ti</font> <a href="hidden.html#R2">in</a> <font color="Maroon">d</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T43">FUNCT_2:43</a></i>;<br/>
<a NAME="E3:30_2_1"/><b>then </b>
<font color="Maroon">s</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">ti</font>
 
<b>by </b><i>, <a class="ref" href="lattice3.html#D9">LATTICE3:def 9</a></i>;<br/>
<b>hence </b><a NAME="E4:30_2_1"/>
<font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">s</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">ti</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:30_2"/><b>then </b>
<font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">s</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">X</font>
 
<b>by </b><i>, <a class="ref" href="yellow_0.html#T30">YELLOW_0:30</a></i>;<br/>
<b>hence </b><a NAME="E4:30_2"/>
<font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">X</font></span>)</span> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">s</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:30"/>
(  <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Maroon">d</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">X</font>,<font color="Maroon">S</font> &amp;  <a href="yellow_0.html#K1">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">X</font></span>)</span>,<font color="Maroon">S</font> <a href="hidden.html#R1">=</a> <font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K1">"\/"</a> <font color="Maroon">X</font>,<font color="Maroon">T</font></span>)</span> )
 <b>by </b><i>, <a class="ref" href="yellow_0.html#T30">YELLOW_0:30</a></i>;<br/>


</div>
