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<b>let </b><font color="Maroon">L</font> be  <a href="lattice3.html#V3">complete</a> <a href="waybel_0.html#NM8">LATTICE</a>;<br/><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>;<br/>



<b>given </b><font color="Maroon">g</font> being   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">L</font>,<font color="Maroon">S</font><b> such that </b><a NAME="E1:58"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">g</font> is <a href="waybel_0.html#V17">infs-preserving</a>
 <b>and </b><br/><a NAME="E2:58"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">g</font> is <a href="funct_2.html#V2">onto</a>
 ;<br/>

<a NAME="E3:58"/>
for <font color="Olive">A</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">S</font> holds   <a href="yellow_0.html#R2">ex_inf_of</a> <font color="Olive">A</font>,<font color="Maroon">S</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">A</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">S</font>;<br/>

<b>set </b><font color="Maroon">Y</font> = <font color="Maroon">g</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">A</font>;<br/>
<a NAME="E1:58_1"/>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>
 
<b>by </b><i><a class="txt" href="waybel_5.html#E4"><i><font color="Green">Lemma40</font></i></a>, <a class="ref" href="funct_2.html#D3">FUNCT_2:def 3</a></i>;<br/>
<a NAME="E2:58_1"/><b>then </b><i><font color="Green">E38</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">A</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_1.html#T147">FUNCT_1:147</a></i>;<br/>
<a NAME="E3:58_1"/><i><font color="Green">E39</font></i>: 
 <a href="yellow_0.html#R2">ex_inf_of</a> <font color="Maroon">g</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">A</font>,<font color="Maroon">L</font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#T17">YELLOW_0:17</a></i>;<br/>
<a NAME="E4:58_1"/>
<font color="Maroon">g</font> <a href="waybel_0.html#R3">preserves_inf_of</a> <font color="Maroon">g</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="txt" href="waybel_5.html#E3:58_1"><i><font color="Green">E39</font></i></a>, <a class="ref" href="waybel_0.html#D32">WAYBEL_0:def 32</a></i>;<br/>
<b>hence </b><a NAME="E5:58_1"/>
 <a href="yellow_0.html#R2">ex_inf_of</a> <font color="Maroon">A</font>,<font color="Maroon">S</font>
 <b>by </b><i>, <a class="txt" href="waybel_5.html#E5"><i><font color="Green">Lemma42</font></i></a>, <a class="ref" href="waybel_0.html#D30">WAYBEL_0:def 30</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:58"/><b>then </b>
<font color="Maroon">S</font> is  non <a href="struct_0.html#V3">empty</a> <a href="lattice3.html#V3">complete</a> <a href="orders_2.html#NM1">Poset</a>
 
<b>by </b><i><a class="ref" href="yellow_2.html#T30">YELLOW_2:30</a></i>;<br/>
<b>hence </b><a NAME="E5:58"/>
<font color="Maroon">S</font> is  <a href="lattice3.html#V3">complete</a> <a href="waybel_0.html#NM8">LATTICE</a>
 ;<br/>


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