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<b>let </b><font color="Maroon">L</font> be  <a href="yellow_0.html#V1">lower-bounded</a> <a href="waybel_0.html#NM8">LATTICE</a>;<br/>

<b>assume </b><a NAME="E1:14"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">L</font> is <a href="waybel_3.html#V3">continuous</a>
 ;<br/>

<b>reconsider </b><font color="Maroon">A</font> =  <a href="yellow_1.html#K2">InclPoset</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K7">Ids</a> <font color="Maroon">L</font></span>)</span> as  <a href="yellow_0.html#V1">lower-bounded</a> <a href="waybel_8.html#V3">arithmetic</a> <a href="waybel_0.html#NM8">LATTICE</a> <b>by </b><i><a class="ref" href="waybel13.html#T14">WAYBEL13:14</a></i>;<br/>
<b>take </b>
<font color="Maroon">A</font>
;<br/>

<b>reconsider </b><font color="Maroon">g</font> =  <a href="yellow_2.html#K2">SupMap</a> <font color="Maroon">L</font> as   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">A</font>,<font color="Maroon">L</font> ;<br/>
<b>take </b>
<font color="Maroon">g</font>
;<br/>

<a NAME="E4:14"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">g</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:14_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">y'</font> = <font color="Maroon">y</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> ;<br/>
<a NAME="E3:14_1"/>
 <a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">y'</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="yellow_2.html#T43">YELLOW_2:43</a></i>;<br/>
<a NAME="E4:14_1"/><b>then </b>
(  <a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">y'</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font> &amp; not the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font> is <a href="xboole_0.html#V1">empty</a> )
 
;<br/>
<a NAME="E5:14_1"/><b>then </b><i><font color="Green">E34</font></i>: 
 <a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">y'</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">y'</font></span>)</span> = 
 <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">y'</font></span>)</span>
<b>by </b><i><a class="ref" href="yellow_2.html#D3">YELLOW_2:def 3</a></i>
<br/>.= 
<font color="Maroon">y'</font>
<b>by </b><i><a class="ref" href="waybel_0.html#T34">WAYBEL_0:34</a></i>
;<br/>
<b>hence </b><a NAME="E7:14_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">g</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:14"/><b>then </b>
 <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">L</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>
<b>hence </b><a NAME="E6:14"/>
<font color="Maroon">g</font> is <a href="funct_2.html#V2">onto</a>
 <b>by </b><i><a class="ref" href="funct_2.html#D3">FUNCT_2:def 3</a></i>;<br/>

<b>thus </b><a NAME="E7:14"/>
<font color="Maroon">g</font> is <a href="waybel_0.html#V17">infs-preserving</a>
 <b>by </b><i>, <a class="ref" href="waybel13.html#T33">WAYBEL13:33</a></i>;<br/>

<a NAME="E8:14"/>
<font color="Maroon">g</font> is <a href="waybel_0.html#V18">sups-preserving</a>
 
<b>by </b><i>, <a class="ref" href="waybel13.html#T33">WAYBEL13:33</a></i>;<br/>
<a NAME="E9:14"/><b>then </b>
for <font color="Olive">X</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">A</font>  st not <font color="Olive">X</font> is <a href="xboole_0.html#V1">empty</a> &amp; <font color="Olive">X</font> is <a href="waybel_0.html#V1">directed</a> holds <br/><font color="Maroon">g</font> <a href="waybel_0.html#R4">preserves_sup_of</a> <font color="Olive">X</font>
 
<b>by </b><i><a class="ref" href="waybel_0.html#D33">WAYBEL_0:def 33</a></i>;<br/>
<b>hence </b><a NAME="E10:14"/>
<font color="Maroon">g</font> is <a href="waybel_0.html#V22">directed-sups-preserving</a>
 <b>by </b><i><a class="ref" href="waybel_0.html#D37">WAYBEL_0:def 37</a></i>;<br/>


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