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<b>let </b><font color="Maroon">L</font> be  <a href="waybel_0.html#V24">up-complete</a> <a href="waybel_0.html#NM6">Semilattice</a>;<br/>

<b>assume </b><a NAME="E1:62"/><i><font color="Green">E29</font></i>: 
for <font color="Olive">I1</font>, <font color="Olive">I2</font> being  <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font> holds  <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Olive">I1</font></span>)</span> <a href="lattice3.html#K12">"/\"</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Olive">I2</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><font color="Olive">I1</font> <a href="yellow_4.html#K4">"/\"</a> <font color="Olive">I2</font></span>)</span>
 ;<br/>

<b>set </b><font color="Maroon">f</font> =  <a href="yellow_2.html#K2">SupMap</a> <font color="Maroon">L</font>;<br/>
<a NAME="E2:62"/><i><font color="Green">E32</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="yellow_2.html#K2">SupMap</a> <font color="Maroon">L</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K7">Ids</a> <font color="Maroon">L</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K7">Ids</a> <font color="Maroon">L</font></span>)</span></span>)</span>, <font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K7">Ids</a> <font color="Maroon">L</font></span>)</span></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_0.html#D34">WAYBEL_0:def 34</a><br/>

<b>assume </b><a NAME="E3:62"/>
 <a href="yellow_0.html#R2">ex_inf_of</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="domain_1.html#K7">}</a></span>, <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>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_0.html#D30">WAYBEL_0:def 30</a><br/>

<a NAME="E4:62"/>
<span class="p1">(<span class="default"><a href="yellow_2.html#K2">SupMap</a> <font color="Maroon">L</font></span>)</span> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="domain_1.html#K7">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="yellow_2.html#K2">SupMap</a> <font color="Maroon">L</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="yellow_2.html#K2">SupMap</a> <font color="Maroon">L</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span></span><a href="domain_1.html#K7">}</a></span>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#T118">FUNCT_1:118</a></i>;<br/>
<b>hence </b><a NAME="E5:62"/>
 <a href="yellow_0.html#R2">ex_inf_of</a> <span class="p1">(<span class="default"><a href="yellow_2.html#K2">SupMap</a> <font color="Maroon">L</font></span>)</span> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="domain_1.html#K7">}</a></span>,<font color="Maroon">L</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T21">YELLOW_0:21</a></i>;<br/>

<b>reconsider </b><font color="Maroon">x1</font> = <font color="Maroon">x</font>, <font color="Maroon">y1</font> = <font color="Maroon">y</font> as   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font> <b>by </b><i><a class="ref" href="yellow_2.html#T43">YELLOW_2:43</a></i>;<br/>
<b>thus </b> <a href="yellow_0.html#NK4" target="_self">inf</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow_2.html#K2">SupMap</a> <font color="Maroon">L</font></span>)</span> <a href="relset_1.html#K10">.:</a> <span class="p2"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="domain_1.html#K7">}</a></span></span>)</span> = 
 <a href="yellow_0.html#NK4" target="_self">inf</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="yellow_2.html#K2">SupMap</a> <font color="Maroon">L</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="yellow_2.html#K2">SupMap</a> <font color="Maroon">L</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span></span><a href="domain_1.html#K7">}</a></span>
<b>by </b><i>, <a class="ref" href="funct_1.html#T118">FUNCT_1:118</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow_2.html#K2">SupMap</a> <font color="Maroon">L</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="lattice3.html#K12">"/\"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow_2.html#K2">SupMap</a> <font color="Maroon">L</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span>
<b>by </b><i><a class="ref" href="yellow_0.html#T40">YELLOW_0:40</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow_2.html#K2">SupMap</a> <font color="Maroon">L</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="lattice3.html#K12">"/\"</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">y1</font></span>)</span>
<b>by </b><i><a class="ref" href="yellow_2.html#D3">YELLOW_2:def 3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">x1</font></span>)</span> <a href="lattice3.html#K12">"/\"</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">y1</font></span>)</span>
<b>by </b><i><a class="ref" href="yellow_2.html#D3">YELLOW_2:def 3</a></i>
<br/>.= 
 <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><font color="Maroon">x1</font> <a href="yellow_4.html#K4">"/\"</a> <font color="Maroon">y1</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="yellow_2.html#K2">SupMap</a> <font color="Maroon">L</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">x1</font> <a href="yellow_4.html#K4">"/\"</a> <font color="Maroon">y1</font></span>)</span>
<b>by </b><i><a class="ref" href="yellow_2.html#D3">YELLOW_2:def 3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="yellow_2.html#K2">SupMap</a> <font color="Maroon">L</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">y</font></span>)</span>
<b>by </b><i><a class="ref" href="yellow_4.html#T58">YELLOW_4:58</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="yellow_2.html#K2">SupMap</a> <font color="Maroon">L</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK4" target="_self">inf</a> <span class="p2"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="domain_1.html#K7">}</a></span></span>)</span>
<b>by </b><i><a class="ref" href="yellow_0.html#T40">YELLOW_0:40</a></i>
;<br/>


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