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<b>let </b><font color="Maroon">L</font> be   <a href="waybel_0.html#NM6">Semilattice</a>;<br/><b>let </b><font color="Maroon">I</font> be   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font>;<br/><b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>, <font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="orders_3.html#D5">ORDERS_3:def 5</a><br/>





<b>assume </b><a NAME="E1:78"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">y</font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="78_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:78_1"/>
( ( <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">I</font> &amp; <font color="Maroon">y</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">I</font> ) or ( <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">I</font> &amp; not <font color="Maroon">y</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">I</font> ) or ( not <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">I</font> &amp; <font color="Maroon">y</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">I</font> ) or ( not <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">I</font> &amp; not <font color="Maroon">y</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">I</font> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="78_1_1"><b>suppose </b></a><a NAME="E1:78_1_1"/>
( <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">I</font> &amp; <font color="Maroon">y</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">I</font> )
 ;<br/><div class="add"><a NAME="E2:78_1_1"/><b>then </b><i><font color="Green">E32</font></i>: 
( <span class="p1">(<span class="default"><a href="waybel_4.html#K11" target="_self">DownMap</a> <font color="Maroon">I</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">x</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">I</font> &amp; <span class="p1">(<span class="default"><a href="waybel_4.html#K11" target="_self">DownMap</a> <font color="Maroon">I</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">y</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">I</font> )
 <b>by </b><i/>;<br/><a NAME="E3:78_1_1"/>
 <a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">x</font> <a href="tarski.html#R1">c=</a>  <a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">y</font>
 <b>by </b><i><a class="ref" href="waybel_4.html#D7" target="_self">Def7</a>, <a class="ref" href="waybel_0.html#T21">WAYBEL_0:21</a></i>;<br/><a NAME="E4:78_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="waybel_4.html#K11" target="_self">DownMap</a> <font color="Maroon">I</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="waybel_4.html#K11" target="_self">DownMap</a> <font color="Maroon">I</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font>
 <b>by </b><i><a class="ref" href="waybel_4.html#D8" target="_self">Def8</a>, <a class="ref" href="xboole_1.html#T26">XBOOLE_1:26</a></i>;<br/><b>hence </b><a NAME="E5:78_1_1"/>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of 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>  holds <br/>(  not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="waybel_4.html#K11" target="_self">DownMap</a> <font color="Maroon">I</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> or  not <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="waybel_4.html#K11" target="_self">DownMap</a> <font color="Maroon">I</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> or <font color="Olive">b<sub>1</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">b<sub>2</sub></font> )
 <b>by </b><i><a class="ref" href="yellow_1.html#T3">YELLOW_1:3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="78_1_2"><b>suppose </b></a><a NAME="E1:78_1_2"/>
( <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">I</font> &amp; not <font color="Maroon">y</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">I</font> )
 ;<br/><div class="add"><a NAME="E2:78_1_2"/><b>then </b><i><font color="Green">E33</font></i>: 
( <span class="p1">(<span class="default"><a href="waybel_4.html#K11" target="_self">DownMap</a> <font color="Maroon">I</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">x</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">I</font> &amp; <span class="p1">(<span class="default"><a href="waybel_4.html#K11" target="_self">DownMap</a> <font color="Maroon">I</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">y</font> )
 <b>by </b><i/>;<br/><a NAME="E3:78_1_2"/><i><font color="Green">E34</font></i>: 
 <a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">x</font> <a href="tarski.html#R1">c=</a>  <a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">y</font>
 <b>by </b><i><a class="ref" href="waybel_4.html#D7" target="_self">Def7</a>, <a class="ref" href="waybel_0.html#T21">WAYBEL_0:21</a></i>;<br/><a NAME="E4:78_1_2"/>
<span class="p1">(<span class="default"><a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">x</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">I</font> <a href="tarski.html#R1">c=</a>  <a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/><a NAME="E5:78_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="waybel_4.html#K11" target="_self">DownMap</a> <font color="Maroon">I</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="waybel_4.html#K11" target="_self">DownMap</a> <font color="Maroon">I</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font>
 <b>by </b><i><a class="ref" href="waybel_4.html#D8" target="_self">Def8</a>, , <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><b>hence </b><a NAME="E6:78_1_2"/>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of 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>  holds <br/>(  not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="waybel_4.html#K11" target="_self">DownMap</a> <font color="Maroon">I</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> or  not <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="waybel_4.html#K11" target="_self">DownMap</a> <font color="Maroon">I</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> or <font color="Olive">b<sub>1</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">b<sub>2</sub></font> )
 <b>by </b><i><a class="ref" href="yellow_1.html#T3">YELLOW_1:3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="78_1_3"><b>suppose </b></a><a NAME="E1:78_1_3"/>
( not <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">I</font> &amp; <font color="Maroon">y</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">I</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:78_1_3"/>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of 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>  holds <br/>(  not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="waybel_4.html#K11" target="_self">DownMap</a> <font color="Maroon">I</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> or  not <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="waybel_4.html#K11" target="_self">DownMap</a> <font color="Maroon">I</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> or <font color="Olive">b<sub>1</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">b<sub>2</sub></font> )
 <b>by </b><i><a class="ref" href="waybel_4.html#D7" target="_self">Def7</a>, <a class="ref" href="orders_2.html#T26">ORDERS_2:26</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="78_1_4"><b>suppose </b></a><a NAME="E1:78_1_4"/>
( not <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">I</font> &amp; not <font color="Maroon">y</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">I</font> )
 ;<br/><div class="add"><a NAME="E2:78_1_4"/><b>then </b>
( <span class="p1">(<span class="default"><a href="waybel_4.html#K11" target="_self">DownMap</a> <font color="Maroon">I</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">x</font> &amp; <span class="p1">(<span class="default"><a href="waybel_4.html#K11" target="_self">DownMap</a> <font color="Maroon">I</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">y</font> )
 <b>by </b><i/>;<br/><a NAME="E3:78_1_4"/><b>then </b>
<span class="p1">(<span class="default"><a href="waybel_4.html#K11" target="_self">DownMap</a> <font color="Maroon">I</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="waybel_4.html#K11" target="_self">DownMap</a> <font color="Maroon">I</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font>
 <b>by </b><i><a class="ref" href="waybel_4.html#D7" target="_self">Def7</a>, <a class="ref" href="waybel_0.html#T21">WAYBEL_0:21</a></i>;<br/><b>hence </b><a NAME="E4:78_1_4"/>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of 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>  holds <br/>(  not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="waybel_4.html#K11" target="_self">DownMap</a> <font color="Maroon">I</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> or  not <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="waybel_4.html#K11" target="_self">DownMap</a> <font color="Maroon">I</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> or <font color="Olive">b<sub>1</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">b<sub>2</sub></font> )
 <b>by </b><i><a class="ref" href="yellow_1.html#T3">YELLOW_1:3</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
