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<b>let </b><font color="Maroon">c<sub>1</sub></font> be  <a href="waybel_1.html#V2">distributive</a> <a href="waybel_3.html#V3">continuous</a> <a href="waybel_0.html#NM8">LATTICE</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<b>set </b><font color="Maroon">c<sub>3</sub></font> =  <a href="waybel_3.html#K1">waybelow</a> <font color="Maroon">c<sub>2</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>4</sub></font> =  <a href="waybel_0.html#K4">uparrow</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K12">fininfs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span>;<br/>
<b>assume </b><a NAME="E1:56"/>
 <a href="waybel_0.html#K4">uparrow</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K12">fininfs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> <a href="subset_1.html#R1">misses</a>  <a href="waybel_3.html#K1">waybelow</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:56"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> is <a href="waybel_7.html#V1" target="_self">prime</a> &amp;  <a href="waybel_3.html#K1">waybelow</a> <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#R1">misses</a>  <a href="waybel_0.html#K4">uparrow</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K12">fininfs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="waybel_7.html#T27" target="_self">Th27</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>5</sub></font> as  <a href="waybel_7.html#V1" target="_self">prime</a> <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="waybel_7.html#E3:56"><i><font color="Green">E34</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>6</sub></font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_7.html#D6" target="_self">WAYBEL_7:def 6</a><br/>

<a NAME="E5:56"/><i><font color="Green">E35</font></i>: 
(  <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>1</sub></font> &amp;  <a href="yellow_0.html#R1">ex_sup_of</a>  <a href="waybel_3.html#K1">waybelow</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font> &amp;  <a href="yellow_0.html#R1">ex_sup_of</a>  <a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="ref" href="waybel_0.html#T75">WAYBEL_0:75</a></i>;<br/>
<a NAME="E6:56"/>
<span class="p1">(<span class="default"><a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K3">`</a>  <a href="tarski.html#R1">c=</a>  <a href="waybel_0.html#K4">uparrow</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K12">fininfs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="waybel_0.html#T62">WAYBEL_0:62</a></i>;<br/>
<a NAME="E7:56"/><b>then </b>
( <font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K4">uparrow</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K12">fininfs</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K3">`</a>  &amp; <span class="p1">(<span class="default"><a href="waybel_0.html#K4">uparrow</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K12">fininfs</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K3">`</a>  <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="subset_1.html#K3">`</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R1">=</a>  <a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="txt" href="waybel_7.html#E3:56"><i><font color="Green">E34</font></i></a>, <a class="ref" href="subset_1.html#T31">SUBSET_1:31</a>, <a class="ref" href="subset_1.html#T43">SUBSET_1:43</a></i>;<br/>
<a NAME="E8:56"/><b>then </b>
( <font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">c=</a>  <a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">c<sub>2</sub></font> &amp;  <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">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><a href="waybel_3.html#K1">waybelow</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> &amp;  <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><a href="waybel_3.html#K1">waybelow</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">c<sub>6</sub></font> )
 
<b>by </b><i><a class="txt" href="waybel_7.html#E3:56"><i><font color="Green">E34</font></i></a>, <a class="txt" href="waybel_7.html#E5:56"><i><font color="Green">E35</font></i></a>, <a class="ref" href="waybel_0.html#T34">WAYBEL_0:34</a>, <a class="ref" href="waybel_3.html#D5">WAYBEL_3:def 5</a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a>, <a class="ref" href="yellow_0.html#T34">YELLOW_0:34</a></i>;<br/>
<a NAME="E9:56"/><b>then </b>
(  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">c<sub>6</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> &amp;  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">c<sub>6</sub></font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="txt" href="waybel_7.html#E5:56"><i><font color="Green">E35</font></i></a>, <a class="ref" href="yellow_0.html#T34">YELLOW_0:34</a></i>;<br/>
<b>hence </b><a NAME="E10:56"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="orders_2.html#T25">ORDERS_2:25</a></i>;<br/>


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