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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="orders_2.html#NM1">Poset</a>;<br/>

<b>assume </b><a NAME="E1:7_1_1"/><i><font color="Green">E3</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> is   <a href="waybel_0.html#NM7">sup-Semilattice</a> &amp; <font color="Maroon">c<sub>1</sub></font> is <a href="abcmiz_0.html#V1" target="_self">Noetherian</a> )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="abcmiz_0.html#D3" target="_self">ABCMIZ_0:def 3</a><br/>

<b>hence </b><a NAME="E2:7_1_1"/>
( <font color="Maroon">c<sub>1</sub></font> is <a href="abcmiz_0.html#V1" target="_self">Noetherian</a> &amp; <font color="Maroon">c<sub>1</sub></font> is <a href="lattice3.html#V1">with_suprema</a> )
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>2</sub></font> = <font color="Maroon">c<sub>1</sub></font> as   <a href="waybel_0.html#NM7">sup-Semilattice</a> <b>by </b><i><a class="txt" href="abcmiz_0.html#E1:7_1_1"><i><font color="Green">E3</font></i></a></i>;<br/>
<a NAME="E4:7_1_1"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>
 
;<br/>
<b>then consider </b><font color="Maroon">c<sub>3</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E6:7_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 <b>and </b><br/><a NAME="E7:7_1_1"/><i><font color="Green">E4</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> holds <br/> not <font color="Maroon">c<sub>3</sub></font> <a href="orders_2.html#R2">&lt;</a> <font color="Olive">b<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="abcmiz_0.html#E1:7_1_1"><i><font color="Green">E3</font></i></a>, <a class="ref" href="abcmiz_0.html#D2" target="_self">Def2</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>3</sub></font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow_0.html#D5">YELLOW_0:def 5</a><br/>

<b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D9">LATTICE3:def 9</a><br/>

<a NAME="E8:7_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="lattice3.html#K10">"\/"</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>
 
;<br/>
<a NAME="E9:7_1_1"/><b>then </b>
( <font color="Maroon">c<sub>3</sub></font> <a href="lattice3.html#K10">"\/"</a> <font color="Maroon">c<sub>4</sub></font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>3</sub></font> &amp; not <font color="Maroon">c<sub>3</sub></font> <a href="orders_2.html#R2">&lt;</a> <font color="Maroon">c<sub>3</sub></font> <a href="lattice3.html#K10">"\/"</a> <font color="Maroon">c<sub>4</sub></font> )
 
<b>by </b><i><a class="txt" href="abcmiz_0.html#E7:7_1_1"><i><font color="Green">E4</font></i></a>, <a class="ref" href="yellow_0.html#T22">YELLOW_0:22</a></i>;<br/>
<a NAME="E10:7_1_1"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="lattice3.html#K10">"\/"</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="orders_2.html#D10">ORDERS_2:def 10</a></i>;<br/>
<b>hence </b><a NAME="E11:7_1_1"/>
(  not <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> or <font color="Maroon">c<sub>4</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="abcmiz_0.html#E1:7_1_1"><i><font color="Green">E3</font></i></a>, <a class="ref" href="yellow_0.html#T22">YELLOW_0:22</a></i>;<br/>


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