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<b>let </b><font color="Maroon">c<sub>1</sub></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="E1:3"/><i><font color="Green">E4</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is <a href="waybel_3.html#V3">continuous</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>reconsider </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> as   <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_5.html#E1:3"><i><font color="Green">E4</font></i></a>, <a class="ref" href="waybel_3.html#D6">WAYBEL_3:def 6</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>3</sub></font>
;<br/>

<b>thus </b><a NAME="E3:3"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">c<sub>3</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Maroon">c<sub>2</sub></font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>1</sub></font> ) )
 <b>by </b><i><a class="txt" href="waybel_5.html#E1:3"><i><font color="Green">E4</font></i></a>, <a class="txt" href="waybel_5.html#E1"><i><font color="Green">Lemma1</font></i></a></i>;<br/>


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