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<b>let </b><font color="Maroon">c<sub>1</sub></font> be  <a href="lattice3.html#V3">complete</a> <a href="waybel_0.html#NM8">LATTICE</a>;<br/>

<b>thus </b><a NAME="E1:61"/>
( <font color="Maroon">c<sub>1</sub></font> is <a href="waybel_5.html#V1">completely-distributive</a> implies ( <font color="Maroon">c<sub>1</sub></font> is <a href="waybel_1.html#V2">distributive</a> &amp; <font color="Maroon">c<sub>1</sub></font> is <a href="waybel_3.html#V3">continuous</a> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="lattice3.html#K7">~</a>  is <a href="waybel_3.html#V3">continuous</a> ) )
 <b>by </b><i><a class="txt" href="waybel_6.html#E48"><i><font color="Green">Lemma37</font></i></a></i>;<br/>

<b>assume </b><a NAME="E2:61"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> is <a href="waybel_1.html#V2">distributive</a> &amp; <font color="Maroon">c<sub>1</sub></font> is <a href="waybel_3.html#V3">continuous</a> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="lattice3.html#K7">~</a>  is <a href="waybel_3.html#V3">continuous</a> )
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>2</sub></font> = <font color="Maroon">c<sub>1</sub></font> as  <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/>
<a NAME="E4:61"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font>ex <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>2</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Olive">b<sub>2</sub></font> &amp; ( for <font color="Olive">b<sub>3</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font> st <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>2</sub></font> holds <br/><font color="Olive">b<sub>3</sub></font> is <a href="waybel_6.html#V6" target="_self">co-prime</a> ) )
 
<b>by </b><i><a class="txt" href="waybel_6.html#E2:61"><i><font color="Green">E39</font></i></a>, <a class="txt" href="waybel_6.html#E49"><i><font color="Green">Lemma38</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:61"/>
<font color="Maroon">c<sub>1</sub></font> is <a href="waybel_5.html#V1">completely-distributive</a>
 <b>by </b><i><a class="txt" href="waybel_6.html#E47"><i><font color="Green">Lemma36</font></i></a></i>;<br/>


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