<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="struct_0.html#V3">empty</a>  <a href="robbins1.html#L2">ComplLattStr</a> ;<br/>

<b>assume </b><a NAME="E1:66_2_1"/>
( <font color="Maroon">c<sub>1</sub></font> is <a href="lattices.html#V4">join-commutative</a> &amp; <font color="Maroon">c<sub>1</sub></font> is <a href="lattices.html#V5">join-associative</a> &amp; <font color="Maroon">c<sub>1</sub></font> is <a href="robbins1.html#V5">Huntington</a> )
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>2</sub></font> = <font color="Maroon">c<sub>1</sub></font> as  non <a href="struct_0.html#V3">empty</a> <a href="lattices.html#V4">join-commutative</a> <a href="lattices.html#V5">join-associative</a> <a href="robbins1.html#V5">Huntington</a>  <a href="robbins1.html#L2">ComplLattStr</a>  ;<br/>
<a NAME="E3:66_2_1"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is <a href="robbins2.html#V2" target="_self">satisfying_MD_1</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font>; <i><font color="Red">:: according to </font></i><a class="ref" href="robbins2.html#D2" target="_self">ROBBINS2:def 2</a><br/>

<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="robbins1.html#K3">`</a> </span>)</span> <a href="robbins1.html#K5">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="robbins1.html#K3">`</a> </span>)</span> <a href="robbins1.html#K5">+</a> <font color="Maroon">c<sub>3</sub></font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="robbins1.html#K3">`</a> </span>)</span> <a href="robbins1.html#K5">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="robbins1.html#K3">`</a> </span>)</span> <a href="robbins1.html#K3">`</a> </span>)</span></span>)</span> <a href="robbins1.html#K3">`</a> </span>)</span> <a href="robbins1.html#K5">+</a> <font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="ref" href="robbins1.html#T3">ROBBINS1:3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="robbins1.html#K6">*'</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="robbins1.html#K3">`</a> </span>)</span></span>)</span> <a href="robbins1.html#K5">+</a> <font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="ref" href="robbins1.html#D4">ROBBINS1:def 4</a></i>
<br/>.= 
<font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="ref" href="robbins1.html#T21">ROBBINS1:21</a></i>
;<br/>
<b>hence </b><a NAME="E2:66_2_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="robbins1.html#K3">`</a> </span>)</span> <a href="robbins1.html#NK4" target="_self">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="robbins1.html#K3">`</a> </span>)</span> <a href="robbins1.html#NK4" target="_self">+</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E4:66_2_1"/>
<font color="Maroon">c<sub>2</sub></font> is <a href="robbins2.html#V3" target="_self">satisfying_MD_2</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_2_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font>; <i><font color="Red">:: according to </font></i><a class="ref" href="robbins2.html#D3" target="_self">ROBBINS2:def 3</a><br/>

<b>set </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>5</sub></font> <a href="robbins1.html#K5">+</a> <font color="Maroon">c<sub>4</sub></font>;<br/>
<b>consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E2:66_2_1_2"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="robbins1.html#K5">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="robbins1.html#K3">`</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="lattices.html#K6">Top</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="robbins1.html#D8">ROBBINS1:def 8</a></i>;<br/>
<i><font color="Green">E62</font></i>: <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="robbins1.html#K5">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="robbins1.html#K5">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="robbins1.html#K3">`</a> </span>)</span> = 
<font color="Maroon">c<sub>5</sub></font> <a href="robbins1.html#K5">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="robbins1.html#K5">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="robbins1.html#K3">`</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="lattices.html#D5">LATTICES:def 5</a></i>
<br/>.= 
<font color="Maroon">c<sub>5</sub></font> <a href="robbins1.html#K5">+</a> <span class="p1">(<span class="default"><a href="lattices.html#K6">Top</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="robbins2.html#E2:66_2_1_2"><i><font color="Green">E61</font></i></a>, <a class="ref" href="robbins1.html#T4">ROBBINS1:4</a></i>
<br/>.= 
 <a href="lattices.html#K6">Top</a> <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="ref" href="robbins1.html#T20">ROBBINS1:20</a></i>
;<br/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="robbins1.html#K3">`</a> </span>)</span> <a href="robbins1.html#K5">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="robbins1.html#K3">`</a> </span>)</span> <a href="robbins1.html#K5">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="robbins1.html#K5">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="robbins1.html#K3">`</a> </span>)</span> <a href="robbins1.html#K5">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="robbins1.html#K3">`</a> </span>)</span> <a href="robbins1.html#K3">`</a> </span>)</span></span>)</span> <a href="robbins1.html#K3">`</a> </span>)</span> <a href="robbins1.html#K5">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="robbins1.html#K5">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="robbins1.html#T3">ROBBINS1:3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="robbins1.html#K6">*'</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="robbins1.html#K3">`</a> </span>)</span></span>)</span> <a href="robbins1.html#K5">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="robbins1.html#K5">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="robbins1.html#D4">ROBBINS1:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="robbins1.html#K5">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="robbins1.html#K5">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="robbins1.html#K6">*'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="robbins1.html#K5">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="robbins1.html#K5">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="robbins1.html#K3">`</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="robbins1.html#T32">ROBBINS1:32</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="robbins1.html#K5">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="robbins1.html#K5">+</a> <font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="txt" href="robbins2.html#E3:66_2_1_2"><i><font color="Green">E62</font></i></a>, <a class="ref" href="robbins1.html#T15">ROBBINS1:15</a></i>
<br/>.= 
<font color="Maroon">c<sub>4</sub></font> <a href="robbins1.html#K5">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="robbins1.html#K5">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="lattices.html#D5">LATTICES:def 5</a></i>
;<br/>
<b>hence </b><a NAME="E5:66_2_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="robbins1.html#K3">`</a> </span>)</span> <a href="robbins1.html#NK4" target="_self">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="robbins1.html#K3">`</a> </span>)</span> <a href="robbins1.html#NK4" target="_self">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="robbins1.html#NK4" target="_self">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="robbins1.html#NK4" target="_self">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="robbins1.html#NK4" target="_self">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:66_2_1"/>
( <font color="Maroon">c<sub>1</sub></font> is <a href="robbins2.html#V2" target="_self">satisfying_MD_1</a> &amp; <font color="Maroon">c<sub>1</sub></font> is <a href="robbins2.html#V3" target="_self">satisfying_MD_2</a> )
 <b>by </b><i><a class="txt" href="robbins2.html#E3:66_2_1"><i><font color="Green">E60</font></i></a></i>;<br/>


</div>
