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<a NAME="E1:4_1_1"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a> ex <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="lattice3.html#R2">is_&lt;=_than</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 <span class="p1">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> st <font color="Olive">b<sub>1</sub></font> <a href="lattice3.html#R2">is_&lt;=_than</a> <font color="Olive">b<sub>3</sub></font> holds <br/><font color="Olive">b<sub>2</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">b<sub>3</sub></font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>2</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>take </b><font color="Maroon">c<sub>3</sub></font> =  <a href="yellow_0.html#K1">"\/"</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/>

<a NAME="E1:4_1_1_1"/><i><font color="Green">E1</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="lattice3.html#K15">"\/"</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="yellow_0.html#T29">YELLOW_0:29</a></i>;<br/>
<a NAME="E2:4_1_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#R4">is_less_than</a>  <a href="lattice3.html#K15">"\/"</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="lattice3.html#D21">LATTICE3:def 21</a></i>;<br/>
<a NAME="E3:4_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#R2">is_&lt;=_than</a> <span class="p1">(<span class="default"><a href="lattice3.html#K15">"\/"</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="lattice3.html#K4">%</a> 
 
<b>by </b><i><a class="ref" href="lattice3.html#T30">LATTICE3:30</a></i>;<br/>
<b>hence </b><a NAME="E4:4_1_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#R2">is_&lt;=_than</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="yellow_1.html#E1:4_1_1_1"><i><font color="Green">E1</font></i></a></i>;<br/>

<b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/>

<b>reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<a NAME="E6:4_1_1_1"/><i><font color="Green">E2</font></i>: 
( <span class="p1">(<span class="default"><a href="lattice3.html#K15">"\/"</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="lattice3.html#K4">%</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="lattice3.html#K4">%</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> )
 
<b>by </b><i><a class="txt" href="yellow_1.html#E1:4_1_1_1"><i><font color="Green">E1</font></i></a></i>;<br/>
<b>assume </b><a NAME="E7:4_1_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#R2">is_&lt;=_than</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<a NAME="E8:4_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#R2">is_&lt;=_than</a> <font color="Maroon">c<sub>5</sub></font> <a href="lattice3.html#K4">%</a> 
 
;<br/>
<a NAME="E9:4_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#R4">is_less_than</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="ref" href="lattice3.html#T30">LATTICE3:30</a></i>;<br/>
<a NAME="E10:4_1_1_1"/><b>then </b>
 <a href="lattice3.html#K15">"\/"</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font> <a href="lattices.html#R3">[=</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="ref" href="lattice3.html#D21">LATTICE3:def 21</a></i>;<br/>
<b>hence </b><a NAME="E11:4_1_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="yellow_1.html#E6:4_1_1_1"><i><font color="Green">E2</font></i></a>, <a class="ref" href="lattice3.html#T7">LATTICE3:7</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:4_1_1"/>
 <a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">c<sub>1</sub></font> is <a href="lattice3.html#V3">complete</a>
 <b>by </b><i><a class="ref" href="lattice3.html#D12">LATTICE3:def 12</a></i>;<br/>


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