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<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="struct_0.html#V3">empty</a>  <a href="orders_2.html#L1">RelStr</a> ;<br/>

<b>assume </b><a NAME="E1:2_1_1"/><i><font color="Green">E1</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> is <a href="orders_2.html#V2">reflexive</a> &amp; <font color="Maroon">c<sub>1</sub></font> is <a href="waybel_0.html#V16">connected</a> )
 ;<br/>

<b>thus </b><a NAME="E2:2_1_1"/>
<font color="Maroon">c<sub>1</sub></font> is <a href="lattice3.html#V2">with_infima</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</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#D11">LATTICE3:def 11</a><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:2_1_1_1_1"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> or <font color="Maroon">c<sub>3</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> )
 </span><b>by </b><i><a class="txt" href="waybel_2.html#E1:2_1_1"><i><font color="Green">E1</font></i></a>, <a class="ref" href="waybel_0.html#D29">WAYBEL_0:def 29</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:2_1_1_1_1_1"/><i><font color="Green">E2</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><div class="add"><b>take </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>2</sub></font>;<br/><b>thus </b><a NAME="E2:2_1_1_1_1_1"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> &amp; <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="waybel_2.html#E1:2_1_1"><i><font color="Green">E1</font></i></a>, <a class="txt" href="waybel_2.html#E1:2_1_1_1_1_1"><i><font color="Green">E2</font></i></a>, <a class="ref" href="yellow_0.html#D1">YELLOW_0:def 1</a></i>;<br/><b>thus </b><a NAME="E3:2_1_1_1_1_1"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> holds <br/>(  not <font color="Olive">b<sub>1</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> or  not <font color="Olive">b<sub>1</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> or <font color="Olive">b<sub>1</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:2_1_1_1_1_2"/><i><font color="Green">E2</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><div class="add"><b>take </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>3</sub></font>;<br/><b>thus </b><a NAME="E2:2_1_1_1_1_2"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> &amp; <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="waybel_2.html#E1:2_1_1"><i><font color="Green">E1</font></i></a>, <a class="txt" href="waybel_2.html#E1:2_1_1_1_1_2"><i><font color="Green">E2</font></i></a>, <a class="ref" href="yellow_0.html#D1">YELLOW_0:def 1</a></i>;<br/><b>thus </b><a NAME="E3:2_1_1_1_1_2"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> holds <br/>(  not <font color="Olive">b<sub>1</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> or  not <font color="Olive">b<sub>1</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> or <font color="Olive">b<sub>1</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>

<b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</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#D10">LATTICE3:def 10</a><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:2_1_1_2"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> or <font color="Maroon">c<sub>3</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> )
 </span><b>by </b><i><a class="txt" href="waybel_2.html#E1:2_1_1"><i><font color="Green">E1</font></i></a>, <a class="ref" href="waybel_0.html#D29">WAYBEL_0:def 29</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_1"><b>suppose </b></a><a NAME="E1:2_1_1_2_1"/><i><font color="Green">E2</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><div class="add"><b>take </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>3</sub></font>;<br/><b>thus </b><a NAME="E2:2_1_1_2_1"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>2</sub></font> &amp; <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> )
 <b>by </b><i><a class="txt" href="waybel_2.html#E1:2_1_1"><i><font color="Green">E1</font></i></a>, <a class="txt" href="waybel_2.html#E1:2_1_1_2_1"><i><font color="Green">E2</font></i></a>, <a class="ref" href="yellow_0.html#D1">YELLOW_0:def 1</a></i>;<br/><b>thus </b><a NAME="E3:2_1_1_2_1"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of 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>2</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> or  not <font color="Maroon">c<sub>3</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> or <font color="Maroon">c<sub>4</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_2"><b>suppose </b></a><a NAME="E1:2_1_1_2_2"/><i><font color="Green">E2</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><div class="add"><b>take </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>2</sub></font>;<br/><b>thus </b><a NAME="E2:2_1_1_2_2"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>2</sub></font> &amp; <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> )
 <b>by </b><i><a class="txt" href="waybel_2.html#E1:2_1_1"><i><font color="Green">E1</font></i></a>, <a class="txt" href="waybel_2.html#E1:2_1_1_2_2"><i><font color="Green">E2</font></i></a>, <a class="ref" href="yellow_0.html#D1">YELLOW_0:def 1</a></i>;<br/><b>thus </b><a NAME="E3:2_1_1_2_2"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of 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>2</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> or  not <font color="Maroon">c<sub>3</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> or <font color="Maroon">c<sub>4</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

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