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<b>let </b><font color="Maroon">c<sub>1</sub></font> be  <a href="orders_2.html#V4">antisymmetric</a>  <a href="orders_2.html#L1">RelStr</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font>, <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>1</sub></font>;<br/>



<div><b>hereby </b>
<div class="add"><b>assume </b><b>that </b><br/><a NAME="E1:31_1"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K11">"/\"</a> <font color="Maroon">c<sub>3</sub></font>
 <b>and </b><br/><a NAME="E2:31_1"/><i><font color="Green">E23</font></i>: 
 <a href="yellow_0.html#R2" target="_self">ex_inf_of</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="tarski.html#K2">}</a></span>,<font color="Maroon">c<sub>1</sub></font>
 ;<br/><b>consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E4:31_1"/><i><font color="Green">E24</font></i>: 
<span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="tarski.html#K2">}</a></span> <a href="lattice3.html#NR3" target="_self">is_&gt;=_than</a> <font color="Maroon">c<sub>5</sub></font>
 <b>and </b><br/><a NAME="E5:31_1"/><i><font color="Green">E25</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="tarski.html#K2">}</a></span> <a href="lattice3.html#NR3" target="_self">is_&gt;=_than</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Maroon">c<sub>5</sub></font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Olive">b<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="yellow_0.html#E2:31_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="yellow_0.html#T16" target="_self">Th16</a></i>;<br/><div><i><font color="Green">E26</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>assume </b><a NAME="E1:31_1_1"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>6</sub></font> )
 ;<br/><a NAME="E2:31_1_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="tarski.html#K2">}</a></span> <a href="lattice3.html#NR3" target="_self">is_&gt;=_than</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T8" target="_self">Th8</a></i>;<br/><b>hence </b><a NAME="E3:31_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="yellow_0.html#E5:31_1"><i><font color="Green">E25</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E7:31_1"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="txt" href="yellow_0.html#E4:31_1"><i><font color="Green">E24</font></i></a>, <a class="ref" href="yellow_0.html#T8" target="_self">Th8</a></i>;<br/><b>hence </b><a NAME="E8:31_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> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> holds <br/><font color="Maroon">c<sub>4</sub></font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Olive">b<sub>1</sub></font> ) )
 <b>by </b><i><a class="txt" href="yellow_0.html#E1:31_1"><i><font color="Green">E22</font></i></a>, <a class="txt" href="yellow_0.html#E6:31_1"><i><font color="Green">E26</font></i></a>, <a class="ref" href="lattice3.html#D14">LATTICE3:def 14</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:31"/><i><font color="Green">E22</font></i>: 
( <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> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> holds <br/><font color="Maroon">c<sub>4</sub></font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Olive">b<sub>1</sub></font> ) )
 ;<br/>

<b>hence </b><a NAME="E3:31"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K11">"/\"</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="lattice3.html#D14">LATTICE3:def 14</a></i>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2"><b>now </b></a><div class="add"><b>take </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</sub></font>;<br/><b>thus </b><a NAME="E1:31_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="lattice3.html#R1">is_&lt;=_than</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="tarski.html#K2">}</a></span>
 <b>by </b><i><a class="txt" href="yellow_0.html#E2:31"><i><font color="Green">E22</font></i></a>, <a class="ref" href="yellow_0.html#T8" target="_self">Th8</a></i>;<br/><b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>assume </b><a NAME="E2:31_2"/>
<font color="Maroon">c<sub>6</sub></font> <a href="lattice3.html#R1">is_&lt;=_than</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="tarski.html#K2">}</a></span>
 ;<br/><a NAME="E3:31_2"/><b>then </b>
( <font color="Maroon">c<sub>6</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="ref" href="yellow_0.html#T8" target="_self">Th8</a></i>;<br/><b>hence </b><a NAME="E4:31_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="yellow_0.html#E2:31"><i><font color="Green">E22</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:31"/>
 <a href="yellow_0.html#R2" target="_self">ex_inf_of</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="tarski.html#K2">}</a></span>,<font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T16" target="_self">Th16</a></i>;<br/>


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