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<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be  non <a href="struct_0.html#V3">empty</a> <a href="orders_2.html#V2">reflexive</a> <a href="orders_2.html#V3">transitive</a> <a href="orders_2.html#V4">antisymmetric</a> <a href="lattice3.html#V1">with_suprema</a>  <a href="orders_2.html#L1">RelStr</a> ;<br/><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>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>





<b>assume </b><a NAME="E1:23"/><i><font color="Green">E17</font></i>: 
( the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <a href="xboole_0.html#K3">/\</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> is  <a href="waybel_0.html#V1">directed</a> <a href="waybel_0.html#V12">lower</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>2</sub></font> &amp; the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <a href="xboole_0.html#K3">/\</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> is  <a href="waybel_0.html#V13">upper</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="latsum_1.html#R1" target="_self">tolerates</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> )
 ;<br/>

<a NAME="E2:23"/>
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <a href="xboole_0.html#K3">/\</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <a href="xboole_0.html#K3">/\</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="txt" href="latsum_1.html#E1:23"><i><font color="Green">E17</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E3:23"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <a href="xboole_0.html#K3">/\</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="latsum_1.html#E1:23"><i><font color="Green">E17</font></i></a>, <a class="ref" href="latsum_1.html#T16" target="_self">Th16</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>5</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>6</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="latsum_1.html#T13" target="_self">Th13</a></i>;<br/>
<a NAME="E5:23"/>
<font color="Maroon">c<sub>5</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>6</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="ref" href="yellow_0.html#T22">YELLOW_0:22</a></i>;<br/>
<a NAME="E6:23"/><b>then </b><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>7</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="latsum_1.html#E1:23"><i><font color="Green">E17</font></i></a>, <a class="ref" href="latsum_1.html#T15" target="_self">Th15</a></i>;<br/>
<a NAME="E7:23"/>
<font color="Maroon">c<sub>5</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>6</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="ref" href="yellow_0.html#T22">YELLOW_0:22</a></i>;<br/>
<a NAME="E8:23"/><b>then </b><i><font color="Green">E19</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="latsum_1.html#E1:23"><i><font color="Green">E17</font></i></a>, <a class="ref" href="latsum_1.html#T15" target="_self">Th15</a></i>;<br/>
<a NAME="E9:23"/>
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#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>4</sub></font> holds <br/><font color="Maroon">c<sub>7</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>8</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="E1:23_1"/><i><font color="Green">E20</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>4</sub></font> )
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>8</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="latsum_1.html#K1" target="_self">[*]</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <b>by </b><i><a class="ref" href="latsum_1.html#T10" target="_self">Th10</a></i>;<br/>
<a NAME="E3:23_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>10</sub></font>
 
<b>by </b><i><a class="txt" href="latsum_1.html#E1:23"><i><font color="Green">E17</font></i></a>, <a class="txt" href="latsum_1.html#E1:23_1"><i><font color="Green">E20</font></i></a>, <a class="ref" href="latsum_1.html#T8" target="_self">Th8</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>8</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="txt" href="latsum_1.html#E1:23"><i><font color="Green">E17</font></i></a>, <a class="ref" href="latsum_1.html#T18" target="_self">Th18</a></i>;<br/>
<a NAME="E5:23_1"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">c<sub>11</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="latsum_1.html#E1:23"><i><font color="Green">E17</font></i></a>, <a class="txt" href="latsum_1.html#E1:23_1"><i><font color="Green">E20</font></i></a>, <a class="ref" href="latsum_1.html#T15" target="_self">Th15</a></i>;<br/>
<a NAME="E6:23_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>11</sub></font>
 
<b>by </b><i><a class="txt" href="latsum_1.html#E1:23"><i><font color="Green">E17</font></i></a>, <a class="txt" href="latsum_1.html#E1:23_1"><i><font color="Green">E20</font></i></a>, <a class="ref" href="latsum_1.html#T15" target="_self">Th15</a></i>;<br/>
<a NAME="E7:23_1"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>6</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>11</sub></font>
 
<b>by </b><i><a class="txt" href="latsum_1.html#E5:23_1"><i><font color="Green">E21</font></i></a>, <a class="ref" href="yellow_5.html#T9">YELLOW_5:9</a></i>;<br/>
<b>hence </b><a NAME="E8:23_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="latsum_1.html#E1:23"><i><font color="Green">E17</font></i></a>, <a class="ref" href="latsum_1.html#T15" target="_self">Th15</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E10:23"/>
<font color="Maroon">c<sub>3</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="latsum_1.html#E6:23"><i><font color="Green">E18</font></i></a>, <a class="txt" href="latsum_1.html#E8:23"><i><font color="Green">E19</font></i></a>, <a class="ref" href="yellow_0.html#T22">YELLOW_0:22</a></i>;<br/>


</div>
