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<b>let </b><font color="Maroon">c<sub>1</sub></font> be  <a href="waybel_2.html#V2">meet-continuous</a> <a href="waybel_0.html#NM6">Semilattice</a>;<br/><b>let </b><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#L1">RelStr</a> ;<br/>



<b>assume </b><a NAME="E1:17"/><i><font color="Green">E11</font></i>: 
 <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font> #) <a href="hidden.html#R1">=</a>  <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>2</sub></font> #)
 ;<br/>

<b>hence </b><a NAME="E2:17"/>
<font color="Maroon">c<sub>2</sub></font> is <a href="waybel_0.html#V24">up-complete</a>
 <b>by </b><i><a class="ref" href="waybel_8.html#T15">WAYBEL_8:15</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_2.html#D7">WAYBEL_2:def 7</a><br/>

<b>let </b><font color="Maroon">c<sub>3</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="waybel_2.html#D6">WAYBEL_2:def 6</a><br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be  non <a href="xboole_0.html#V1">empty</a> <a href="waybel_0.html#V1">directed</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>



<b>reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>3</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="txt" href="yellow12.html#E1:17"><i><font color="Green">E11</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>4</sub></font> as  non <a href="xboole_0.html#V1">empty</a> <a href="waybel_0.html#V1">directed</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="yellow12.html#E1:17"><i><font color="Green">E11</font></i></a>, <a class="ref" href="waybel_0.html#T3">WAYBEL_0:3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K6">}</a></span> as  non <a href="xboole_0.html#V1">empty</a> <a href="waybel_0.html#V1">directed</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="waybel_0.html#T5">WAYBEL_0:5</a></i>;<br/>
<a NAME="E6:17"/><i><font color="Green">E12</font></i>: 
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="domain_1.html#K6">}</a></span> <a href="yellow_4.html#K3">"/\"</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K6">}</a></span> <a href="yellow_4.html#K4">"/\"</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="yellow12.html#E1:17"><i><font color="Green">E11</font></i></a>, <a class="ref" href="yellow12.html#T11" target="_self">Th11</a></i>;<br/>
<a NAME="E7:17"/><i><font color="Green">E13</font></i>: 
 <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Maroon">c<sub>7</sub></font> <a href="yellow_4.html#K4">"/\"</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="waybel_0.html#T75">WAYBEL_0:75</a></i>;<br/>
<a NAME="E8:17"/>
 <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="waybel_0.html#T75">WAYBEL_0:75</a></i>;<br/>
<a NAME="E9:17"/><b>then </b>
 <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="yellow12.html#E1:17"><i><font color="Green">E11</font></i></a>, <a class="ref" href="yellow_0.html#T26">YELLOW_0:26</a></i>;<br/>
<b>hence </b><font color="Maroon">c<sub>3</sub></font> <a href="lattice3.html#K11">"/\"</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> = 
<font color="Maroon">c<sub>5</sub></font> <a href="lattice3.html#K12">"/\"</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="yellow12.html#E1:17"><i><font color="Green">E11</font></i></a>, <a class="ref" href="yellow12.html#T9" target="_self">Th9</a></i>
<br/>.= 
 <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K6">}</a></span> <a href="yellow_4.html#K4">"/\"</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="waybel_2.html#D6">WAYBEL_2:def 6</a></i>
<br/>.= 
 <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="domain_1.html#K6">}</a></span> <a href="yellow_4.html#K3">"/\"</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="yellow12.html#E1:17"><i><font color="Green">E11</font></i></a>, <a class="txt" href="yellow12.html#E6:17"><i><font color="Green">E12</font></i></a>, <a class="txt" href="yellow12.html#E7:17"><i><font color="Green">E13</font></i></a>, <a class="ref" href="yellow_0.html#T26">YELLOW_0:26</a></i>
;<br/><br/>


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