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<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be  <a href="waybel_0.html#V24">up-complete</a> <a href="waybel_0.html#V25">/\-complete</a> <a href="waybel_0.html#NM6">Semilattice</a>;<br/>

<b>assume </b><a NAME="E1:11"/><i><font color="Green">E7</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>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="waybel_0.html#NM2">net</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="waybel_0.html#NM2">net</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>



<b>assume </b><a NAME="E2:11"/><i><font color="Green">E8</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>3</sub></font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>3</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>4</sub></font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>4</sub></font> #) &amp; the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#R1">=</a> the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">c<sub>4</sub></font> )
 ;<br/>

<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>3</sub></font>) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="waybel_0.html#K2">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>3</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">a<sub>1</sub></font> </span>}</span> ;<br/>
<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>4</sub></font>) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="waybel_0.html#K2">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>4</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">a<sub>1</sub></font> </span>}</span> ;<br/>
<b>set </b><font color="Maroon">c<sub>5</sub></font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_0.html#K2">"/\"</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>),<font color="Maroon">c<sub>1</sub></font></span>)</span> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>3</sub></font> : verum </span>}</span> ;<br/>
<b>set </b><font color="Maroon">c<sub>6</sub></font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_0.html#K2">"/\"</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">b<sub>1</sub></font>),<font color="Maroon">c<sub>2</sub></font></span>)</span> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>4</sub></font> : verum </span>}</span> ;<br/>
<a NAME="E3:11"/><i><font color="Green">E9</font></i>: 
 <a href="waybel11.html#K1">lim_inf</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K1">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_0.html#K2">"/\"</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>),<font color="Maroon">c<sub>1</sub></font></span>)</span> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>3</sub></font> : verum </span>}</span> ,<font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="waybel11.html#D6">WAYBEL11:def 6</a></i>;<br/>
<a NAME="E4:11"/><i><font color="Green">E10</font></i>: 
 <a href="waybel11.html#K1">lim_inf</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K1">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_0.html#K2">"/\"</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">b<sub>1</sub></font>),<font color="Maroon">c<sub>2</sub></font></span>)</span> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>4</sub></font> : verum </span>}</span> ,<font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="waybel11.html#D6">WAYBEL11:def 6</a></i>;<br/>
<a NAME="E5:11"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_0.html#K2">"/\"</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>),<font color="Maroon">c<sub>1</sub></font></span>)</span> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>3</sub></font> : verum </span>}</span>  <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>7</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:11_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_0.html#K2">"/\"</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>),<font color="Maroon">c<sub>1</sub></font></span>)</span> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>3</sub></font> : verum </span>}</span> 
 ;<br/>

<a NAME="E2:11_1"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>3</sub></font> st <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K2">"/\"</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>),<font color="Maroon">c<sub>1</sub></font>
 
;<br/>
<b>hence </b><a NAME="E3:11_1"/>
<font color="Maroon">c<sub>7</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>
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>7</sub></font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_0.html#K2">"/\"</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>),<font color="Maroon">c<sub>1</sub></font></span>)</span> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>3</sub></font> : verum </span>}</span>  as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<a NAME="E7:11"/>
( not <font color="Maroon">c<sub>7</sub></font> is <a href="xboole_0.html#V1">empty</a> &amp; <font color="Maroon">c<sub>7</sub></font> is <a href="waybel_0.html#V1">directed</a> )
 
<b>by </b><i><a class="ref" href="waybel32.html#T11">WAYBEL32:11</a></i>;<br/>
<a NAME="E8:11"/><b>then </b><i><font color="Green">E11</font></i>: 
 <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Maroon">c<sub>7</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="E9:11"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_0.html#K2">"/\"</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">b<sub>1</sub></font>),<font color="Maroon">c<sub>2</sub></font></span>)</span> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>4</sub></font> : verum </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:11_2"/>
( <font color="Maroon">c<sub>7</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_0.html#K2">"/\"</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">b<sub>1</sub></font>),<font color="Maroon">c<sub>2</sub></font></span>)</span> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>4</sub></font> : verum </span>}</span>  &amp; <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_0.html#K2">"/\"</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">b<sub>1</sub></font>),<font color="Maroon">c<sub>2</sub></font></span>)</span> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>4</sub></font> : verum </span>}</span>  <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>7</sub></font> )
 <b>by </b><i><a class="txt" href="waybel33.html#E1:11"><i><font color="Green">E7</font></i></a>, <a class="txt" href="waybel33.html#E2:11"><i><font color="Green">E8</font></i></a>, <a class="txt" href="waybel33.html#E7"><i><font color="Green">Lemma5</font></i></a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E10:11"/>
 <a href="waybel11.html#K1">lim_inf</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="waybel11.html#K1">lim_inf</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="waybel33.html#E1:11"><i><font color="Green">E7</font></i></a>, <a class="txt" href="waybel33.html#E3:11"><i><font color="Green">E9</font></i></a>, <a class="txt" href="waybel33.html#E4:11"><i><font color="Green">E10</font></i></a>, <a class="txt" href="waybel33.html#E8:11"><i><font color="Green">E11</font></i></a>, <a class="ref" href="yellow_0.html#T26">YELLOW_0:26</a></i>;<br/>


</div>
