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<b>let </b><font color="Maroon">L1</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>, <font color="Maroon">L2</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="E1:13"/><i><font color="Green">E46</font></i>: 
 <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L1</font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">L1</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">L2</font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">L2</font> #)
 ;<br/>

<b>let </b><font color="Maroon">N1</font> be    <a href="waybel_0.html#L1">NetStr</a> of <font color="Maroon">L1</font>;<br/><b>let </b><font color="Maroon">a</font> be    <a href="hidden.html#M1">set</a> ;<br/>



<b>assume </b><a NAME="E2:13"/><i><font color="Green">E47</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">N1</font>,<font color="Maroon">a</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="waybel28.html#K3">lim_inf-Convergence</a> <font color="Maroon">L1</font>
 ;<br/>

<a NAME="E3:13"/>
 <a href="waybel28.html#K3">lim_inf-Convergence</a> <font color="Maroon">L1</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_6.html#K7">NetUniv</a> <font color="Maroon">L1</font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L1</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="yellow_6.html#D21">YELLOW_6:def 21</a></i>;<br/>
<b>then consider </b><font color="Maroon">N</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:13"/><i><font color="Green">E52</font></i>: 
( <font color="Maroon">N</font> <a href="hidden.html#R2">in</a>  <a href="yellow_6.html#K7">NetUniv</a> <font color="Maroon">L1</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L1</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">N1</font>,<font color="Maroon">a</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">N</font>,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span> )
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L1</font> <b>by </b><i/>;<br/>
<a NAME="E7:13"/><i><font color="Green">E53</font></i>: 
( <font color="Maroon">N</font> <a href="hidden.html#R1">=</a> <font color="Maroon">N1</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> )
 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>
<b>then consider </b><font color="Maroon">N2</font> being  <a href="waybel_0.html#V6">strict</a> <a href="waybel_0.html#NM2">net</a> of <font color="Maroon">L2</font><b> such that </b><br/><a NAME="E9:13"/><i><font color="Green">E54</font></i>: 
( <font color="Maroon">N2</font> <a href="hidden.html#R2">in</a>  <a href="yellow_6.html#K7">NetUniv</a> <font color="Maroon">L2</font> &amp;  <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">N1</font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">N1</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">N2</font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">N2</font> #) &amp; the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">N1</font> <a href="funct_2.html#R1">=</a> the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">N2</font> )
 <b>by </b><i>, , </i>;<br/>
<b>take </b>
<font color="Maroon">N2</font>
;<br/>

<a NAME="E10:13"/>
 ex <font color="Olive">N</font> being <a href="waybel_0.html#V6">strict</a> <a href="waybel_0.html#NM2">net</a> of <font color="Maroon">L1</font> st <br/>( <font color="Olive">N</font> <a href="hidden.html#R1">=</a> <font color="Maroon">N1</font> &amp; the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">N</font> <a href="hidden.html#R2">in</a>  <a href="yellow_6.html#K2">the_universe_of</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L1</font> )
 
<b>by </b><i>, , <a class="ref" href="yellow_6.html#D14">YELLOW_6:def 14</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">N1</font> = <font color="Maroon">N1</font> as  <a href="waybel_0.html#V6">strict</a> <a href="waybel_0.html#NM2">net</a> of <font color="Maroon">L1</font> ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">M</font> be    <a href="yellow_6.html#M2">subnet</a> of <font color="Maroon">N2</font>;<br/><b>consider </b><font color="Maroon">M1</font> being  <a href="waybel_0.html#V6">strict</a>  <a href="yellow_6.html#M2">subnet</a> of <font color="Maroon">N1</font><b> such that </b><br/><a NAME="E2:13_1"/><i><font color="Green">E55</font></i>: 
(  <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">M</font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">M</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">M1</font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">M1</font> #) &amp; the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">M</font> <a href="funct_2.html#R1">=</a> the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">M1</font> )
 <b>by </b><i>, , </i>;<br/><b>thus </b><font color="Maroon">x</font> = 
 <a href="waybel11.html#K1">lim_inf</a> <font color="Maroon">M1</font>
<b>by </b><i>, , , <a class="ref" href="waybel28.html#D3">WAYBEL28:def 3</a></i>
<br/>.= 
 <a href="waybel11.html#K1">lim_inf</a> <font color="Maroon">M</font>
<b>by </b><i>, , </i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E13:13"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">N2</font>,<font color="Maroon">a</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="waybel28.html#K3">lim_inf-Convergence</a> <font color="Maroon">L2</font> &amp;  <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">N1</font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">N1</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">N2</font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">N2</font> #) &amp; the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">N1</font> <a href="funct_2.html#R1">=</a> the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">N2</font> )
 <b>by </b><i>, , , <a class="ref" href="waybel28.html#D3">WAYBEL28:def 3</a></i>;<br/>


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