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<b>let </b><font color="Maroon" title="c1">R1</font>, <font color="Maroon" title="c2">R2</font> be  <a href="lattice3.html#V3" title="LATTICE3:attr.3">complete</a> <a href="waybel_0.html#NM8" title="WAYBEL_0:NM.8">LATTICE</a>;<br/>

<b>assume </b><a NAME="E1:91"/><i><font color="Green" title="E60">A1</font></i>: 
 <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R1</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R1</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">R2</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c2">R2</font> #)
 ;<br/>

<a NAME="E2:91"/><i><font color="Green" title="E61">A2</font></i>: 
 <a href="waybel11.html#K2" title="WAYBEL11:func.2">Scott-Convergence</a> <font color="Maroon" title="c1">R1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_6.html#K7" title="YELLOW_6:func.7">NetUniv</a> <font color="Maroon" title="c1">R1</font></span>)</span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R1</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="ref" href="yellow_6.html#D21" title="YELLOW_6:def.21">YELLOW_6:def 21</a></i>;<br/>
<a NAME="E3:91"/>
for <font color="Olive" title="b1">e</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel11.html#K2" title="WAYBEL11:func.2">Scott-Convergence</a> <font color="Maroon" title="c1">R1</font> holds <br/><font color="Olive" title="b1">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel11.html#K2" title="WAYBEL11:func.2">Scott-Convergence</a> <font color="Maroon" title="c2">R2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="91_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">e</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:91_1"/><i><font color="Green" title="E62">A3</font></i>: 
<font color="Maroon" title="c3">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel11.html#K2" title="WAYBEL11:func.2">Scott-Convergence</a> <font color="Maroon" title="c1">R1</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c4">N1</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="yellow_6.html#K7" title="YELLOW_6:func.7">NetUniv</a> <font color="Maroon" title="c1">R1</font>, <font color="Maroon" title="c5">p1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R1</font><b> such that </b><br/><a NAME="E3:91_1"/><i><font color="Green" title="E63">A4</font></i>: 
<font color="Maroon" title="c3">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c4">N1</font>,<font color="Maroon" title="c5">p1</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 <b>by </b><i><a class="txt" href="waybel11.html#E2:91"><i><font color="Green" title="E61">A2</font></i></a>, <a class="ref" href="domain_1.html#T9" title="DOMAIN_1:th.9">DOMAIN_1:9</a></i>;<br/>
<a NAME="E4:91_1"/><i><font color="Green" title="E64">A5</font></i>: 
<font color="Maroon" title="c4">N1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="yellow_6.html#K7" title="YELLOW_6:func.7">NetUniv</a> <font color="Maroon" title="c1">R1</font>
 
;<br/>
<b>consider </b><font color="Maroon" title="c6">N</font> being  <a href="waybel_0.html#V6" title="WAYBEL_0:attr.6">strict</a> <a href="waybel_0.html#NM2" title="WAYBEL_0:NM.2">net</a> of <font color="Maroon" title="c1">R1</font><b> such that </b><br/><a NAME="E6:91_1"/><i><font color="Green" title="E65">A6</font></i>: 
<font color="Maroon" title="c4">N1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">N</font>
 <b>and </b><br/><a NAME="E7:91_1"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c6">N</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="yellow_6.html#K2" title="YELLOW_6:func.2">the_universe_of</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R1</font>
 <b>by </b><i><a class="ref" href="yellow_6.html#D14" title="YELLOW_6:def.14">YELLOW_6:def 14</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">N2</font> = <font color="Maroon" title="c4">N1</font> as  <a href="waybel_0.html#V6" title="WAYBEL_0:attr.6">strict</a> <a href="waybel_0.html#NM2" title="WAYBEL_0:NM.2">net</a> of <font color="Maroon" title="c2">R2</font> <b>by </b><i><a class="txt" href="waybel11.html#E1:91"><i><font color="Green" title="E60">A1</font></i></a>, <a class="txt" href="waybel11.html#E6:91_1"><i><font color="Green" title="E65">A6</font></i></a>, <a class="txt" href="waybel11.html#E53"><i><font color="Green" title="E44">Lm4</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c8">N1</font> = <font color="Maroon" title="c4">N1</font> as  <a href="waybel_0.html#V6" title="WAYBEL_0:attr.6">strict</a> <a href="waybel_0.html#NM2" title="WAYBEL_0:NM.2">net</a> of <font color="Maroon" title="c1">R1</font> <b>by </b><i><a class="txt" href="waybel11.html#E6:91_1"><i><font color="Green" title="E65">A6</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c9">p2</font> = <font color="Maroon" title="c5">p1</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">R2</font> <b>by </b><i><a class="txt" href="waybel11.html#E1:91"><i><font color="Green" title="E60">A1</font></i></a></i>;<br/>
<a NAME="E11:91_1"/><i><font color="Green" title="E66">A7</font></i>: 
<font color="Maroon" title="c7">N2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="yellow_6.html#K7" title="YELLOW_6:func.7">NetUniv</a> <font color="Maroon" title="c2">R2</font>
 
<b>by </b><i><a class="txt" href="waybel11.html#E1:91"><i><font color="Green" title="E60">A1</font></i></a>, <a class="txt" href="waybel11.html#E4:91_1"><i><font color="Green" title="E64">A5</font></i></a>, <a class="txt" href="waybel11.html#E54"><i><font color="Green" title="E45">Lm5</font></i></a></i>;<br/>
<a NAME="E12:91_1"/><i><font color="Green" title="E67">A8</font></i>: 
 <a href="waybel11.html#K1" title="WAYBEL11:func.1">lim_inf</a> <font color="Maroon" title="c8">N1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="waybel11.html#K1" title="WAYBEL11:func.1">lim_inf</a> <font color="Maroon" title="c7">N2</font>
 
<b>by </b><i><a class="txt" href="waybel11.html#E1:91"><i><font color="Green" title="E60">A1</font></i></a>, <a class="txt" href="waybel11.html#E65"><i><font color="Green" title="E56">Lm16</font></i></a></i>;<br/>
<a NAME="E13:91_1"/>
<font color="Maroon" title="c5">p1</font> <a href="waybel11.html#R1" title="WAYBEL11:pred.1">is_S-limit_of</a> <font color="Maroon" title="c8">N1</font>
 
<b>by </b><i><a class="txt" href="waybel11.html#E1:91_1"><i><font color="Green" title="E62">A3</font></i></a>, <a class="txt" href="waybel11.html#E3:91_1"><i><font color="Green" title="E63">A4</font></i></a>, <a class="ref" href="waybel11.html#D8" target="_self" title="WAYBEL11:def.8">Def8</a></i>;<br/>
<a NAME="E14:91_1"/><b>then </b>
<font color="Maroon" title="c5">p1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a>  <a href="waybel11.html#K1" title="WAYBEL11:func.1">lim_inf</a> <font color="Maroon" title="c8">N1</font>
 
<b>by </b><i><a class="ref" href="waybel11.html#D7" target="_self" title="WAYBEL11:def.7">Def7</a></i>;<br/>
<a NAME="E15:91_1"/><b>then </b>
<font color="Maroon" title="c9">p2</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a>  <a href="waybel11.html#K1" title="WAYBEL11:func.1">lim_inf</a> <font color="Maroon" title="c7">N2</font>
 
<b>by </b><i><a class="txt" href="waybel11.html#E1:91"><i><font color="Green" title="E60">A1</font></i></a>, <a class="txt" href="waybel11.html#E12:91_1"><i><font color="Green" title="E67">A8</font></i></a>, <a class="txt" href="waybel11.html#E1"><i><font color="Green" title="E1">Lm1</font></i></a></i>;<br/>
<a NAME="E16:91_1"/><b>then </b>
<font color="Maroon" title="c9">p2</font> <a href="waybel11.html#R1" title="WAYBEL11:pred.1">is_S-limit_of</a> <font color="Maroon" title="c7">N2</font>
 
<b>by </b><i><a class="ref" href="waybel11.html#D7" target="_self" title="WAYBEL11:def.7">Def7</a></i>;<br/>
<b>hence </b><a NAME="E17:91_1"/>
<font color="Maroon" title="c3">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel11.html#K2" title="WAYBEL11:func.2">Scott-Convergence</a> <font color="Maroon" title="c2">R2</font>
 <b>by </b><i><a class="txt" href="waybel11.html#E3:91_1"><i><font color="Green" title="E63">A4</font></i></a>, <a class="txt" href="waybel11.html#E11:91_1"><i><font color="Green" title="E66">A7</font></i></a>, <a class="ref" href="waybel11.html#D8" target="_self" title="WAYBEL11:def.8">Def8</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:91"/>
 <a href="waybel11.html#K2" title="WAYBEL11:func.2">Scott-Convergence</a> <font color="Maroon" title="c1">R1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="waybel11.html#K2" title="WAYBEL11:func.2">Scott-Convergence</a> <font color="Maroon" title="c2">R2</font>
 <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>


</div>
