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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="lattice3.html#V3">complete</a> <a href="waybel_0.html#NM8">LATTICE</a>;<br/>

<b>assume </b><a NAME="E1:91"/><i><font color="Green">E60</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/>

<a NAME="E2:91"/><i><font color="Green">E61</font></i>: 
 <a href="waybel11.html#K2" target="_self">Scott-Convergence</a> <font color="Maroon">c<sub>1</sub></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">c<sub>1</sub></font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></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/>
<a NAME="E3:91"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="waybel11.html#K2" target="_self">Scott-Convergence</a> <font color="Maroon">c<sub>1</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="waybel11.html#K2" target="_self">Scott-Convergence</a> <font color="Maroon">c<sub>2</sub></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">c<sub>3</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:91_1"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="waybel11.html#K2" target="_self">Scott-Convergence</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>4</sub></font> being    <a href="subset_1.html#M1">Element</a> of  <a href="yellow_6.html#K7">NetUniv</a> <font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>5</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:91_1"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K1">]</a></span>
 <b>by </b><i><a class="txt" href="waybel11.html#E2:91"><i><font color="Green">E61</font></i></a>, <a class="ref" href="domain_1.html#T9">DOMAIN_1:9</a></i>;<br/>
<a NAME="E4:91_1"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="yellow_6.html#K7">NetUniv</a> <font color="Maroon">c<sub>1</sub></font>
 
;<br/>
<b>consider </b><font color="Maroon">c<sub>6</sub></font> being  <a href="waybel_0.html#V6">strict</a> <a href="waybel_0.html#NM2">net</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E6:91_1"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font>
 <b>and </b><br/><a NAME="E7:91_1"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>6</sub></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">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="yellow_6.html#D14">YELLOW_6:def 14</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>4</sub></font> as  <a href="waybel_0.html#V6">strict</a> <a href="waybel_0.html#NM2">net</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="txt" href="waybel11.html#E1:91"><i><font color="Green">E60</font></i></a>, <a class="txt" href="waybel11.html#E6:91_1"><i><font color="Green">E65</font></i></a>, <a class="txt" href="waybel11.html#E53"><i><font color="Green">Lemma44</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>4</sub></font> as  <a href="waybel_0.html#V6">strict</a> <a href="waybel_0.html#NM2">net</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="waybel11.html#E6:91_1"><i><font color="Green">E65</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>5</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="waybel11.html#E1:91"><i><font color="Green">E60</font></i></a></i>;<br/>
<a NAME="E11:91_1"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="yellow_6.html#K7">NetUniv</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="waybel11.html#E1:91"><i><font color="Green">E60</font></i></a>, <a class="txt" href="waybel11.html#E4:91_1"><i><font color="Green">E64</font></i></a>, <a class="txt" href="waybel11.html#E54"><i><font color="Green">Lemma45</font></i></a></i>;<br/>
<a NAME="E12:91_1"/><i><font color="Green">E67</font></i>: 
 <a href="waybel11.html#K1" target="_self">lim_inf</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="waybel11.html#K1" target="_self">lim_inf</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="waybel11.html#E1:91"><i><font color="Green">E60</font></i></a>, <a class="txt" href="waybel11.html#E65"><i><font color="Green">Lemma56</font></i></a></i>;<br/>
<a NAME="E13:91_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="waybel11.html#R1" target="_self">is_S-limit_of</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="txt" href="waybel11.html#E1:91_1"><i><font color="Green">E62</font></i></a>, <a class="txt" href="waybel11.html#E3:91_1"><i><font color="Green">E63</font></i></a>, <a class="ref" href="waybel11.html#D8" target="_self">Def8</a></i>;<br/>
<a NAME="E14:91_1"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="orders_2.html#R3">&lt;=</a>  <a href="waybel11.html#K1" target="_self">lim_inf</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="ref" href="waybel11.html#D7" target="_self">Def7</a></i>;<br/>
<a NAME="E15:91_1"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font> <a href="orders_2.html#R3">&lt;=</a>  <a href="waybel11.html#K1" target="_self">lim_inf</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="waybel11.html#E1:91"><i><font color="Green">E60</font></i></a>, <a class="txt" href="waybel11.html#E12:91_1"><i><font color="Green">E67</font></i></a>, <a class="txt" href="waybel11.html#E1"><i><font color="Green">Lemma1</font></i></a></i>;<br/>
<a NAME="E16:91_1"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font> <a href="waybel11.html#R1" target="_self">is_S-limit_of</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="ref" href="waybel11.html#D7" target="_self">Def7</a></i>;<br/>
<b>hence </b><a NAME="E17:91_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="waybel11.html#K2" target="_self">Scott-Convergence</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="waybel11.html#E3:91_1"><i><font color="Green">E63</font></i></a>, <a class="txt" href="waybel11.html#E11:91_1"><i><font color="Green">E66</font></i></a>, <a class="ref" href="waybel11.html#D8" target="_self">Def8</a></i>;<br/>


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


</div>
