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<span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <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> #) implies  <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> )</span><br/>

<span class="kw">assume </span><a NAME="E1:90"/><span class="lab"><font color="Green" title="E60">A1</font></span>: 
 <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> #)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <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></span><br/>

<a NAME="E2:90"/><span class="lab"><font color="Green" title="E61">A2</font></span>: 
 <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#K6" title="YELLOW_6:func.6">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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_6.html#D21" title="YELLOW_6:def.21">YELLOW_6:def 21</a></span>;<br/>
<a NAME="E3:90"/>
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="90_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">e</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <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> implies <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> )</span><br/>

<span class="kw">assume </span><a NAME="E1:90_1"/><span class="lab"><font color="Green" title="E62">A3</font></span>: 
<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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <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></span><br/>

<span class="kw">then </span><span class="kw">consider </span><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#K6" title="YELLOW_6:func.6">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><span class="kw"> such that </span><br/><a NAME="E3:90_1"/><span class="lab"><font color="Green" title="E63">A4</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:90"><span class="lab"><font color="Green" title="E61">A2</font></span></a>, <a class="ref" href="domain_1.html#T9" title="DOMAIN_1:th.9">DOMAIN_1:9</a></span>;<br/>
<a NAME="E4:90_1"/><span class="lab"><font color="Green" title="E64">A5</font></span>: 
<font color="Maroon" title="c4">N1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="yellow_6.html#K6" title="YELLOW_6:func.6">NetUniv</a> <font color="Maroon" title="c1">R1</font>
 
;<br/>
<a NAME="E5:90_1"/><span class="lab"><font color="Green" title="E65">A6</font></span>: 
 ex <font color="Olive" title="b1">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> st <br/>( <font color="Maroon" title="c4">N1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">N</font> &amp; the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">N</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="yellow_6.html#K1" title="YELLOW_6:func.1">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> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_6.html#D14" title="YELLOW_6:def.14">YELLOW_6:def 14</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c6">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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:90"><span class="lab"><font color="Green" title="E60">A1</font></span></a>, <a class="txt" href="waybel11.html#E53"><span class="lab"><font color="Green" title="E44">Lm4</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c7">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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:90_1"><span class="lab"><font color="Green" title="E65">A6</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c8">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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:90"><span class="lab"><font color="Green" title="E60">A1</font></span></a></span>;<br/>
<a NAME="E9:90_1"/><span class="lab"><font color="Green" title="E66">A7</font></span>: 
<font color="Maroon" title="c6">N2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="yellow_6.html#K6" title="YELLOW_6:func.6">NetUniv</a> <font color="Maroon" title="c2">R2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:90"><span class="lab"><font color="Green" title="E60">A1</font></span></a>, <a class="txt" href="#E4:90_1"><span class="lab"><font color="Green" title="E64">A5</font></span></a>, <a class="txt" href="waybel11.html#E54"><span class="lab"><font color="Green" title="E45">Lm5</font></span></a></span>;<br/>
<a NAME="E10:90_1"/><span class="lab"><font color="Green" title="E67">A8</font></span>: 
 <a href="waybel11.html#K1" title="WAYBEL11:func.1">lim_inf</a> <font color="Maroon" title="c7">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="c6">N2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:90"><span class="lab"><font color="Green" title="E60">A1</font></span></a>, <a class="txt" href="waybel11.html#E65"><span class="lab"><font color="Green" title="E56">Lm16</font></span></a></span>;<br/>
<a NAME="E11:90_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="c7">N1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:90_1"><span class="lab"><font color="Green" title="E62">A3</font></span></a>, <a class="txt" href="#E3:90_1"><span class="lab"><font color="Green" title="E63">A4</font></span></a>, <a class="ref" href="waybel11.html#D8" target="_self" title="WAYBEL11:def.8">Def8</a></span>;<br/>
<a NAME="E12:90_1"/><span class="kw">then </span>
<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="c7">N1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="waybel11.html#D7" target="_self" title="WAYBEL11:def.7">Def7</a></span>;<br/>
<a NAME="E13:90_1"/><span class="kw">then </span>
<font color="Maroon" title="c8">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="c6">N2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:90"><span class="lab"><font color="Green" title="E60">A1</font></span></a>, <a class="txt" href="#E10:90_1"><span class="lab"><font color="Green" title="E67">A8</font></span></a>, <a class="txt" href="waybel11.html#E1"><span class="lab"><font color="Green" title="E1">Lm1</font></span></a></span>;<br/>
<a NAME="E14:90_1"/><span class="kw">then </span>
<font color="Maroon" title="c8">p2</font> <a href="waybel11.html#R1" title="WAYBEL11:pred.1">is_S-limit_of</a> <font color="Maroon" title="c6">N2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="waybel11.html#D7" target="_self" title="WAYBEL11:def.7">Def7</a></span>;<br/>
<span class="kw">hence </span><a NAME="E15:90_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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:90_1"><span class="lab"><font color="Green" title="E63">A4</font></span></a>, <a class="txt" href="#E9:90_1"><span class="lab"><font color="Green" title="E66">A7</font></span></a>, <a class="ref" href="waybel11.html#D8" target="_self" title="WAYBEL11:def.8">Def8</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E4:90"/>
 <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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
