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<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">R</font> be   <a href="waybel_0.html#V25" title="WAYBEL_0:attr.25">/\-complete</a>  <a href="waybel_0.html#NM6" title="WAYBEL_0:NM.6">Semilattice</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">S</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">R</font> holds <br/> ( <font color="Olive" title="b1">S</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K12" title="YELLOW_6:func.12">ConvergenceSpace</a> <span class="p2">(<span class="default"><a href="waybel11.html#K2" title="WAYBEL11:func.2">Scott-Convergence</a> <font color="Maroon" title="c1">R</font></span>)</span></span>)</span> iff ( <font color="Olive" title="b1">S</font> is  <a href="waybel11.html#NV4" title="WAYBEL11:NV.4">inaccessible</a>  &amp; <font color="Olive" title="b1">S</font> is  <a href="waybel_0.html#V13" title="WAYBEL_0:attr.13">upper</a>  ) )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c2">SC</font> =  <a href="waybel11.html#K2" title="WAYBEL11:func.2">Scott-Convergence</a> <font color="Maroon" title="c1">R</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c3">T</font> =  <a href="yellow_6.html#K12" title="YELLOW_6:func.12">ConvergenceSpace</a> <span class="p1">(<span class="default"><a href="waybel11.html#K2" title="WAYBEL11:func.2">Scott-Convergence</a> <font color="Maroon" title="c1">R</font></span>)</span>;<br/>
<a NAME="E1:16"/><span class="lab"><font color="Green" title="E11">A1</font></span>: 
 the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K12" title="YELLOW_6:func.12">ConvergenceSpace</a> <span class="p2">(<span class="default"><a href="waybel11.html#K2" title="WAYBEL11:func.2">Scott-Convergence</a> <font color="Maroon" title="c1">R</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">V</font> where <font color="Olive" title="b1">V</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">R</font> :  for <font color="Olive" title="b2">p</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font>  st <font color="Olive" title="b2">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">V</font> holds <br/> for <font color="Olive" title="b3">N</font> being   <a href="waybel_0.html#NM2" title="WAYBEL_0:NM.2">net</a> of <font color="Maroon" title="c1">R</font>  st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b3">N</font>,<font color="Olive" title="b2">p</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <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">R</font> holds <br/><font color="Olive" title="b3">N</font> <a href="waybel_0.html#R1" title="WAYBEL_0:pred.1">is_eventually_in</a> <font color="Olive" title="b1">V</font> </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_6.html#D24" onmouseover="rs('yellow_6/D24')" onmouseout="rh()">YELLOW_6:def 24</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c4">S</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">R</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">S</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K12" title="YELLOW_6:func.12">ConvergenceSpace</a> <span class="p2">(<span class="default"><a href="waybel11.html#K2" title="WAYBEL11:func.2">Scott-Convergence</a> <font color="Maroon" title="c1">R</font></span>)</span></span>)</span> iff ( <font color="Maroon" title="c4">S</font> is  <a href="waybel11.html#NV4" title="WAYBEL11:NV.4">inaccessible</a>  &amp; <font color="Maroon" title="c4">S</font> is  <a href="waybel_0.html#V13" title="WAYBEL_0:attr.13">upper</a>  ) )</span><br/>

<div><span class="kw">hereby </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">S</font> is  <a href="waybel11.html#NV4" title="WAYBEL11:NV.4">inaccessible</a>  &amp; <font color="Maroon" title="c4">S</font> is  <a href="waybel_0.html#V13" title="WAYBEL_0:attr.13">upper</a>  implies <font color="Maroon" title="c4">S</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K12" title="YELLOW_6:func.12">ConvergenceSpace</a> <span class="p2">(<span class="default"><a href="waybel11.html#K2" title="WAYBEL11:func.2">Scott-Convergence</a> <font color="Maroon" title="c1">R</font></span>)</span></span>)</span> )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:16_1"/>
<font color="Maroon" title="c4">S</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K12" title="YELLOW_6:func.12">ConvergenceSpace</a> <span class="p2">(<span class="default"><a href="waybel11.html#K2" title="WAYBEL11:func.2">Scott-Convergence</a> <font color="Maroon" title="c1">R</font></span>)</span></span>)</span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">S</font> is  <a href="waybel11.html#NV4" title="WAYBEL11:NV.4">inaccessible</a>  &amp; <font color="Maroon" title="c4">S</font> is  <a href="waybel_0.html#V13" title="WAYBEL_0:attr.13">upper</a>  )</span><br/><a NAME="E2:16_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E12">A2</font></span>: 
 ex <font color="Olive" title="b1">V</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">R</font> st <br/>( <font color="Maroon" title="c4">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">V</font> &amp; (  for <font color="Olive" title="b2">p</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font>  st <font color="Olive" title="b2">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">V</font> holds <br/> for <font color="Olive" title="b3">N</font> being   <a href="waybel_0.html#NM2" title="WAYBEL_0:NM.2">net</a> of <font color="Maroon" title="c1">R</font>  st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b3">N</font>,<font color="Olive" title="b2">p</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <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">R</font> holds <br/><font color="Olive" title="b3">N</font> <a href="waybel_0.html#R1" title="WAYBEL_0:pred.1">is_eventually_in</a> <font color="Olive" title="b1">V</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:16"><span class="lab"><font color="Green" title="E11">A1</font></span></a></span>;<br/><span class="kw">thus </span><a NAME="E3:16_1"/>
<font color="Maroon" title="c4">S</font> is  <a href="waybel11.html#NV4" title="WAYBEL11:NV.4">inaccessible</a> 
  <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">S</font> is  <a href="waybel_0.html#V13" title="WAYBEL_0:attr.13">upper</a> </span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="16_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">D</font> be   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="waybel_0.html#V1" title="WAYBEL_0:attr.1">directed</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">R</font>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="waybel11.html#D1" onmouseover="rs('waybel11/D1')" onmouseout="rh()">WAYBEL11:def 1</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not  <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> (<font color="Maroon" title="c5">D</font>,<font color="Maroon" title="c1">R</font>) <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">S</font> or  not <font color="Maroon" title="c5">D</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c4">S</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:16_1_1"/><span class="lab"><font color="Green" title="E13">A3</font></span>: 
 <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c5">D</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">S</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  not <font color="Maroon" title="c5">D</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c4">S</font></span><br/>

<a NAME="E2:16_1_1"/><span class="lab"><font color="Green" title="E14">A4</font></span>: 
 <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1">(<span class="default"><a href="relat_1.html#K4" title="RELAT_1:func.4">id</a> <font color="Maroon" title="c5">D</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">D</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T45" onmouseover="rs('relat_1/T45')" onmouseout="rh()">RELAT_1:45</a></span>;<br/>
<a NAME="E3:16_1_1"/><span class="lab"><font color="Green" title="E15">A5</font></span>: 
 <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <span class="p1">(<span class="default"><a href="relat_1.html#K4" title="RELAT_1:func.4">id</a> <font color="Maroon" title="c5">D</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">D</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T45" onmouseover="rs('relat_1/T45')" onmouseout="rh()">RELAT_1:45</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c6">f</font> =  <a href="relat_1.html#K4" title="RELAT_1:func.4">id</a> <font color="Maroon" title="c5">D</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c5">D</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:16_1_1"><span class="lab"><font color="Green" title="E14">A4</font></span></a>, <a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T4" onmouseover="rs('relset_1/T4')" onmouseout="rh()">RELSET_1:4</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c7">N</font> =  <a href="waybel11.html#K3" title="WAYBEL11:func.3">Net-Str</a> (<font color="Maroon" title="c5">D</font>,<font color="Maroon" title="c6">f</font>) as   <a href="orders_2.html#V3" title="ORDERS_2:attr.3">reflexive</a>   <a href="waybel_0.html#V6" title="WAYBEL_0:attr.6">strict</a>   <a href="waybel_0.html#V8" title="WAYBEL_0:attr.8">monotone</a>  <a href="waybel_0.html#NM2" title="WAYBEL_0:NM.2">net</a> of <font color="Maroon" title="c1">R</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:16_1_1"><span class="lab"><font color="Green" title="E15">A5</font></span></a>, <a class="ref" href="waybel11.html#T20" onmouseover="rs('waybel11/T20')" onmouseout="rh()">WAYBEL11:20</a></span>;<br/>
<a NAME="E6:16_1_1"/><span class="lab"><font color="Green" title="E16">A6</font></span>: 
<font color="Maroon" title="c7">N</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="yellow_6.html#K5" title="YELLOW_6:func.5">NetUniv</a> <font color="Maroon" title="c1">R</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="waybel11.html#T21" onmouseover="rs('waybel11/T21')" onmouseout="rh()">WAYBEL11:21</a></span>;<br/>
 <a href="waybel11.html#K1" title="WAYBEL11:func.1">lim_inf</a> <font color="Maroon" title="c7">N</font> = 
 <a href="waybel_2.html#K1" title="WAYBEL_2:func.1">sup</a> <font color="Maroon" title="c7">N</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="waybel32.html#T9" target="_self" onmouseover="rs('waybel32/T9')" onmouseout="rh()">Th9</a></span>
<br/>.= 
 <a href="yellow_2.html#NK6" title="YELLOW_2:NK.6">Sup</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_2.html#D1" onmouseover="rs('waybel_2/D1')" onmouseout="rh()">WAYBEL_2:def 1</a></span>
<br/>.= 
 <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> (<span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a>  the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Maroon" title="c7">N</font></span>)</span>,<font color="Maroon" title="c1">R</font>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_2.html#D5" onmouseover="rs('yellow_2/D5')" onmouseout="rh()">YELLOW_2:def 5</a></span>
<br/>.= 
 <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> (<span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c6">f</font></span>)</span>,<font color="Maroon" title="c1">R</font>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="waybel11.html#D10" onmouseover="rs('waybel11/D10')" onmouseout="rh()">WAYBEL11:def 10</a></span>
<br/>.= 
 <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c5">D</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T45" onmouseover="rs('relat_1/T45')" onmouseout="rh()">RELAT_1:45</a></span>
;<br/>
<a NAME="E8:16_1_1"/><span class="kw">then </span>
 <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c5">D</font> <a href="waybel11.html#R1" title="WAYBEL11:pred.1">is_S-limit_of</a> <font color="Maroon" title="c7">N</font>
 
;<br/>
<a NAME="E9:16_1_1"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">N</font>,<span class="p2">(<span class="default"><a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c5">D</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <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">R</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:16_1_1"><span class="lab"><font color="Green" title="E16">A6</font></span></a>, <a class="ref" href="waybel11.html#D8" onmouseover="rs('waybel11/D8')" onmouseout="rh()">WAYBEL11:def 8</a></span>;<br/>
<a NAME="E10:16_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">N</font> <a href="waybel_0.html#R1" title="WAYBEL_0:pred.1">is_eventually_in</a> <font color="Maroon" title="c4">S</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:16_1"><span class="lab"><font color="Green" title="E12">A2</font></span></a>, <a class="txt" href="#E1:16_1_1"><span class="lab"><font color="Green" title="E13">A3</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">i0</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c7">N</font><span class="kw"> such that </span><br/><a NAME="E12:16_1_1"/><span class="lab"><font color="Green" title="E17">A7</font></span>: 
 for <font color="Olive" title="b1">j</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c7">N</font>  st <font color="Maroon" title="c8">i0</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b1">j</font> holds <br/><font color="Maroon" title="c7">N</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Olive" title="b1">j</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">S</font>
 ;<br/>
<span class="kw">consider </span><font color="Maroon" title="c9">j0</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c7">N</font><span class="kw"> such that </span><br/><a NAME="E14:16_1_1"/><span class="lab"><font color="Green" title="E18">A8</font></span>: 
<font color="Maroon" title="c9">j0</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c8">i0</font>
 <span class="kw">and </span><br/><a NAME="E15:16_1_1"/>
<font color="Maroon" title="c9">j0</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c8">i0</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_6.html#D3" onmouseover="rs('yellow_6/D3')" onmouseout="rh()">YELLOW_6:def 3</a></span>;<br/>
<a NAME="E16:16_1_1"/><span class="lab"><font color="Green" title="E19">A9</font></span>: 
<font color="Maroon" title="c7">N</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c9">j0</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">S</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:16_1_1"><span class="lab"><font color="Green" title="E17">A7</font></span></a>, <a class="txt" href="#E14:16_1_1"><span class="lab"><font color="Green" title="E18">A8</font></span></a></span>;<br/>
<a NAME="E17:16_1_1"/><span class="lab"><font color="Green" title="E20">A10</font></span>: 
<font color="Maroon" title="c5">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">N</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="waybel11.html#D10" onmouseover="rs('waybel11/D10')" onmouseout="rh()">WAYBEL11:def 10</a></span>;<br/>
<font color="Maroon" title="c7">N</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c9">j0</font> = 
<span class="p1">(<span class="default"><a href="relat_1.html#K4" title="RELAT_1:func.4">id</a> <font color="Maroon" title="c5">D</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">j0</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="waybel11.html#D10" onmouseover="rs('waybel11/D10')" onmouseout="rh()">WAYBEL11:def 10</a></span>
<br/>.= 
<font color="Maroon" title="c9">j0</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:16_1_1"><span class="lab"><font color="Green" title="E20">A10</font></span></a></span>
;<br/>
<span class="kw">hence </span><a NAME="E19:16_1_1"/>
 not <font color="Maroon" title="c5">D</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c4">S</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:16_1_1"><span class="lab"><font color="Green" title="E19">A9</font></span></a>, <a class="txt" href="#E17:16_1_1"><span class="lab"><font color="Green" title="E20">A10</font></span></a>, <a class="ref" href="xboole_0.html#T3" onmouseover="rs('xboole_0/T3')" onmouseout="rh()">XBOOLE_0:3</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">thus </span><a NAME="E4:16_1"/>
<font color="Maroon" title="c4">S</font> is  <a href="waybel_0.html#V13" title="WAYBEL_0:attr.13">upper</a> 
  <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="16_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">x</font>, <font color="Maroon" title="c6">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="waybel_0.html#D20" onmouseover="rs('waybel_0/D20')" onmouseout="rh()">WAYBEL_0:def 20</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c5">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">S</font> or  not <font color="Maroon" title="c5">x</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c6">y</font> or <font color="Maroon" title="c6">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">S</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:16_1_2"/><span class="lab"><font color="Green" title="E13">A11</font></span>: 
<font color="Maroon" title="c5">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">S</font>
 <span class="kw">and </span><br/><a NAME="E2:16_1_2"/><span class="lab"><font color="Green" title="E14">A12</font></span>: 
<font color="Maroon" title="c5">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">y</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">S</font></span><br/>

<a NAME="E3:16_1_2"/><span class="lab"><font color="Green" title="E15">A13</font></span>: 
 <a href="waybel11.html#K4" title="WAYBEL11:func.4">Net-Str</a> <font color="Maroon" title="c6">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="yellow_6.html#K5" title="YELLOW_6:func.5">NetUniv</a> <font color="Maroon" title="c1">R</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="waybel11.html#T29" onmouseover="rs('waybel11/T29')" onmouseout="rh()">WAYBEL11:29</a></span>;<br/>
<a NAME="E4:16_1_2"/>
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="waybel11.html#K1" title="WAYBEL11:func.1">lim_inf</a> <span class="p1">(<span class="default"><a href="waybel11.html#K4" title="WAYBEL11:func.4">Net-Str</a> <font color="Maroon" title="c6">y</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="waybel11.html#T28" onmouseover="rs('waybel11/T28')" onmouseout="rh()">WAYBEL11:28</a></span>;<br/>
<a NAME="E5:16_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c5">x</font> <a href="waybel11.html#R1" title="WAYBEL11:pred.1">is_S-limit_of</a>  <a href="waybel11.html#K4" title="WAYBEL11:func.4">Net-Str</a> <font color="Maroon" title="c6">y</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:16_1_2"><span class="lab"><font color="Green" title="E14">A12</font></span></a></span>;<br/>
<a NAME="E6:16_1_2"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="waybel11.html#K4" title="WAYBEL11:func.4">Net-Str</a> <font color="Maroon" title="c6">y</font></span>)</span>,<font color="Maroon" title="c5">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <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">R</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:16_1_2"><span class="lab"><font color="Green" title="E15">A13</font></span></a>, <a class="ref" href="waybel11.html#D8" onmouseover="rs('waybel11/D8')" onmouseout="rh()">WAYBEL11:def 8</a></span>;<br/>
<a NAME="E7:16_1_2"/><span class="kw">then </span>
 <a href="waybel11.html#K4" title="WAYBEL11:func.4">Net-Str</a> <font color="Maroon" title="c6">y</font> <a href="waybel_0.html#R1" title="WAYBEL_0:pred.1">is_eventually_in</a> <font color="Maroon" title="c4">S</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:16_1"><span class="lab"><font color="Green" title="E12">A2</font></span></a>, <a class="txt" href="#E1:16_1_2"><span class="lab"><font color="Green" title="E13">A11</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:16_1_2"/>
<font color="Maroon" title="c6">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">S</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="waybel11.html#T27" onmouseover="rs('waybel11/T27')" onmouseout="rh()">WAYBEL11:27</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div></div>
<span class="kw">end;</span></div>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E3:16"/><span class="lab"><font color="Green" title="E12">A14</font></span>: 
<font color="Maroon" title="c4">S</font> is  <a href="waybel11.html#NV4" title="WAYBEL11:NV.4">inaccessible</a> 
 <span class="kw">and </span><br/><a NAME="E4:16"/><span class="lab"><font color="Green" title="E13">A15</font></span>: 
<font color="Maroon" title="c4">S</font> is  <a href="waybel_0.html#V13" title="WAYBEL_0:attr.13">upper</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">S</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K12" title="YELLOW_6:func.12">ConvergenceSpace</a> <span class="p2">(<span class="default"><a href="waybel11.html#K2" title="WAYBEL11:func.2">Scott-Convergence</a> <font color="Maroon" title="c1">R</font></span>)</span></span>)</span></span><br/>

<a NAME="E5:16"/>
 for <font color="Olive" title="b1">p</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font>  st <font color="Olive" title="b1">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">S</font> holds <br/> for <font color="Olive" title="b2">N</font> being   <a href="waybel_0.html#NM2" title="WAYBEL_0:NM.2">net</a> of <font color="Maroon" title="c1">R</font>  st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">N</font>,<font color="Olive" title="b1">p</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <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">R</font> holds <br/><font color="Olive" title="b2">N</font> <a href="waybel_0.html#R1" title="WAYBEL_0:pred.1">is_eventually_in</a> <font color="Maroon" title="c4">S</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="16_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">p</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">S</font> implies  for <font color="Olive" title="b1">N</font> being   <a href="waybel_0.html#NM2" title="WAYBEL_0:NM.2">net</a> of <font color="Maroon" title="c1">R</font>  st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">N</font>,<font color="Maroon" title="c5">p</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <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">R</font> holds <br/><font color="Olive" title="b1">N</font> <a href="waybel_0.html#R1" title="WAYBEL_0:pred.1">is_eventually_in</a> <font color="Maroon" title="c4">S</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:16_2"/><span class="lab"><font color="Green" title="E14">A16</font></span>: 
<font color="Maroon" title="c5">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">S</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">N</font> being   <a href="waybel_0.html#NM2" title="WAYBEL_0:NM.2">net</a> of <font color="Maroon" title="c1">R</font>  st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">N</font>,<font color="Maroon" title="c5">p</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <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">R</font> holds <br/><font color="Olive" title="b1">N</font> <a href="waybel_0.html#R1" title="WAYBEL_0:pred.1">is_eventually_in</a> <font color="Maroon" title="c4">S</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c6">p9</font> = <font color="Maroon" title="c5">p</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font> ;<br/>
<span class="kw">let </span><font color="Maroon" title="c7">N</font> be   <a href="waybel_0.html#NM2" title="WAYBEL_0:NM.2">net</a> of <font color="Maroon" title="c1">R</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">N</font>,<font color="Maroon" title="c5">p</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <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">R</font> implies <font color="Maroon" title="c7">N</font> <a href="waybel_0.html#R1" title="WAYBEL_0:pred.1">is_eventually_in</a> <font color="Maroon" title="c4">S</font> )</span><br/>

<span class="kw">assume </span><a NAME="E3:16_2"/><span class="lab"><font color="Green" title="E15">A17</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">N</font>,<font color="Maroon" title="c5">p</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <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">R</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">N</font> <a href="waybel_0.html#R1" title="WAYBEL_0:pred.1">is_eventually_in</a> <font color="Maroon" title="c4">S</font></span><br/>

<a NAME="E4:16_2"/>
 <a href="waybel11.html#K2" title="WAYBEL11:func.2">Scott-Convergence</a> <font color="Maroon" title="c1">R</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#K5" title="YELLOW_6:func.5">NetUniv</a> <font color="Maroon" title="c1">R</font></span>)</span>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</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#D18" onmouseover="rs('yellow_6/D18')" onmouseout="rh()">YELLOW_6:def 18</a></span>;<br/>
<a NAME="E5:16_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E16">A18</font></span>: 
<font color="Maroon" title="c7">N</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="yellow_6.html#K5" title="YELLOW_6:func.5">NetUniv</a> <font color="Maroon" title="c1">R</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:16_2"><span class="lab"><font color="Green" title="E15">A17</font></span></a>, <a class="ref" href="zfmisc_1.html#T87" onmouseover="rs('zfmisc_1/T87')" onmouseout="rh()">ZFMISC_1:87</a></span>;<br/>
<a NAME="E6:16_2"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">N9</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">R</font> st <br/>( <font color="Olive" title="b1">N9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">N</font> &amp;  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">N9</font> <a href="tarski.html#R2" title="TARSKI: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">R</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_6.html#D11" onmouseover="rs('yellow_6/D11')" onmouseout="rh()">YELLOW_6:def 11</a></span>;<br/>
<a NAME="E7:16_2"/><span class="kw">then </span>
<font color="Maroon" title="c5">p</font> <a href="waybel11.html#R1" title="WAYBEL11:pred.1">is_S-limit_of</a> <font color="Maroon" title="c7">N</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:16_2"><span class="lab"><font color="Green" title="E15">A17</font></span></a>, <a class="txt" href="#E5:16_2"><span class="lab"><font color="Green" title="E16">A18</font></span></a>, <a class="ref" href="waybel11.html#D8" onmouseover="rs('waybel11/D8')" onmouseout="rh()">WAYBEL11:def 8</a></span>;<br/>
<a NAME="E8:16_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E17">A19</font></span>: 
<font color="Maroon" title="c6">p9</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">N</font>
 
;<br/>
<span class="kw">deffunc </span><font color="Maroon">H<sub>1</sub></font>(  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c7">N</font>) <span class="kw">-&gt; </span>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font> =  <a href="yellow_0.html#K2" title="YELLOW_0:func.2">"/\"</a> (<span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c7">N</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Olive" title="b1">i</font></span>)</span> where <font color="Olive" title="b1">i</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c7">N</font> : <font color="Olive" title="b1">i</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> $1 </span> } </span> ,<font color="Maroon" title="c1">R</font>);<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ] <span class="kw">means </span>verum;<br/>
<span class="kw">set </span><font color="Maroon" title="c8">X</font> = <span class="p1"> { <span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">j</font>) where <font color="Olive" title="b1">j</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c7">N</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">j</font>] </span> } </span> ;<br/>
<a NAME="E9:16_2"/>
<span class="p1"> { <span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">j</font>) where <font color="Olive" title="b1">j</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c7">N</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">j</font>] </span> } </span>  is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">R</font>
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="domain_1.html#S8" onmouseover="rs('domain_1/S8')" onmouseout="rh()">DOMAIN_1:sch 8</a>();<br/></span>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c9">D</font> = <span class="p1"> { <span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">j</font>) where <font color="Olive" title="b1">j</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c7">N</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">j</font>] </span> } </span>  as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">R</font> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c10">D</font> = <font color="Maroon" title="c9">D</font> as   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="waybel_0.html#V1" title="WAYBEL_0:attr.1">directed</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">R</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="waybel32.html#T7" target="_self" onmouseover="rs('waybel32/T7')" onmouseout="rh()">Th7</a></span>;<br/>
<a NAME="E12:16_2"/>
 <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c10">D</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">S</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:16"><span class="lab"><font color="Green" title="E13">A15</font></span></a>, <a class="txt" href="#E1:16_2"><span class="lab"><font color="Green" title="E14">A16</font></span></a>, <a class="txt" href="#E8:16_2"><span class="lab"><font color="Green" title="E17">A19</font></span></a></span>;<br/>
<a NAME="E13:16_2"/><span class="kw">then </span>
<font color="Maroon" title="c10">D</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c4">S</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:16"><span class="lab"><font color="Green" title="E12">A14</font></span></a></span>;<br/>
<a NAME="E14:16_2"/><span class="kw">then </span>
<font color="Maroon" title="c10">D</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">S</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">d</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font><span class="kw"> such that </span><br/><a NAME="E16:16_2"/><span class="lab"><font color="Green" title="E18">A20</font></span>: 
<font color="Maroon" title="c11">d</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c10">D</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">S</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="subset_1.html#T4" onmouseover="rs('subset_1/T4')" onmouseout="rh()">SUBSET_1:4</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c12">d</font> = <font color="Maroon" title="c11">d</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font> ;<br/>
<a NAME="E18:16_2"/>
<font color="Maroon" title="c12">d</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c10">D</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:16_2"><span class="lab"><font color="Green" title="E18">A20</font></span></a>, <a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">j</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c7">N</font><span class="kw"> such that </span><br/><a NAME="E20:16_2"/><span class="lab"><font color="Green" title="E19">A21</font></span>: 
<font color="Maroon" title="c12">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K2" title="YELLOW_0:func.2">"/\"</a> (<span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c7">N</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Olive" title="b1">i</font></span>)</span> where <font color="Olive" title="b1">i</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c7">N</font> : <font color="Olive" title="b1">i</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c13">j</font> </span> } </span> ,<font color="Maroon" title="c1">R</font>)
 ;<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>2</sub></font>[  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c7">N</font>] <span class="kw">means </span>$1 <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c13">j</font>;<br/>
<span class="kw">deffunc </span><font color="Maroon">H<sub>2</sub></font>(  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c7">N</font>) <span class="kw">-&gt; </span>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font> = <font color="Maroon" title="c7">N</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> $1;<br/>
<a NAME="E21:16_2"/>
<span class="p1"> { <span class="default"> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">i</font>) where <font color="Olive" title="b1">i</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c7">N</font> : <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">i</font>] </span> } </span>  is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">R</font>
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="domain_1.html#S8" onmouseover="rs('domain_1/S8')" onmouseout="rh()">DOMAIN_1:sch 8</a>();<br/></span>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c14">Y</font> = <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c7">N</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Olive" title="b1">i</font></span>)</span> where <font color="Olive" title="b1">i</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c7">N</font> : <font color="Olive" title="b1">i</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c13">j</font> </span> } </span>  as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">R</font> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c15">j</font> = <font color="Maroon" title="c13">j</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c7">N</font> ;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c15">j</font>
; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="waybel_0.html#D11" onmouseover="rs('waybel_0/D11')" onmouseout="rh()">WAYBEL_0:def 11</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive">b<sub>1</sub></font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">N</font>  holds <br/>(  not <font color="Maroon" title="c15">j</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> or <font color="Maroon" title="c7">N</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">S</font> )</span><br/>

<span class="kw">let </span><font color="Maroon" title="c16">i0</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c7">N</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c15">j</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c16">i0</font> or <font color="Maroon" title="c7">N</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c16">i0</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">S</font> )</span><br/>

<a NAME="E24:16_2"/><span class="lab"><font color="Green" title="E20">A22</font></span>: 
<font color="Maroon" title="c12">d</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">S</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:16_2"><span class="lab"><font color="Green" title="E18">A20</font></span></a>, <a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/>
<span class="kw">assume </span><a NAME="E25:16_2"/>
<font color="Maroon" title="c15">j</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c16">i0</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">N</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c16">i0</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">S</font></span><br/>

<a NAME="E26:16_2"/><span class="kw">then </span>
<font color="Maroon" title="c7">N</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c16">i0</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c14">Y</font>
 
;<br/>
<a NAME="E27:16_2"/><span class="kw">then </span>
<font color="Maroon" title="c12">d</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c7">N</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c16">i0</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:16_2"><span class="lab"><font color="Green" title="E19">A21</font></span></a>, <a class="ref" href="waybel32.html#T8" target="_self" onmouseover="rs('waybel32/T8')" onmouseout="rh()">Th8</a></span>;<br/>
<span class="kw">hence </span><a NAME="E28:16_2"/>
<font color="Maroon" title="c7">N</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c16">i0</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">S</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:16"><span class="lab"><font color="Green" title="E13">A15</font></span></a>, <a class="txt" href="#E24:16_2"><span class="lab"><font color="Green" title="E20">A22</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E6:16"/>
<font color="Maroon" title="c4">S</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K12" title="YELLOW_6:func.12">ConvergenceSpace</a> <span class="p2">(<span class="default"><a href="waybel11.html#K2" title="WAYBEL11:func.2">Scott-Convergence</a> <font color="Maroon" title="c1">R</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:16"><span class="lab"><font color="Green" title="E11">A1</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
