<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">L</font> be  <a href="lattice3.html#V3">complete</a> <a href="waybel_0.html#NM8">LATTICE</a>;<br/>

<b>assume </b><a NAME="E1:59"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">S<sub>8</sub></font>[<font color="Maroon">L</font>]
 ;<br/>

<b>consider </b><font color="Maroon">SL</font> being  <a href="waybel11.html#V4">Scott</a>  <a href="yellow_9.html#M1">TopAugmentation</a> of <font color="Maroon">L</font>;<br/>
<a NAME="E3:59"/><i><font color="Green">E47</font></i>: 
 <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">SL</font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">SL</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">L</font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">L</font> #)
 
<b>by </b><i><a class="ref" href="yellow_9.html#D4">YELLOW_9:def 4</a></i>;<br/>
<a NAME="E4:59"/>
 <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">SL</font>,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">SL</font> #) <a href="hidden.html#R1">=</a>  <a href="yellow_6.html#K14">ConvergenceSpace</a> <span class="p1">(<span class="default"><a href="waybel11.html#K2">Scott-Convergence</a> <font color="Maroon">SL</font></span>)</span>
 
<b>by </b><i><a class="ref" href="waybel11.html#T32">WAYBEL11:32</a></i>;<br/>
<a NAME="E5:59"/><b>then </b><i><font color="Green">E48</font></i>: 
the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">SL</font> <a href="hidden.html#R1">=</a>  <a href="waybel11.html#K5">sigma</a> <font color="Maroon">SL</font>
 
;<br/>
<a NAME="E6:59"/><b>then </b><i><font color="Green">E50</font></i>: 
 <a href="yellow_1.html#K2">InclPoset</a> <span class="p1">(<span class="default"><a href="waybel11.html#K5">sigma</a> <font color="Maroon">L</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="yellow_1.html#K2">InclPoset</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">SL</font>
 
<b>by </b><i>, <a class="ref" href="yellow_9.html#T52">YELLOW_9:52</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">S</font> =  <a href="yellow_1.html#K2">InclPoset</a> <span class="p1">(<span class="default"><a href="waybel11.html#K5">sigma</a> <font color="Maroon">L</font></span>)</span> as  <a href="lattice3.html#V3">complete</a> <a href="waybel_0.html#NM8">LATTICE</a> ;<br/>
<b>consider </b><font color="Maroon">SS</font> being  <a href="waybel11.html#V4">Scott</a>  <a href="yellow_9.html#M1">TopAugmentation</a> of <font color="Maroon">S</font>;<br/>
<a NAME="E9:59"/><i><font color="Green">E53</font></i>: 
 <a href="waybel11.html#K5">sigma</a> <span class="p1"><a href="yellow_3.html#K3">[:</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">L</font></span><a href="yellow_3.html#K3">:]</a></span> <a href="hidden.html#R1">=</a> the <a href="pre_topc.html#U1">topology</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">SS</font>,<font color="Maroon">SL</font></span><a href="borsuk_1.html#K5">:]</a></span>
 
<b>by </b><i/>;<br/>
<a NAME="E10:59"/><i><font color="Green">E54</font></i>: 
( <font color="Maroon">S<sub>5</sub></font>[<font color="Maroon">SL</font>] implies  <a href="yellow_1.html#K2">InclPoset</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">SL</font> is <a href="waybel_3.html#V3">continuous</a> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:59_1"/>
<font color="Maroon">S<sub>5</sub></font>[<font color="Maroon">SL</font>]
 ;<br/>

<a NAME="E2:59_1"/><b>then </b>
<font color="Maroon">S<sub>6</sub></font>[<font color="Maroon">SL</font>]
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:59_1"/>
 <a href="yellow_1.html#K2">InclPoset</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">SL</font> is <a href="waybel_3.html#V3">continuous</a>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E11:59"/>
<font color="Maroon">S<sub>5</sub></font>[<font color="Maroon">SL</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">T</font> be  <a href="waybel11.html#V4">Scott</a>  <a href="yellow_9.html#M1">TopAugmentation</a> of  <a href="yellow_1.html#K2">InclPoset</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">SL</font>;<br/>

<b>reconsider </b><font color="Maroon">T1</font> = <font color="Maroon">T</font> as  <a href="waybel11.html#V4">Scott</a>  <a href="yellow_9.html#M1">TopAugmentation</a> of <font color="Maroon">S</font> <b>by </b><i/>;<br/>
<b>set </b><font color="Maroon">Wy</font> = <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">W</font>,<font color="Olive">y</font></span><a href="tarski.html#K4">]</a></span> where <font color="Olive">W</font> is  <a href="pre_topc.html#V3">open</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">SL</font>, <font color="Olive">y</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">SL</font> : <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Olive">W</font> </span>}</span> ;<br/>
<a NAME="E2:59_2"/>
<span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">W</font>,<font color="Olive">y</font></span><a href="tarski.html#K4">]</a></span> where <font color="Olive">W</font> is  <a href="pre_topc.html#V3">open</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">SL</font>, <font color="Olive">y</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">SL</font> : <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Olive">W</font> </span>}</span>  <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">T</font>,<font color="Maroon">SL</font></span><a href="borsuk_1.html#K5">:]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:59_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">W</font>,<font color="Olive">y</font></span><a href="tarski.html#K4">]</a></span> where <font color="Olive">W</font> is  <a href="pre_topc.html#V3">open</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">SL</font>, <font color="Olive">y</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">SL</font> : <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Olive">W</font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">W</font> being  <a href="pre_topc.html#V3">open</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">SL</font>, <font color="Maroon">y</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">SL</font><b> such that </b><br/><a NAME="E3:59_2_1"/><i><font color="Green">E72</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">W</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font> )
 ;<br/>
<a NAME="E4:59_2_1"/>
<font color="Maroon">W</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">SL</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/>
<a NAME="E5:59_2_1"/><b>then </b><i><font color="Green">E74</font></i>: 
<font color="Maroon">W</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">SL</font></span>)</span>
 
<b>by </b><i><a class="ref" href="yellow_1.html#T1">YELLOW_1:1</a></i>;<br/>
<a NAME="E6:59_2_1"/>
 <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">T</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 <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">SL</font></span>)</span>,the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">SL</font></span>)</span> #)
 
<b>by </b><i><a class="ref" href="yellow_9.html#D4">YELLOW_9:def 4</a></i>;<br/>
<a NAME="E7:59_2_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">W</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">SL</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<b>hence </b><a NAME="E8:59_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">T</font>,<font color="Maroon">SL</font></span><a href="borsuk_1.html#K5">:]</a></span>
 <b>by </b><i><a class="ref" href="waybel29.html#T1" target="_self">Th1</a>, <a class="ref" href="borsuk_1.html#D5">BORSUK_1:def 5</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">WY</font> = <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">W</font>,<font color="Olive">y</font></span><a href="tarski.html#K4">]</a></span> where <font color="Olive">W</font> is  <a href="pre_topc.html#V3">open</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">SL</font>, <font color="Olive">y</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">SL</font> : <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Olive">W</font> </span>}</span>  as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">T</font>,<font color="Maroon">SL</font></span><a href="borsuk_1.html#K5">:]</a></span> ;<br/>
<i><font color="Green">E75</font></i>:  <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">SS</font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">SS</font> #) = 
 <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">SL</font></span>)</span>,the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">SL</font></span>)</span> #)
<b>by </b><i>, <a class="ref" href="yellow_9.html#D4">YELLOW_9:def 4</a></i>
<br/>.= 
 <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">T</font> #)
<b>by </b><i><a class="ref" href="yellow_9.html#D4">YELLOW_9:def 4</a></i>
;<br/>
the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">SS</font> = 
 <a href="waybel11.html#K5">sigma</a> <font color="Maroon">S</font>
<b>by </b><i><a class="ref" href="yellow_9.html#T51">YELLOW_9:51</a></i>
<br/>.= 
the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T1</font>
<b>by </b><i><a class="ref" href="yellow_9.html#T51">YELLOW_9:51</a></i>
<br/>.= 
the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font>

;<br/>
<a NAME="E6:59_2"/><b>then </b><i><font color="Green">E76</font></i>: 
 <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">SS</font>,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">SS</font> #) <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font> #)
 
<b>by </b><i><a class="ref" href="waybel29.html#T1" target="_self">Th1</a></i>;<br/>
<a NAME="E7:59_2"/>
 <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">SL</font>,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">SL</font> #) <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">SL</font>,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">SL</font> #)
 
;<br/>
<a NAME="E8:59_2"/><b>then </b><i><font color="Green">E77</font></i>: 
<span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">SS</font>,<font color="Maroon">SL</font></span><a href="borsuk_1.html#K5">:]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">T</font>,<font color="Maroon">SL</font></span><a href="borsuk_1.html#K5">:]</a></span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E9:59_2"/><i><font color="Green">E78</font></i>: 
 <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">SS</font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">SS</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">S</font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">S</font> #)
 
<b>by </b><i><a class="ref" href="yellow_9.html#D4">YELLOW_9:def 4</a></i>;<br/>
the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">T</font>,<font color="Maroon">SL</font></span><a href="borsuk_1.html#K5">:]</a></span> = 
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">SL</font></span><a href="zfmisc_1.html#K2">:]</a></span>
<b>by </b><i><a class="ref" href="borsuk_1.html#D5">BORSUK_1:def 5</a></i>
<br/>.= 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="yellow_3.html#K3">[:</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">L</font></span><a href="yellow_3.html#K3">:]</a></span>
<b>by </b><i>, <a class="ref" href="waybel29.html#T1" target="_self">Th1</a>, , <a class="ref" href="yellow_3.html#D2">YELLOW_3:def 2</a></i>
;<br/>
<b>then reconsider </b><font color="Maroon">wy</font> = <font color="Maroon">WY</font> as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1"><a href="yellow_3.html#K3">[:</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">L</font></span><a href="yellow_3.html#K3">:]</a></span> ;<br/>
<a NAME="E12:59_2"/><i><font color="Green">E79</font></i>: 
<font color="Maroon">wy</font> is <a href="waybel_0.html#V13">upper</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_2_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1"><a href="yellow_3.html#K3">[:</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">L</font></span><a href="yellow_3.html#K3">:]</a></span>, <font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1"><a href="yellow_3.html#K3">[:</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">L</font></span><a href="yellow_3.html#K3">:]</a></span>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_0.html#D20">WAYBEL_0:def 20</a><br/>

<b>assume </b><a NAME="E1:59_2_5"/><i><font color="Green">E80</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">wy</font> &amp; <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">y</font> )
 ;<br/>

<a NAME="E2:59_2_5"/>
the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="yellow_3.html#K3">[:</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">L</font></span><a href="yellow_3.html#K3">:]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="yellow_3.html#D2">YELLOW_3:def 2</a></i>;<br/>
<b>then consider </b><font color="Maroon">y1</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">y2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:59_2_5"/><i><font color="Green">E81</font></i>: 
( <font color="Maroon">y1</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> &amp; <font color="Maroon">y2</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y1</font>,<font color="Maroon">y2</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/>
<a NAME="E5:59_2_5"/>
<font color="Maroon">y1</font> <a href="hidden.html#R2">in</a>  <a href="waybel11.html#K5">sigma</a> <font color="Maroon">L</font>
 
<b>by </b><i>, <a class="ref" href="yellow_1.html#T1">YELLOW_1:1</a></i>;<br/>
<a NAME="E6:59_2_5"/><b>then </b>
<font color="Maroon">y1</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">SL</font>
 
<b>by </b><i>, , <a class="ref" href="yellow_9.html#T52">YELLOW_9:52</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">y1</font> = <font color="Maroon">y1</font> as  <a href="pre_topc.html#V3">open</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">SL</font> <b>by </b><i><a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">y2</font> = <font color="Maroon">y2</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">SL</font> <b>by </b><i>, </i>;<br/>
<b>consider </b><font color="Maroon">x1</font> being  <a href="pre_topc.html#V3">open</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">SL</font>, <font color="Maroon">x2</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">SL</font><b> such that </b><br/><a NAME="E10:59_2_5"/><i><font color="Green">E82</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x1</font> )
 <b>by </b><i/>;<br/>
<a NAME="E11:59_2_5"/>
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">SL</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/>
<a NAME="E12:59_2_5"/><b>then </b>
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="waybel11.html#K5">sigma</a> <font color="Maroon">L</font>
 
<b>by </b><i>, , <a class="ref" href="yellow_9.html#T52">YELLOW_9:52</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">u1</font> = <font color="Maroon">x1</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font> <b>by </b><i><a class="ref" href="yellow_1.html#T1">YELLOW_1:1</a></i>;<br/>
<a NAME="E14:59_2_5"/>
<font color="Maroon">y1</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">SL</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/>
<a NAME="E15:59_2_5"/><b>then </b>
<font color="Maroon">y1</font> <a href="hidden.html#R2">in</a>  <a href="waybel11.html#K5">sigma</a> <font color="Maroon">L</font>
 
<b>by </b><i>, , <a class="ref" href="yellow_9.html#T52">YELLOW_9:52</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">v1</font> = <font color="Maroon">y1</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font> <b>by </b><i><a class="ref" href="yellow_1.html#T1">YELLOW_1:1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">u2</font> = <font color="Maroon">x2</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">v2</font> = <font color="Maroon">y2</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> <b>by </b><i/>;<br/>
<a NAME="E19:59_2_5"/>
<span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">u1</font>,<font color="Maroon">u2</font></span><a href="yellow_3.html#K7">]</a></span> <a href="orders_2.html#R3">&lt;=</a> <span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">v1</font>,<font color="Maroon">v2</font></span><a href="yellow_3.html#K7">]</a></span>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E20:59_2_5"/><b>then </b><i><font color="Green">E119</font></i>: 
( <font color="Maroon">u1</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">v1</font> &amp; <font color="Maroon">u2</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">v2</font> )
 
<b>by </b><i><a class="ref" href="yellow_3.html#T11">YELLOW_3:11</a></i>;<br/>
<a NAME="E21:59_2_5"/><b>then </b><i><font color="Green">E121</font></i>: 
<font color="Maroon">x2</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">y2</font>
 
<b>by </b><i>, <a class="ref" href="yellow_0.html#T1">YELLOW_0:1</a></i>;<br/>
<a NAME="E22:59_2_5"/><i><font color="Green">E122</font></i>: 
<font color="Maroon">x1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">y1</font>
 
<b>by </b><i><a class="txt" href="waybel29.html#E1:59"><i><font color="Green">E46</font></i></a>, <a class="ref" href="yellow_1.html#T3">YELLOW_1:3</a></i>;<br/>
<a NAME="E23:59_2_5"/>
<font color="Maroon">y2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x1</font>
 
<b>by </b><i>, <a class="txt" href="waybel29.html#E3:59"><i><font color="Green">E47</font></i></a>, <a class="ref" href="waybel_0.html#D20">WAYBEL_0:def 20</a></i>;<br/>
<b>hence </b><a NAME="E24:59_2_5"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">wy</font>
 <b>by </b><i>, <a class="txt" href="waybel29.html#E5:59"><i><font color="Green">E48</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E13:59_2"/>
<font color="Maroon">wy</font> is <a href="waybel11.html#V1">inaccessible_by_directed_joins</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_2_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">D</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="waybel_0.html#V1">directed</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1"><a href="yellow_3.html#K3">[:</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">L</font></span><a href="yellow_3.html#K3">:]</a></span>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel11.html#D1">WAYBEL11:def 1</a><br/>

<b>assume </b><a NAME="E1:59_2_6"/>
 <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">D</font> <a href="hidden.html#R2">in</a> <font color="Maroon">wy</font>
 ;<br/>

<a NAME="E2:59_2_6"/><b>then </b>
<span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p3">(<span class="default"><a href="yellow_3.html#K4">proj1</a> <font color="Maroon">D</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p3">(<span class="default"><a href="yellow_3.html#K5">proj2</a> <font color="Maroon">D</font></span>)</span></span>)</span></span><a href="yellow_3.html#K7">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">wy</font>
 
<b>by </b><i><a class="ref" href="yellow_3.html#T46">YELLOW_3:46</a></i>;<br/>
<b>then consider </b><font color="Maroon">sp1D</font> being  <a href="pre_topc.html#V3">open</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">SL</font>, <font color="Maroon">sp2D</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">SL</font><b> such that </b><br/><a NAME="E4:59_2_6"/><i><font color="Green">E123</font></i>: 
( <span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p3">(<span class="default"><a href="yellow_3.html#K4">proj1</a> <font color="Maroon">D</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p3">(<span class="default"><a href="yellow_3.html#K5">proj2</a> <font color="Maroon">D</font></span>)</span></span>)</span></span><a href="yellow_3.html#K7">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">sp1D</font>,<font color="Maroon">sp2D</font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">sp2D</font> <a href="hidden.html#R2">in</a> <font color="Maroon">sp1D</font> )
 ;<br/>
<b>reconsider </b><font color="Maroon">proj2D</font> =  <a href="yellow_3.html#K5">proj2</a> <font color="Maroon">D</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="waybel_0.html#V1">directed</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font> <b>by </b><i><a class="ref" href="yellow_3.html#T21">YELLOW_3:21</a>, <a class="ref" href="yellow_3.html#T22">YELLOW_3:22</a></i>;<br/>
<b>reconsider </b><font color="Maroon">sp1D'</font> = <font color="Maroon">sp1D</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font> <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">sp1D'</font> = <font color="Maroon">sp1D'</font> as  <a href="waybel_0.html#V13">upper</a> <a href="waybel11.html#V1">inaccessible</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font> <b>by </b><i>, <a class="ref" href="waybel_0.html#T25">WAYBEL_0:25</a>, <a class="ref" href="yellow_9.html#T47">YELLOW_9:47</a></i>;<br/>
<a NAME="E8:59_2_6"/>
 <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">proj2D</font> <a href="hidden.html#R2">in</a> <font color="Maroon">sp1D'</font>
 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>
<a NAME="E9:59_2_6"/><b>then </b>
 <a href="yellow_3.html#K5">proj2</a> <font color="Maroon">D</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">sp1D</font>
 
<b>by </b><i><a class="ref" href="waybel11.html#D1">WAYBEL11:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon">d</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E11:59_2_6"/><i><font color="Green">E124</font></i>: 
( <font color="Maroon">d</font> <a href="hidden.html#R2">in</a>  <a href="yellow_3.html#K5">proj2</a> <font color="Maroon">D</font> &amp; <font color="Maroon">d</font> <a href="hidden.html#R2">in</a> <font color="Maroon">sp1D</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">d</font> = <font color="Maroon">d</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">Y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E14:59_2_6"/><i><font color="Green">E125</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">Y</font>,<font color="Maroon">d</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font>
 <b>by </b><i>, <a class="ref" href="funct_5.html#D2">FUNCT_5:def 2</a></i>;<br/>
<a NAME="E15:59_2_6"/>
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a>  <a href="yellow_3.html#K4">proj1</a> <font color="Maroon">D</font>
 
<b>by </b><i>, <a class="ref" href="funct_5.html#D1">FUNCT_5:def 1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">Y</font> = <font color="Maroon">Y</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font> ;<br/>
<b>reconsider </b><font color="Maroon">pD</font> =  <a href="yellow_3.html#K4">proj1</a> <font color="Maroon">D</font> as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">SL</font></span>)</span> <b>by </b><i>, , <a class="ref" href="yellow_9.html#T52">YELLOW_9:52</a></i>;<br/>
<a NAME="E18:59_2_6"/><i><font color="Green">E126</font></i>: 
 <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><a href="yellow_3.html#K4">proj1</a> <font color="Maroon">D</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">pD</font>
 
<b>by </b><i>, <a class="ref" href="yellow_1.html#T22">YELLOW_1:22</a></i>;<br/>
<a NAME="E19:59_2_6"/>
<font color="Maroon">d</font> <a href="hidden.html#R2">in</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><a href="yellow_3.html#K4">proj1</a> <font color="Maroon">D</font></span>)</span>
 
<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">V</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E21:59_2_6"/><i><font color="Green">E127</font></i>: 
( <font color="Maroon">d</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font> &amp; <font color="Maroon">V</font> <a href="hidden.html#R2">in</a>  <a href="yellow_3.html#K4">proj1</a> <font color="Maroon">D</font> )
 <b>by </b><i><a class="txt" href="waybel29.html#E1:59"><i><font color="Green">E46</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>reconsider </b><font color="Maroon">V</font> = <font color="Maroon">V</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font> <b>by </b><i><a class="txt" href="waybel29.html#E3:59"><i><font color="Green">E47</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">e</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E24:59_2_6"/><i><font color="Green">E129</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">V</font>,<font color="Maroon">e</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font>
 <b>by </b><i><a class="txt" href="waybel29.html#E3:59"><i><font color="Green">E47</font></i></a>, <a class="ref" href="funct_5.html#D1">FUNCT_5:def 1</a></i>;<br/>
<a NAME="E25:59_2_6"/>
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a>  <a href="yellow_3.html#K5">proj2</a> <font color="Maroon">D</font>
 
<b>by </b><i><a class="txt" href="waybel29.html#E5:59"><i><font color="Green">E48</font></i></a>, <a class="ref" href="funct_5.html#D2">FUNCT_5:def 2</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">e</font> = <font color="Maroon">e</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> ;<br/>
<b>consider </b><font color="Maroon">DD</font> being   <a href="struct_0.html#NM1">Element</a> of <span class="p1"><a href="yellow_3.html#K3">[:</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">L</font></span><a href="yellow_3.html#K3">:]</a></span><b> such that </b><br/><a NAME="E28:59_2_6"/><i><font color="Green">E130</font></i>: 
( <font color="Maroon">DD</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> &amp; <span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">V</font>,<font color="Maroon">e</font></span><a href="yellow_3.html#K7">]</a></span> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">DD</font> &amp; <span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">Y</font>,<font color="Maroon">d</font></span><a href="yellow_3.html#K7">]</a></span> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">DD</font> )
 <b>by </b><i>, <a class="txt" href="waybel29.html#E5:59"><i><font color="Green">E48</font></i></a>, <a class="ref" href="waybel_0.html#D1">WAYBEL_0:def 1</a></i>;<br/>
<a NAME="E29:59_2_6"/>
<font color="Maroon">D</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="yellow_3.html#K3">[:</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">L</font></span><a href="yellow_3.html#K3">:]</a></span>
 
;<br/>
<a NAME="E30:59_2_6"/><b>then </b>
<font color="Maroon">D</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="yellow_3.html#D2">YELLOW_3:def 2</a></i>;<br/>
<b>then consider </b><font color="Maroon">DD1</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">DD2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E32:59_2_6"/><i><font color="Green">E131</font></i>: 
<font color="Maroon">DD</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">DD1</font>,<font color="Maroon">DD2</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#T102">ZFMISC_1:102</a></i>;<br/>
<a NAME="E33:59_2_6"/>
<font color="Maroon">DD1</font> <a href="hidden.html#R2">in</a>  <a href="yellow_3.html#K4">proj1</a> <font color="Maroon">D</font>
 
<b>by </b><i>, <a class="txt" href="waybel29.html#E6:59"><i><font color="Green">E50</font></i></a>, <a class="ref" href="funct_5.html#D1">FUNCT_5:def 1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">DD1</font> = <font color="Maroon">DD1</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font> ;<br/>
<a NAME="E35:59_2_6"/>
<font color="Maroon">DD2</font> <a href="hidden.html#R2">in</a>  <a href="yellow_3.html#K5">proj2</a> <font color="Maroon">D</font>
 
<b>by </b><i>, <a class="txt" href="waybel29.html#E6:59"><i><font color="Green">E50</font></i></a>, <a class="ref" href="funct_5.html#D2">FUNCT_5:def 2</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">DD2</font> = <font color="Maroon">DD2</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> ;<br/>
<a NAME="E37:59_2_6"/>
( <span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">V</font>,<font color="Maroon">e</font></span><a href="yellow_3.html#K7">]</a></span> <a href="orders_2.html#R3">&lt;=</a> <span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">DD1</font>,<font color="Maroon">DD2</font></span><a href="yellow_3.html#K7">]</a></span> &amp; <span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">Y</font>,<font color="Maroon">d</font></span><a href="yellow_3.html#K7">]</a></span> <a href="orders_2.html#R3">&lt;=</a> <span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">DD1</font>,<font color="Maroon">DD2</font></span><a href="yellow_3.html#K7">]</a></span> )
 
<b>by </b><i>, <a class="txt" href="waybel29.html#E6:59"><i><font color="Green">E50</font></i></a></i>;<br/>
<a NAME="E38:59_2_6"/><b>then </b>
( <font color="Maroon">V</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">DD1</font> &amp; <font color="Maroon">Y</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">DD1</font> )
 
<b>by </b><i><a class="ref" href="yellow_3.html#T11">YELLOW_3:11</a></i>;<br/>
<a NAME="E39:59_2_6"/><b>then </b><i><font color="Green">E132</font></i>: 
( <font color="Maroon">V</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">DD1</font> &amp; <font color="Maroon">Y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">DD1</font> )
 
<b>by </b><i><a class="ref" href="yellow_1.html#T3">YELLOW_1:3</a></i>;<br/>
<a NAME="E40:59_2_6"/>
<font color="Maroon">DD1</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>
 
;<br/>
<a NAME="E41:59_2_6"/><b>then </b>
<font color="Maroon">DD1</font> <a href="hidden.html#R2">in</a>  <a href="waybel11.html#K5">sigma</a> <font color="Maroon">L</font>
 
<b>by </b><i><a class="ref" href="yellow_1.html#T1">YELLOW_1:1</a></i>;<br/>
<a NAME="E42:59_2_6"/><b>then </b>
<font color="Maroon">DD1</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">SL</font>
 
<b>by </b><i>, , <a class="ref" href="yellow_9.html#T52">YELLOW_9:52</a></i>;<br/>
<a NAME="E43:59_2_6"/><b>then </b>
<font color="Maroon">DD1</font> is  <a href="pre_topc.html#V3">open</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">SL</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/>
<a NAME="E44:59_2_6"/><b>then </b><i><font color="Green">E133</font></i>: 
<font color="Maroon">DD1</font> is  <a href="waybel_0.html#V13">upper</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font>
 
<b>by </b><i>, <a class="ref" href="waybel_0.html#T25">WAYBEL_0:25</a></i>;<br/>
<a NAME="E45:59_2_6"/>
( <span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">V</font>,<font color="Maroon">e</font></span><a href="yellow_3.html#K7">]</a></span> <a href="orders_2.html#R3">&lt;=</a> <span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">DD1</font>,<font color="Maroon">DD2</font></span><a href="yellow_3.html#K7">]</a></span> &amp; <span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">Y</font>,<font color="Maroon">d</font></span><a href="yellow_3.html#K7">]</a></span> <a href="orders_2.html#R3">&lt;=</a> <span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">DD1</font>,<font color="Maroon">DD2</font></span><a href="yellow_3.html#K7">]</a></span> )
 
<b>by </b><i>, <a class="txt" href="waybel29.html#E6:59"><i><font color="Green">E50</font></i></a></i>;<br/>
<a NAME="E46:59_2_6"/><b>then </b>
( <font color="Maroon">e</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">DD2</font> &amp; <font color="Maroon">d</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">DD2</font> )
 
<b>by </b><i><a class="ref" href="yellow_3.html#T11">YELLOW_3:11</a></i>;<br/>
<a NAME="E47:59_2_6"/><b>then </b><i><font color="Green">E134</font></i>: 
<font color="Maroon">DD2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">DD1</font>
 
<b>by </b><i><a class="txt" href="waybel29.html#E3:59"><i><font color="Green">E47</font></i></a>, , , <a class="ref" href="waybel_0.html#D20">WAYBEL_0:def 20</a></i>;<br/>
<a NAME="E48:59_2_6"/>
<font color="Maroon">DD1</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>
 
;<br/>
<a NAME="E49:59_2_6"/><b>then </b>
<font color="Maroon">DD1</font> <a href="hidden.html#R2">in</a>  <a href="waybel11.html#K5">sigma</a> <font color="Maroon">L</font>
 
<b>by </b><i><a class="ref" href="yellow_1.html#T1">YELLOW_1:1</a></i>;<br/>
<a NAME="E50:59_2_6"/><b>then </b>
<font color="Maroon">DD1</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">SL</font>
 
<b>by </b><i>, , <a class="ref" href="yellow_9.html#T52">YELLOW_9:52</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">DD1</font> = <font color="Maroon">DD1</font> as  <a href="pre_topc.html#V3">open</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">SL</font> <b>by </b><i><a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">DD2</font> = <font color="Maroon">DD2</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">SL</font> <b>by </b><i/>;<br/>
<a NAME="E53:59_2_6"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">DD1</font>,<font color="Maroon">DD2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">wy</font>
 
<b>by </b><i><a class="txt" href="waybel29.html#E9:59"><i><font color="Green">E53</font></i></a></i>;<br/>
<b>hence </b><a NAME="E54:59_2_6"/>
<font color="Maroon">D</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">wy</font>
 <b>by </b><i>, <a class="txt" href="waybel29.html#E6:59"><i><font color="Green">E50</font></i></a>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E14:59_2"/><b>then </b>
<font color="Maroon">wy</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K14">ConvergenceSpace</a> <span class="p2">(<span class="default"><a href="waybel11.html#K2">Scott-Convergence</a> <span class="p3"><a href="yellow_3.html#K3">[:</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">L</font></span><a href="yellow_3.html#K3">:]</a></span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="waybel29.html#T2" target="_self">Th2</a>, <a class="ref" href="waybel11.html#T31">WAYBEL11:31</a></i>;<br/>
<a NAME="E15:59_2"/><b>then </b>
<font color="Maroon">WY</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">SS</font>,<font color="Maroon">SL</font></span><a href="borsuk_1.html#K5">:]</a></span>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E16:59_2"/>
<span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">W</font>,<font color="Olive">y</font></span><a href="tarski.html#K4">]</a></span> where <font color="Olive">W</font> is  <a href="pre_topc.html#V3">open</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">SL</font>, <font color="Olive">y</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">SL</font> : <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Olive">W</font> </span>}</span>  is  <a href="pre_topc.html#V3">open</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">T</font>,<font color="Maroon">SL</font></span><a href="borsuk_1.html#K5">:]</a></span>
 <b>by </b><i>, <a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E12:59"/>
 <a href="yellow_1.html#K2">InclPoset</a> <span class="p1">(<span class="default"><a href="waybel11.html#K5">sigma</a> <font color="Maroon">L</font></span>)</span> is <a href="waybel_3.html#V3">continuous</a>
 <b>by </b><i>, , , <a class="ref" href="yellow_9.html#T52">YELLOW_9:52</a></i>;<br/>


</div>
