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<b>let </b><font color="Maroon">L</font> be  <a href="lattice3.html#V3">complete</a> <a href="waybel11.html#V4">Scott</a> <a href="waybel_9.html#NM1">TopLattice</a>;<br/>

<b>set </b><font color="Maroon">IPs</font> =  <a href="yellow_1.html#K2">InclPoset</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">L</font>;<br/>
<a NAME="E1:45"/><i><font color="Green">E36</font></i>: 
 <a href="waybel11.html#K5">sigma</a> <font color="Maroon">L</font> <a href="hidden.html#R1">=</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">L</font>
 
<b>by </b><i/>;<br/>
<a NAME="E2:45"/><b>then </b><i><font color="Green">E37</font></i>: 
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">L</font></span>)</span> <a href="hidden.html#R1">=</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/>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:45_1"/><i><font color="Green">E38</font></i>: 
for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>  ex <font color="Olive">X</font> being   <a href="yellow_8.html#M1">Basis</a> of <font color="Olive">x</font> st <br/>for <font color="Olive">Y</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font>  st <font color="Olive">Y</font> <a href="hidden.html#R2">in</a> <font color="Olive">X</font> holds <br/>( <font color="Olive">Y</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Olive">Y</font> is <a href="waybel_0.html#V2">filtered</a> )
 ;<br/><b>let </b><font color="Maroon">V</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel11.html#K5">sigma</a> <font color="Maroon">L</font></span>)</span></span>)</span>;<br/><a NAME="E2:45_1"/>
<font color="Maroon">V</font> <a href="hidden.html#R2">in</a>  <a href="waybel11.html#K5">sigma</a> <font color="Maroon">L</font>
 <b>by </b><i>, </i>;<br/><b>then reconsider </b><font color="Maroon">Vl</font> = <font color="Maroon">V</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font> ;<br/><a NAME="E4:45_1"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">Vl</font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i>, , </i>;<br/><b>set </b><font color="Maroon">X</font> = <span class="p1">{<span class="default"> <font color="Olive">Y</font> where <font color="Olive">Y</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font> : ( <font color="Olive">Y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">V</font> &amp;  ex <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> ex <font color="Olive">bas</font> being   <a href="yellow_8.html#M1">Basis</a> of <font color="Olive">x</font> st <br/>( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font> &amp; <font color="Olive">Y</font> <a href="hidden.html#R2">in</a> <font color="Olive">bas</font> &amp; ( for <font color="Olive">Yx</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font>  st <font color="Olive">Yx</font> <a href="hidden.html#R2">in</a> <font color="Olive">bas</font> holds <br/>( <font color="Olive">Yx</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Olive">Yx</font> is <a href="waybel_0.html#V2">filtered</a> ) ) ) ) </span>}</span> ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">YY</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:45_1_1"/>
<font color="Maroon">YY</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">Y</font> where <font color="Olive">Y</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font> : ( <font color="Olive">Y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">V</font> &amp;  ex <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> ex <font color="Olive">bas</font> being   <a href="yellow_8.html#M1">Basis</a> of <font color="Olive">x</font> st <br/>( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font> &amp; <font color="Olive">Y</font> <a href="hidden.html#R2">in</a> <font color="Olive">bas</font> &amp; ( for <font color="Olive">Yx</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font>  st <font color="Olive">Yx</font> <a href="hidden.html#R2">in</a> <font color="Olive">bas</font> holds <br/>( <font color="Olive">Yx</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Olive">Yx</font> is <a href="waybel_0.html#V2">filtered</a> ) ) ) ) </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon">Y</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E3:45_1_1"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">Y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">YY</font>
 <b>and </b><br/><a NAME="E4:45_1_1"/>
<font color="Maroon">Y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">V</font>
 <b>and </b><br/><a NAME="E5:45_1_1"/><i><font color="Green">E42</font></i>: 
 ex <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> ex <font color="Olive">bas</font> being   <a href="yellow_8.html#M1">Basis</a> of <font color="Olive">x</font> st <br/>( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font> &amp; <font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Olive">bas</font> &amp; ( for <font color="Olive">Yx</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font>  st <font color="Olive">Yx</font> <a href="hidden.html#R2">in</a> <font color="Olive">bas</font> holds <br/>( <font color="Olive">Yx</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Olive">Yx</font> is <a href="waybel_0.html#V2">filtered</a> ) ) )
 ;<br/><b>consider </b><font color="Maroon">x</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>, <font color="Maroon">bas</font> being    <a href="yellow_8.html#M1">Basis</a> of <font color="Maroon">x</font><b> such that </b><br/><a NAME="E7:45_1_1"/><i><font color="Green">E44</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font> &amp; <font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">bas</font> )
 <b>by </b><i><a class="txt" href="waybel14.html#E1:45"><i><font color="Green">E36</font></i></a></i>;<br/><a NAME="E8:45_1_1"/>
<font color="Maroon">Y</font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i><a class="txt" href="waybel14.html#E2:45"><i><font color="Green">E37</font></i></a>, <a class="ref" href="yellow_8.html#T21">YELLOW_8:21</a></i>;<br/><a NAME="E9:45_1_1"/><b>then </b>
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a>  <a href="waybel11.html#K5">sigma</a> <font color="Maroon">L</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E10:45_1_1"/>
<font color="Maroon">YY</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> <span class="p2">(<span class="default"><a href="waybel11.html#K5">sigma</a> <font color="Maroon">L</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="yellow_1.html#T1">YELLOW_1:1</a></i>;<br/></div><b>end;</b></div><b>then reconsider </b><font color="Maroon">X</font> = <span class="p1">{<span class="default"> <font color="Olive">Y</font> where <font color="Olive">Y</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font> : ( <font color="Olive">Y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">V</font> &amp;  ex <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> ex <font color="Olive">bas</font> being   <a href="yellow_8.html#M1">Basis</a> of <font color="Olive">x</font> st <br/>( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font> &amp; <font color="Olive">Y</font> <a href="hidden.html#R2">in</a> <font color="Olive">bas</font> &amp; ( for <font color="Olive">Yx</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font>  st <font color="Olive">Yx</font> <a href="hidden.html#R2">in</a> <font color="Olive">bas</font> holds <br/>( <font color="Olive">Yx</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Olive">Yx</font> is <a href="waybel_0.html#V2">filtered</a> ) ) ) ) </span>}</span>  as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel11.html#K5">sigma</a> <font color="Maroon">L</font></span>)</span></span>)</span> <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><b>take </b><font color="Maroon">X</font> = <font color="Maroon">X</font>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:45_1_2_1"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font>
 ;<br/><a NAME="E2:45_1_2_1"/>
<font color="Maroon">Vl</font> <a href="hidden.html#R1">=</a> <font color="Maroon">V</font>
 ;<br/><b>then reconsider </b><font color="Maroon">d</font> = <font color="Maroon">x</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">bas</font> being    <a href="yellow_8.html#M1">Basis</a> of <font color="Maroon">d</font><b> such that </b><br/><a NAME="E5:45_1_2_1"/><i><font color="Green">E46</font></i>: 
for <font color="Olive">Y</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font>  st <font color="Olive">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">bas</font> holds <br/>( <font color="Olive">Y</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Olive">Y</font> is <a href="waybel_0.html#V2">filtered</a> )
 <b>by </b><i/>;<br/><b>consider </b><font color="Maroon">Y</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E7:45_1_2_1"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">bas</font> &amp; <font color="Maroon">Y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">V</font> )
 <b>by </b><i>, , <a class="ref" href="yellow_8.html#T22">YELLOW_8:22</a></i>;<br/><a NAME="E8:45_1_2_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> )
 <b>by </b><i>, <a class="txt" href="waybel14.html#E1:45"><i><font color="Green">E36</font></i></a>, <a class="txt" href="waybel14.html#E2:45"><i><font color="Green">E37</font></i></a>, <a class="ref" href="yellow_8.html#T21">YELLOW_8:21</a></i>;<br/><b>hence </b><a NAME="E9:45_1_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/></div>
<b>end;</b></div><b>assume </b><a NAME="E2:45_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">X</font>
 ;<br/><b>then consider </b><font color="Maroon">YY</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:45_1_2"/><i><font color="Green">E48</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">YY</font> &amp; <font color="Maroon">YY</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><b>consider </b><font color="Maroon">Y</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E6:45_1_2"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">Y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">YY</font>
 <b>and </b><br/><a NAME="E7:45_1_2"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">Y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">V</font>
 <b>and </b><br/><a NAME="E8:45_1_2"/>
 ex <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> ex <font color="Olive">bas</font> being   <a href="yellow_8.html#M1">Basis</a> of <font color="Olive">x</font> st <br/>( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font> &amp; <font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Olive">bas</font> &amp; ( for <font color="Olive">Yx</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font>  st <font color="Olive">Yx</font> <a href="hidden.html#R2">in</a> <font color="Olive">bas</font> holds <br/>( <font color="Olive">Yx</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Olive">Yx</font> is <a href="waybel_0.html#V2">filtered</a> ) ) )
 <b>by </b><i/>;<br/><b>thus </b><a NAME="E9:45_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font>
 <b>by </b><i>, <a class="txt" href="waybel14.html#E1:45"><i><font color="Green">E36</font></i></a>, <a class="txt" href="waybel14.html#E2:45"><i><font color="Green">E37</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E8:45_1"/><b>then </b>
<font color="Maroon">V</font> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/><b>hence </b><a NAME="E9:45_1"/>
<font color="Maroon">V</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">X</font>
 <b>by </b><i>, <a class="ref" href="yellow_1.html#T22">YELLOW_1:22</a></i>;<br/><b>let </b><font color="Maroon">Yp</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel11.html#K5">sigma</a> <font color="Maroon">L</font></span>)</span></span>)</span>;<br/><b>assume </b><a NAME="E10:45_1"/>
<font color="Maroon">Yp</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/><b>then consider </b><font color="Maroon">Y</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E12:45_1"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">Y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Yp</font>
 <b>and </b><br/><a NAME="E13:45_1"/>
<font color="Maroon">Y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">V</font>
 <b>and </b><br/><a NAME="E14:45_1"/><i><font color="Green">E53</font></i>: 
 ex <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> ex <font color="Olive">bas</font> being   <a href="yellow_8.html#M1">Basis</a> of <font color="Olive">x</font> st <br/>( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font> &amp; <font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Olive">bas</font> &amp; ( for <font color="Olive">Yx</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font>  st <font color="Olive">Yx</font> <a href="hidden.html#R2">in</a> <font color="Olive">bas</font> holds <br/>( <font color="Olive">Yx</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Olive">Yx</font> is <a href="waybel_0.html#V2">filtered</a> ) ) )
 ;<br/><b>consider </b><font color="Maroon">x</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>, <font color="Maroon">bas</font> being    <a href="yellow_8.html#M1">Basis</a> of <font color="Maroon">x</font><b> such that </b><br/><a NAME="E16:45_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font>
 <b>and </b><br/><a NAME="E17:45_1"/><i><font color="Green">E118</font></i>: 
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">bas</font>
 <b>and </b><br/><a NAME="E18:45_1"/><i><font color="Green">E119</font></i>: 
for <font color="Olive">Yx</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font>  st <font color="Olive">Yx</font> <a href="hidden.html#R2">in</a> <font color="Maroon">bas</font> holds <br/>( <font color="Olive">Yx</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Olive">Yx</font> is <a href="waybel_0.html#V2">filtered</a> )
 <b>by </b><i><a class="txt" href="waybel14.html#E1:45"><i><font color="Green">E36</font></i></a></i>;<br/><a NAME="E19:45_1"/><i><font color="Green">E120</font></i>: 
( <font color="Maroon">Y</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">Y</font> is <a href="waybel_0.html#V2">filtered</a> )
 <b>by </b><i><a class="txt" href="waybel14.html#E2:45"><i><font color="Green">E37</font></i></a>, <a class="txt" href="waybel14.html#E1:45_1"><i><font color="Green">E38</font></i></a></i>;<br/><a NAME="E20:45_1"/><b>then </b>
<font color="Maroon">Y</font> is <a href="waybel_0.html#V13">upper</a>
 <b>by </b><i><a class="ref" href="waybel11.html#D4">WAYBEL11:def 4</a></i>;<br/><b>hence </b><a NAME="E21:45_1"/>
<font color="Maroon">Yp</font> is <a href="waybel_6.html#V6">co-prime</a>
 <b>by </b><i>, , </i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E4:45"/><i><font color="Green">E121</font></i>: 
for <font color="Olive">V</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel11.html#K5">sigma</a> <font color="Maroon">L</font></span>)</span></span>)</span>  ex <font color="Olive">X</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel11.html#K5">sigma</a> <font color="Maroon">L</font></span>)</span></span>)</span> st <br/>( <font color="Olive">V</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Olive">X</font> &amp; ( for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel11.html#K5">sigma</a> <font color="Maroon">L</font></span>)</span></span>)</span>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">X</font> holds <br/><font color="Olive">x</font> is <a href="waybel_6.html#V6">co-prime</a> ) )
 ;<br/>

<b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/>

<b>set </b><font color="Maroon">bas</font> = <span class="p1">{<span class="default"> <font color="Olive">V</font> where <font color="Olive">V</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel11.html#K5">sigma</a> <font color="Maroon">L</font></span>)</span></span>)</span> : ( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">V</font> &amp; <font color="Olive">V</font> is <a href="waybel_6.html#V6">co-prime</a> ) </span>}</span> ;<br/>
<a NAME="E5:45"/>
<span class="p1">{<span class="default"> <font color="Olive">V</font> where <font color="Olive">V</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel11.html#K5">sigma</a> <font color="Maroon">L</font></span>)</span></span>)</span> : ( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">V</font> &amp; <font color="Olive">V</font> is <a href="waybel_6.html#V6">co-prime</a> ) </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="zfmisc_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">VV</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="E1:45_2"/>
<font color="Maroon">VV</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">V</font> where <font color="Olive">V</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel11.html#K5">sigma</a> <font color="Maroon">L</font></span>)</span></span>)</span> : ( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">V</font> &amp; <font color="Olive">V</font> is <a href="waybel_6.html#V6">co-prime</a> ) </span>}</span> 
 ;<br/>

<a NAME="E2:45_2"/><b>then </b>
 ex <font color="Olive">V</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel11.html#K5">sigma</a> <font color="Maroon">L</font></span>)</span></span>)</span> st <br/>( <font color="Maroon">VV</font> <a href="hidden.html#R1">=</a> <font color="Olive">V</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">V</font> &amp; <font color="Olive">V</font> is <a href="waybel_6.html#V6">co-prime</a> )
 
;<br/>
<a NAME="E3:45_2"/><b>then </b>
<font color="Maroon">VV</font> <a href="hidden.html#R2">in</a>  <a href="waybel11.html#K5">sigma</a> <font color="Maroon">L</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E4:45_2"/>
<font color="Maroon">VV</font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">bas</font> = <span class="p1">{<span class="default"> <font color="Olive">V</font> where <font color="Olive">V</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel11.html#K5">sigma</a> <font color="Maroon">L</font></span>)</span></span>)</span> : ( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">V</font> &amp; <font color="Olive">V</font> is <a href="waybel_6.html#V6">co-prime</a> ) </span>}</span>  as   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">L</font> ;<br/>
<b>reconsider </b><font color="Maroon">bas</font> = <font color="Maroon">bas</font> as   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">L</font> ;<br/>
<a NAME="E8:45"/>
<font color="Maroon">bas</font> is    <a href="yellow_8.html#M1">Basis</a> of <font color="Maroon">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_3"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:45_3"/>
<font color="Maroon">bas</font> <a href="tarski.html#R1">c=</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">L</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="yellow_8.html#D2">YELLOW_8:def 2</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">VV</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="E1:45_3_1"/>
<font color="Maroon">VV</font> <a href="hidden.html#R2">in</a> <font color="Maroon">bas</font>
 ;<br/>

<a NAME="E2:45_3_1"/><b>then </b>
 ex <font color="Olive">V</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel11.html#K5">sigma</a> <font color="Maroon">L</font></span>)</span></span>)</span> st <br/>( <font color="Maroon">VV</font> <a href="hidden.html#R1">=</a> <font color="Olive">V</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">V</font> &amp; <font color="Olive">V</font> is <a href="waybel_6.html#V6">co-prime</a> )
 
;<br/>
<b>hence </b><a NAME="E3:45_3_1"/>
<font color="Maroon">VV</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">L</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_3_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_3_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:45_3_2_1"/>
( <font color="Maroon">bas</font> is <a href="xboole_0.html#V1">empty</a> or  not <font color="Maroon">bas</font> is <a href="xboole_0.html#V1">empty</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_3_2_1_1"><b>suppose </b></a><a NAME="E1:45_3_2_1_1"/>
<font color="Maroon">bas</font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/><div class="add"><a NAME="E2:45_3_2_1_1"/><b>then </b>
 <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">bas</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>
 <b>by </b><i><a class="ref" href="setfam_1.html#D10">SETFAM_1:def 10</a></i>;<br/><b>hence </b><a NAME="E3:45_3_2_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">bas</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_3_2_1_2"><b>suppose </b></a><a NAME="E1:45_3_2_1_2"/><i><font color="Green">E122</font></i>: 
not <font color="Maroon">bas</font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/><div class="add"><a NAME="E2:45_3_2_1_2"/><b>then </b><i><font color="Green">E123</font></i>: 
 <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">bas</font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K6">meet</a> <font color="Maroon">bas</font>
 <b>by </b><i><a class="ref" href="setfam_1.html#D10">SETFAM_1:def 10</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_3_2_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">Y</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:45_3_2_1_2_1"/>
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">bas</font>
 ;<br/><a NAME="E2:45_3_2_1_2_1"/><b>then </b>
 ex <font color="Olive">V</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel11.html#K5">sigma</a> <font color="Maroon">L</font></span>)</span></span>)</span> st <br/>( <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a> <font color="Olive">V</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">V</font> &amp; <font color="Olive">V</font> is <a href="waybel_6.html#V6">co-prime</a> )
 ;<br/><b>hence </b><a NAME="E3:45_3_2_1_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:45_3_2_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">bas</font>
 <b>by </b><i>, , <a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:45_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">bas</font>
 ;<br/>

<b>let </b><font color="Maroon">S</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font>;<br/>

<b>assume </b><a NAME="E4:45_3"/><i><font color="Green">E124</font></i>: 
( <font color="Maroon">S</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> )
 ;<br/>

<b>then reconsider </b><font color="Maroon">S'</font> = <font color="Maroon">S</font> as   <a href="struct_0.html#NM1">Element</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">L</font></span>)</span> <b>by </b><i>, </i>;<br/>
<b>consider </b><font color="Maroon">X</font> being   <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">L</font></span>)</span><b> such that </b><br/><a NAME="E7:45_3"/><i><font color="Green">E125</font></i>: 
<font color="Maroon">S'</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">X</font>
 <b>and </b><br/><a NAME="E8:45_3"/><i><font color="Green">E126</font></i>: 
for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</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">L</font></span>)</span>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> holds <br/><font color="Olive">x</font> is <a href="waybel_6.html#V6">co-prime</a>
 <b>by </b><i>, </i>;<br/>
<a NAME="E9:45_3"/>
<font color="Maroon">S'</font> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">X</font>
 
<b>by </b><i>, <a class="ref" href="yellow_1.html#T22">YELLOW_1:22</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="E11:45_3"/><i><font color="Green">E127</font></i>: 
( <font color="Maroon">x</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> <font color="Maroon">X</font> )
 <b>by </b><i>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<a NAME="E12:45_3"/>
<font color="Maroon">V</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="txt" href="waybel14.html#E2:45"><i><font color="Green">E37</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon">V</font> = <font color="Maroon">V</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font> ;<br/>
<b>reconsider </b><font color="Maroon">Vp</font> = <font color="Maroon">V</font> as   <a href="struct_0.html#NM1">Element</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">L</font></span>)</span> <b>by </b><i><a class="txt" href="waybel14.html#E2:45"><i><font color="Green">E37</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon">V</font>
;<br/>

<a NAME="E15:45_3"/>
<font color="Maroon">Vp</font> is <a href="waybel_6.html#V6">co-prime</a>
 
<b>by </b><i><a class="txt" href="waybel14.html#E1:45"><i><font color="Green">E36</font></i></a>, <a class="txt" href="waybel14.html#E2:45"><i><font color="Green">E37</font></i></a></i>;<br/>
<b>hence </b><a NAME="E16:45_3"/>
<font color="Maroon">V</font> <a href="hidden.html#R2">in</a> <font color="Maroon">bas</font>
 <b>by </b><i>, <a class="txt" href="waybel14.html#E2:45"><i><font color="Green">E37</font></i></a></i>;<br/>

<a NAME="E17:45_3"/>
 <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">X</font> <a href="lattice3.html#NR4" target="_self">is_&gt;=_than</a> <font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#T32">YELLOW_0:32</a></i>;<br/>
<a NAME="E18:45_3"/><b>then </b>
<font color="Maroon">Vp</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">X</font>
 
<b>by </b><i><a class="txt" href="waybel14.html#E2:45"><i><font color="Green">E37</font></i></a>, <a class="ref" href="lattice3.html#D9">LATTICE3:def 9</a></i>;<br/>
<b>hence </b><a NAME="E19:45_3"/>
<font color="Maroon">V</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">S</font>
 <b>by </b><i>, <a class="ref" href="yellow_1.html#T3">YELLOW_1:3</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">bas</font> = <font color="Maroon">bas</font> as    <a href="yellow_8.html#M1">Basis</a> of <font color="Maroon">x</font> ;<br/>
<b>take </b>
<font color="Maroon">bas</font>
;<br/>

<b>let </b><font color="Maroon">V</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font>;<br/>

<b>assume </b><a NAME="E10:45"/>
<font color="Maroon">V</font> <a href="hidden.html#R2">in</a> <font color="Maroon">bas</font>
 ;<br/>

<b>then consider </b><font color="Maroon">Vp</font> being   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel11.html#K5">sigma</a> <font color="Maroon">L</font></span>)</span></span>)</span><b> such that </b><br/><a NAME="E12:45"/><i><font color="Green">E144</font></i>: 
<font color="Maroon">V</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Vp</font>
 <b>and </b><br/><a NAME="E13:45"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Vp</font>
 <b>and </b><br/><a NAME="E14:45"/><i><font color="Green">E145</font></i>: 
<font color="Maroon">Vp</font> is <a href="waybel_6.html#V6">co-prime</a>
 ;<br/>
<b>thus </b><a NAME="E15:45"/>
( <font color="Maroon">V</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">V</font> is <a href="waybel_0.html#V2">filtered</a> )
 <b>by </b><i>, , , , , </i>;<br/>


</div>
