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

<b>let </b><font color="Maroon">c<sub>1</sub></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>assume </b><a NAME="E1:48"/>
<font color="Maroon">c<sub>1</sub></font> is <a href="waybel_8.html#V2">algebraic</a>
 ;<br/>

<a NAME="E2:48"/><b>then </b><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is <a href="waybel_8.html#V1">satisfying_axiom_K</a>
 
<b>by </b><i><a class="ref" href="waybel_8.html#D4">WAYBEL_8:def 4</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>2</sub></font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></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="waybel_8.html#K1">CompactSublatt</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> </span>}</span> ;<br/>
<a NAME="E3:48"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></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="waybel_8.html#K1">CompactSublatt</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> </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">c<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="hidden.html#M1">set</a> ; <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="E1:48_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></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="waybel_8.html#K1">CompactSublatt</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> </span>}</span> 
 ;<br/>

<a NAME="E2:48_1"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="waybel_0.html#K6">uparrow</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></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="waybel_8.html#K1">CompactSublatt</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> )
 
;<br/>
<b>hence </b><a NAME="E3:48_1"/>
<font color="Maroon">c<sub>3</sub></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">c<sub>1</sub></font>
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>3</sub></font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></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="waybel_8.html#K1">CompactSublatt</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> </span>}</span>  as   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>3</sub></font> as   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<a NAME="E6:48"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</sub></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="E1:48_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>6</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:48_2"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>6</sub></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="waybel_8.html#K1">CompactSublatt</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> )
 ;<br/>
<a NAME="E4:48_2"/>
<font color="Maroon">c<sub>6</sub></font> is <a href="waybel_3.html#V1">compact</a>
 
<b>by </b><i><a class="txt" href="waybel14.html#E3:48_2"><i><font color="Green">E33</font></i></a>, <a class="ref" href="waybel_8.html#D1">WAYBEL_8:def 1</a></i>;<br/>
<a NAME="E5:48_2"/><b>then </b>
 <a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">c<sub>6</sub></font> is <a href="waybel_6.html#V1">Open</a>
 
<b>by </b><i><a class="ref" href="waybel_8.html#T2">WAYBEL_8:2</a></i>;<br/>
<a NAME="E6:48_2"/><b>then </b>
 <a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">c<sub>6</sub></font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="ref" href="waybel11.html#T41">WAYBEL11:41</a></i>;<br/>
<b>hence </b><a NAME="E7:48_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="waybel14.html#E3:48_2"><i><font color="Green">E33</font></i></a>, <a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/>


</div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>set </b><font color="Maroon">c<sub>6</sub></font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : (  <a href="waybel_0.html#K6">uparrow</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K6">uparrow</a> <font color="Olive">b<sub>1</sub></font> ) </span>}</span> ;<br/><a NAME="E1:48_3"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : (  <a href="waybel_0.html#K6">uparrow</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K6">uparrow</a> <font color="Olive">b<sub>1</sub></font> ) </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">c<sub>1</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>7</sub></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="E1:48_3_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : (  <a href="waybel_0.html#K6">uparrow</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K6">uparrow</a> <font color="Olive">b<sub>1</sub></font> ) </span>}</span> 
 ;<br/>

<a NAME="E2:48_3_1"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="waybel_0.html#K6">uparrow</a> <font color="Olive">b<sub>1</sub></font> &amp;  <a href="waybel_0.html#K6">uparrow</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K6">uparrow</a> <font color="Olive">b<sub>1</sub></font> )
 
;<br/>
<b>hence </b><a NAME="E3:48_3_1"/>
<font color="Maroon">c<sub>7</sub></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">c<sub>1</sub></font>
 ;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">c<sub>7</sub></font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : (  <a href="waybel_0.html#K6">uparrow</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K6">uparrow</a> <font color="Olive">b<sub>1</sub></font> ) </span>}</span>  as   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/><b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>7</sub></font> as   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/><a NAME="E4:48_3"/>
<font color="Maroon">c<sub>8</sub></font> is    <a href="yellow_8.html#M1">Basis</a> of <font color="Maroon">c<sub>5</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_3_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:48_3_2"/>
<font color="Maroon">c<sub>8</sub></font> <a href="tarski.html#R1">c=</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>1</sub></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="48_3_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>9</sub></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="E1:48_3_2_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/>

<a NAME="E2:48_3_2_1"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a>  <a href="waybel_0.html#K6">uparrow</a> <font color="Olive">b<sub>1</sub></font> &amp;  <a href="waybel_0.html#K6">uparrow</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K6">uparrow</a> <font color="Olive">b<sub>1</sub></font> )
 
;<br/>
<b>hence </b><a NAME="E3:48_3_2_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="waybel14.html#E6:48"><i><font color="Green">E32</font></i></a></i>;<br/>


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

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_3_2_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_3_2_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:48_3_2_2_1"/>
( <font color="Maroon">c<sub>8</sub></font> is <a href="xboole_0.html#V1">empty</a> or  not <font color="Maroon">c<sub>8</sub></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="48_3_2_2_1_1"><b>suppose </b></a><a NAME="E1:48_3_2_2_1_1"/>
<font color="Maroon">c<sub>8</sub></font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/><div class="add"><a NAME="E2:48_3_2_2_1_1"/><b>then </b>
 <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="setfam_1.html#D10">SETFAM_1:def 10</a></i>;<br/><b>hence </b><a NAME="E3:48_3_2_2_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_3_2_2_1_2"><b>suppose </b></a><a NAME="E1:48_3_2_2_1_2"/><i><font color="Green">E33</font></i>: 
not <font color="Maroon">c<sub>8</sub></font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/><div class="add"><a NAME="E2:48_3_2_2_1_2"/><b>then </b><i><font color="Green">E34</font></i>: 
 <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K6">meet</a> <font color="Maroon">c<sub>8</sub></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="48_3_2_2_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>9</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:48_3_2_2_1_2_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/><a NAME="E2:48_3_2_2_1_2_1"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a>  <a href="waybel_0.html#K6">uparrow</a> <font color="Olive">b<sub>1</sub></font> &amp;  <a href="waybel_0.html#K6">uparrow</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K6">uparrow</a> <font color="Olive">b<sub>1</sub></font> )
 ;<br/><b>hence </b><a NAME="E3:48_3_2_2_1_2_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font>
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:48_3_2_2_1_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="waybel14.html#E1:48_3_2_2_1_2"><i><font color="Green">E33</font></i></a>, <a class="txt" href="waybel14.html#E2:48_3_2_2_1_2"><i><font color="Green">E34</font></i></a>, <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="E3:48_3_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>9</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E4:48_3_2"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">c<sub>9</sub></font> is <a href="pre_topc.html#V3">open</a>
 <b>and </b><br/><a NAME="E5:48_3_2"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font>
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>5</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<a NAME="E7:48_3_2"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><a href="waybel_8.html#K2">compactbelow</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel14.html#E2:48"><i><font color="Green">E31</font></i></a>, <a class="ref" href="waybel_8.html#D3">WAYBEL_8:def 3</a></i>;<br/>
<a NAME="E8:48_3_2"/>
<font color="Maroon">c<sub>9</sub></font> is <a href="waybel11.html#V1">inaccessible_by_directed_joins</a>
 
<b>by </b><i><a class="txt" href="waybel14.html#E4:48_3_2"><i><font color="Green">E33</font></i></a>, <a class="ref" href="waybel11.html#D4">WAYBEL11:def 4</a></i>;<br/>
<a NAME="E9:48_3_2"/><b>then </b>
 <a href="waybel_8.html#K2">compactbelow</a> <font color="Maroon">c<sub>10</sub></font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="txt" href="waybel14.html#E5:48_3_2"><i><font color="Green">E34</font></i></a>, <a class="txt" href="waybel14.html#E7:48_3_2"><i><font color="Green">E35</font></i></a>, <a class="ref" href="waybel11.html#D1">WAYBEL11:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>11</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E11:48_3_2"/><i><font color="Green">E36</font></i>: 
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="waybel_8.html#K2">compactbelow</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>
<a NAME="E12:48_3_2"/><i><font color="Green">E37</font></i>: 
 <a href="waybel_8.html#K2">compactbelow</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="subset_1.html#K8">/\</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel_8.html#K1">CompactSublatt</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="waybel_8.html#T5">WAYBEL_8:5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>11</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="waybel14.html#E11:48_3_2"><i><font color="Green">E36</font></i></a></i>;<br/>
<b>take </b><font color="Maroon">c<sub>13</sub></font> =  <a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">c<sub>12</sub></font>;<br/>

<a NAME="E14:48_3_2"/>
<font color="Maroon">c<sub>12</sub></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="waybel_8.html#K1">CompactSublatt</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel14.html#E11:48_3_2"><i><font color="Green">E36</font></i></a>, <a class="txt" href="waybel14.html#E12:48_3_2"><i><font color="Green">E37</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E15:48_3_2"/><b>then </b><i><font color="Green">E38</font></i>: 
 <a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 
;<br/>
<a NAME="E16:48_3_2"/>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">c<sub>10</sub></font>
 
<b>by </b><i><a class="txt" href="waybel14.html#E11:48_3_2"><i><font color="Green">E36</font></i></a>, <a class="txt" href="waybel14.html#E12:48_3_2"><i><font color="Green">E37</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E17:48_3_2"/><b>then </b>
<font color="Maroon">c<sub>12</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>10</sub></font>
 
<b>by </b><i><a class="ref" href="waybel_0.html#T17">WAYBEL_0:17</a></i>;<br/>
<a NAME="E18:48_3_2"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">c<sub>12</sub></font>
 
<b>by </b><i><a class="ref" href="waybel_0.html#T18">WAYBEL_0:18</a></i>;<br/>
<b>hence </b><a NAME="E19:48_3_2"/>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="waybel14.html#E15:48_3_2"><i><font color="Green">E38</font></i></a></i>;<br/>

<a NAME="E20:48_3_2"/>
<font color="Maroon">c<sub>9</sub></font> is <a href="waybel_0.html#V13">upper</a>
 
<b>by </b><i><a class="txt" href="waybel14.html#E4:48_3_2"><i><font color="Green">E33</font></i></a>, <a class="ref" href="waybel11.html#D4">WAYBEL11:def 4</a></i>;<br/>
<b>hence </b><a NAME="E21:48_3_2"/>
<font color="Maroon">c<sub>13</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="waybel14.html#E11:48_3_2"><i><font color="Green">E36</font></i></a>, <a class="ref" href="waybel11.html#T42">WAYBEL11:42</a></i>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>8</sub></font> as    <a href="yellow_8.html#M1">Basis</a> of <font color="Maroon">c<sub>5</sub></font> ;<br/><b>take </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>9</sub></font>;<br/><b>thus </b><a NAME="E6:48_3"/>
<font color="Maroon">c<sub>10</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_3_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>11</sub></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="E1:48_3_3"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font>
 ;<br/>

<a NAME="E2:48_3_3"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a>  <a href="waybel_0.html#K6">uparrow</a> <font color="Olive">b<sub>1</sub></font> &amp;  <a href="waybel_0.html#K6">uparrow</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K6">uparrow</a> <font color="Olive">b<sub>1</sub></font> )
 
;<br/>
<b>hence </b><a NAME="E3:48_3_3"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E8:48"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> is    <a href="cantor_1.html#M1">Basis</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="waybel14.html#E6:48"><i><font color="Green">E32</font></i></a>, <a class="ref" href="yellow_8.html#T23">YELLOW_8:23</a></i>;<br/>
<b>hence </b><a NAME="E9:48"/>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="cantor_1.html#M1">Basis</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Olive">b<sub>2</sub></font></span>)</span> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>2</sub></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="waybel_8.html#K1">CompactSublatt</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> </span>}</span> 
 ;<br/>


</div>
