<?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="waybel_0.html#NM8">LATTICE</a>;<br/>

<b>set </b><font color="Maroon">c<sub>2</sub></font> = <font color="Maroon">c<sub>1</sub></font> <a href="waybel_4.html#K1" target="_self">-waybelow</a> ;<br/>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:103_1"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is <a href="waybel_3.html#V3">continuous</a>
 ;<br/><b>then reconsider </b><font color="Maroon">c<sub>3</sub></font> = <font color="Maroon">c<sub>1</sub></font> as  <a href="yellow_0.html#V1">lower-bounded</a> <a href="waybel_2.html#V2">meet-continuous</a> <a href="waybel_0.html#NM8">LATTICE</a> <b>by </b><i><a class="ref" href="waybel_4.html#T38" target="_self">Th38</a></i>;<br/><b>thus </b><a NAME="E3:103_1"/>
for <font color="Olive">b<sub>1</sub></font> being <a href="waybel_4.html#V5" target="_self">auxiliary</a> <a href="waybel_4.html#V6" target="_self">approximating</a> <a href="alg_1.html#NM1">Relation</a> of <font color="Maroon">c<sub>1</sub></font> holds <br/>( <font color="Maroon">c<sub>1</sub></font> <a href="waybel_4.html#K1" target="_self">-waybelow</a>  <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="waybel_4.html#K1" target="_self">-waybelow</a>  is <a href="waybel_4.html#V6" target="_self">approximating</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="103_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font> be  <a href="waybel_4.html#V5" target="_self">auxiliary</a> <a href="waybel_4.html#V6" target="_self">approximating</a> <a href="alg_1.html#NM1">Relation</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</sub></font> as  <a href="waybel_4.html#V5" target="_self">auxiliary</a> <a href="waybel_4.html#V6" target="_self">approximating</a> <a href="alg_1.html#NM1">Relation</a> of <font color="Maroon">c<sub>3</sub></font> ;<br/>
<a NAME="E2:103_1_1"/>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1">set</a>  st <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> <a href="waybel_4.html#K1" target="_self">-waybelow</a>  holds <br/><span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="103_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>7</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:103_1_1_1"/><i><font color="Green">E64</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> <a href="waybel_4.html#K1" target="_self">-waybelow</a> 
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>7</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>3</sub></font> <b>by </b><i><a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E3:103_1_1_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="waybel_3.html#NR2" target="_self">&lt;&lt;</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="txt" href="waybel_4.html#E1:103_1_1_1"><i><font color="Green">E64</font></i></a>, <a class="ref" href="waybel_4.html#D2" target="_self">Def2</a></i>;<br/>
<a NAME="E4:103_1_1_1"/><b>then </b><i><font color="Green">E65</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="waybel_3.html#K1">waybelow</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="ref" href="waybel_3.html#T7">WAYBEL_3:7</a></i>;<br/>
<a NAME="E5:103_1_1_1"/><i><font color="Green">E66</font></i>: 
 <a href="setfam_1.html#K1">meet</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="waybel_4.html#K5" target="_self">-below</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> where B is  <a href="waybel_4.html#V5" target="_self">auxiliary</a> <a href="alg_1.html#NM1">Relation</a> of <font color="Maroon">c<sub>3</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="waybel_4.html#K12" target="_self">App</a> <font color="Maroon">c<sub>3</sub></font> </span>}</span>  <a href="hidden.html#R1">=</a>  <a href="waybel_3.html#K1">waybelow</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="ref" href="waybel_4.html#T44" target="_self">Th44</a></i>;<br/>
<a NAME="E6:103_1_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="waybel_4.html#K12" target="_self">App</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="waybel_4.html#D20" target="_self">Def20</a></i>;<br/>
<a NAME="E7:103_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="waybel_4.html#K5" target="_self">-below</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="waybel_4.html#K5" target="_self">-below</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> where B is  <a href="waybel_4.html#V5" target="_self">auxiliary</a> <a href="alg_1.html#NM1">Relation</a> of <font color="Maroon">c<sub>3</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="waybel_4.html#K12" target="_self">App</a> <font color="Maroon">c<sub>3</sub></font> </span>}</span> 
 
;<br/>
<a NAME="E8:103_1_1_1"/><b>then </b>
 <a href="waybel_3.html#K1">waybelow</a> <font color="Maroon">c<sub>9</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font> <a href="waybel_4.html#K5" target="_self">-below</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="txt" href="waybel_4.html#E5:103_1_1_1"><i><font color="Green">E66</font></i></a>, <a class="ref" href="setfam_1.html#T4">SETFAM_1:4</a></i>;<br/>
<b>hence </b><a NAME="E9:103_1_1_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="waybel_4.html#E4:103_1_1_1"><i><font color="Green">E65</font></i></a>, <a class="ref" href="waybel_4.html#T13" target="_self">Th13</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:103_1_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="waybel_4.html#K1" target="_self">-waybelow</a>  <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="relat_1.html#D3">RELAT_1:def 3</a></i>;<br/>

<b>thus </b><a NAME="E4:103_1_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="waybel_4.html#K1" target="_self">-waybelow</a>  is <a href="waybel_4.html#V6" target="_self">approximating</a>
 <b>by </b><i><a class="txt" href="waybel_4.html#E1:103_1"><i><font color="Green">E63</font></i></a>, <a class="txt" href="waybel_4.html#E73"><i><font color="Green">Lemma57</font></i></a></i>;<br/>


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

<b>assume </b><a NAME="E2:103"/><i><font color="Green">E63</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being <a href="waybel_4.html#V5" target="_self">auxiliary</a> <a href="waybel_4.html#V6" target="_self">approximating</a> <a href="alg_1.html#NM1">Relation</a> of <font color="Maroon">c<sub>1</sub></font> holds <br/>( <font color="Maroon">c<sub>1</sub></font> <a href="waybel_4.html#K1" target="_self">-waybelow</a>  <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="waybel_4.html#K1" target="_self">-waybelow</a>  is <a href="waybel_4.html#V6" target="_self">approximating</a> )
 ;<br/>

<a NAME="E3:103"/>
for <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> holds <font color="Olive">b<sub>1</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_3.html#K1">waybelow</a> <font color="Olive">b<sub>1</sub></font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="103_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<a NAME="E1:103_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="waybel_4.html#K1" target="_self">-waybelow</a> </span>)</span> <a href="waybel_4.html#K5" target="_self">-below</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel_4.html#E2:103"><i><font color="Green">E63</font></i></a>, <a class="ref" href="waybel_4.html#D18" target="_self">Def18</a></i>;<br/>
<b>hence </b><a NAME="E2:103_2"/>
<font color="Maroon">c<sub>3</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_3.html#K1">waybelow</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="waybel_4.html#T40" target="_self">Th40</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:103"/><b>then </b><i><font color="Green">E64</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is <a href="waybel_3.html#V2">satisfying_axiom_of_approximation</a>
 
<b>by </b><i><a class="ref" href="waybel_3.html#D5">WAYBEL_3:def 5</a></i>;<br/>
<a NAME="E5:103"/>
for <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> holds <br/>( not  <a href="waybel_3.html#K1">waybelow</a> <font color="Olive">b<sub>1</sub></font> is <a href="xboole_0.html#V1">empty</a> &amp;  <a href="waybel_3.html#K1">waybelow</a> <font color="Olive">b<sub>1</sub></font> is <a href="waybel_0.html#V1">directed</a> )
 
;<br/>
<b>hence </b><a NAME="E6:103"/>
<font color="Maroon">c<sub>1</sub></font> is <a href="waybel_3.html#V3">continuous</a>
 <b>by </b><i><a class="txt" href="waybel_4.html#E4:103"><i><font color="Green">E64</font></i></a>, <a class="ref" href="waybel_3.html#D6">WAYBEL_3:def 6</a></i>;<br/>


</div>
