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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  <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>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<b>set </b><font color="Maroon">c<sub>3</sub></font> = <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>2</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>1</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>1</sub></font> </span>}</span> ;<br/>
<b>consider </b><font color="Maroon">c<sub>4</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>;<br/>
<a NAME="E2:102"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="waybel_4.html#K12" target="_self">App</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="waybel_4.html#D20" target="_self">Def20</a></i>;<br/>
<a NAME="E3:102"/><b>then </b><i><font color="Green">E62</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="waybel_4.html#K5" target="_self">-below</a> <font color="Maroon">c<sub>2</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>2</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>1</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>1</sub></font> </span>}</span> 
 
;<br/>
<a NAME="E4:102"/><i><font color="Green">E63</font></i>: 
 <a href="setfam_1.html#K1">meet</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Maroon">c<sub>2</sub></font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Olive">b<sub>1</sub></font> </span>}</span>  <a href="hidden.html#R1">=</a>  <a href="waybel_3.html#K1">waybelow</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="waybel_4.html#T34" target="_self">Th34</a></i>;<br/>
<a NAME="E5:102"/><i><font color="Green">E64</font></i>: 
 <a href="setfam_1.html#K1">meet</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel_4.html#K11" target="_self">DownMap</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> where B is   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">c<sub>1</sub></font> : verum </span>}</span>  <a href="hidden.html#R1">=</a>  <a href="waybel_3.html#K1">waybelow</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="waybel_4.html#T43" target="_self">Th43</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>
<a NAME="E7:102"/><i><font color="Green">E65</font></i>: 
<span class="p1">(<span class="default"><a href="waybel_4.html#K11" target="_self">DownMap</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel_4.html#K11" target="_self">DownMap</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> where B is   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">c<sub>1</sub></font> : verum </span>}</span> 
 
;<br/>
<a NAME="E8:102"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel_4.html#K11" target="_self">DownMap</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> where B is   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">c<sub>1</sub></font> : verum </span>}</span>  <a href="tarski.html#R1">c=</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>2</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>1</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>1</sub></font> </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="102_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>6</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="E1:102_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel_4.html#K11" target="_self">DownMap</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> where B is   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">c<sub>1</sub></font> : verum </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:102_1"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="waybel_4.html#K11" target="_self">DownMap</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>
<a NAME="E4:102_1"/><i><font color="Green">E67</font></i>: 
 <a href="waybel_4.html#K11" target="_self">DownMap</a> <font color="Maroon">c<sub>7</sub></font> is <a href="waybel_4.html#V7" target="_self">approximating</a>
 
<b>by </b><i><a class="ref" href="waybel_4.html#T37" target="_self">Th37</a></i>;<br/>
<a NAME="E5:102_1"/><i><font color="Green">E68</font></i>: 
 <a href="waybel_4.html#K11" target="_self">DownMap</a> <font color="Maroon">c<sub>7</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_4.html#K8" target="_self">MonSet</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="waybel_4.html#T35" target="_self">Th35</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>8</sub></font> being  <a href="waybel_4.html#V5" target="_self">auxiliary</a> <a href="alg_1.html#NM1">Relation</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E7:102_1"/><i><font color="Green">E69</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="waybel_4.html#K10" target="_self">Map2Rel</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="waybel_4.html#K11" target="_self">DownMap</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> &amp; ( 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>  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>8</sub></font> iff ex <font color="Olive">b<sub>3</sub></font>, <font color="Olive">b<sub>4</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>ex <font color="Olive">b<sub>5</sub></font> being  <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K7">Ids</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> st <br/>( <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>5</sub></font> <a href="funct_2.html#R2">=</a>  <a href="waybel_4.html#K11" target="_self">DownMap</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>4</sub></font> ) ) ) )
 <b>by </b><i><a class="ref" href="waybel_4.html#D16" target="_self">Def16</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>9</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><b> such that </b><br/><a NAME="E9:102_1"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="waybel_4.html#K10" target="_self">Map2Rel</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="waybel_4.html#K11" target="_self">DownMap</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="waybel_4.html#E4:102_1"><i><font color="Green">E67</font></i></a>, <a class="txt" href="waybel_4.html#E5:102_1"><i><font color="Green">E68</font></i></a>, <a class="ref" href="waybel_4.html#T42" target="_self">Th42</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>10</sub></font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E11:102_1"/><i><font color="Green">E71</font></i>: 
( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="waybel_4.html#K11" target="_self">DownMap</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">c<sub>10</sub></font> )
 <b>by </b><i><a class="txt" href="waybel_4.html#E4:102_1"><i><font color="Green">E67</font></i></a>, <a class="ref" href="waybel_4.html#D19" target="_self">Def19</a></i>;<br/>
<a NAME="E12:102_1"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="waybel_4.html#K5" target="_self">-below</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="waybel_4.html#K11" target="_self">DownMap</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="102_1_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:102_1_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="waybel_4.html#K5" target="_self">-below</a> <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="waybel_4.html#K11" target="_self">DownMap</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="102_1_1_1"><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="E1:102_1_1_1"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> <a href="waybel_4.html#K5" target="_self">-below</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<a NAME="E2:102_1_1_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="ref" href="waybel_4.html#T13" target="_self">Th13</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>12</sub></font>, <font color="Maroon">c<sub>13</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>14</sub></font> being   <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K7">Ids</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span><b> such that </b><br/><a NAME="E4:102_1_1_1"/><i><font color="Green">E73</font></i>: 
( <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="funct_2.html#R2">=</a>  <a href="waybel_4.html#K11" target="_self">DownMap</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>14</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>13</sub></font> )
 <b>by </b><i><a class="txt" href="waybel_4.html#E7:102_1"><i><font color="Green">E69</font></i></a></i>;<br/>
<b>thus </b><a NAME="E5:102_1_1_1"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="waybel_4.html#K11" target="_self">DownMap</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="waybel_4.html#E4:102_1_1_1"><i><font color="Green">E73</font></i></a></i>;<br/>


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

<b>thus </b><a NAME="E2:102_1_1"/>
<span class="p1">(<span class="default"><a href="waybel_4.html#K11" target="_self">DownMap</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>8</sub></font> <a href="waybel_4.html#K5" target="_self">-below</a> <font color="Maroon">c<sub>2</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="102_1_1_2"><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="E1:102_1_1_2"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="waybel_4.html#K11" target="_self">DownMap</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<b>then 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="waybel_4.html#E11:102_1"><i><font color="Green">E71</font></i></a></i>;<br/>
<a NAME="E3:102_1_1_2"/>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="txt" href="waybel_4.html#E7:102_1"><i><font color="Green">E69</font></i></a>, <a class="txt" href="waybel_4.html#E1:102_1_1_2"><i><font color="Green">E73</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:102_1_1_2"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> <a href="waybel_4.html#K5" target="_self">-below</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>


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


</div><b>end;</b></div>
<a NAME="E13:102_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="waybel_4.html#K12" target="_self">App</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="waybel_4.html#E7:102_1"><i><font color="Green">E69</font></i></a>, <a class="txt" href="waybel_4.html#E9:102_1"><i><font color="Green">E70</font></i></a>, <a class="ref" href="waybel_4.html#D20" target="_self">Def20</a></i>;<br/>
<b>hence </b><a NAME="E14:102_1"/>
<font color="Maroon">c<sub>6</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>2</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>1</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>1</sub></font> </span>}</span> 
 <b>by </b><i><a class="txt" href="waybel_4.html#E3:102_1"><i><font color="Green">E66</font></i></a>, <a class="txt" href="waybel_4.html#E12:102_1"><i><font color="Green">E72</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E9:102"/>
 <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>2</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>1</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>1</sub></font> </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="waybel_3.html#K1">waybelow</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="waybel_4.html#E5:102"><i><font color="Green">E64</font></i></a>, <a class="txt" href="waybel_4.html#E7:102"><i><font color="Green">E65</font></i></a>, <a class="ref" href="setfam_1.html#T7">SETFAM_1:7</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><br/>

<b>thus </b><a NAME="E10:102"/>
 <a href="waybel_3.html#K1">waybelow</a> <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a>  <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>2</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>1</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>1</sub></font> </span>}</span> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="102_2"><b>proof </b></a><div class="add">

<a NAME="E1:102_2"/>
<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>2</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>1</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>1</sub></font> </span>}</span>  <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Maroon">c<sub>2</sub></font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Olive">b<sub>1</sub></font> </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="102_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>6</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="E1:102_2_1"/>
<font color="Maroon">c<sub>6</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>2</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>1</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>1</sub></font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>7</sub></font> being  <a href="waybel_4.html#V5" target="_self">auxiliary</a> <a href="alg_1.html#NM1">Relation</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:102_2_1"/><i><font color="Green">E66</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="waybel_4.html#K5" target="_self">-below</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="waybel_4.html#K12" target="_self">App</a> <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="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> <b>by </b><i><a class="txt" href="waybel_4.html#E3:102_2_1"><i><font color="Green">E66</font></i></a>, <a class="ref" href="waybel_4.html#D20" target="_self">Def20</a></i>;<br/>
<a NAME="E5:102_2_1"/>
 <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="waybel_4.html#K5" target="_self">-below</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="waybel_4.html#D18" target="_self">Def18</a></i>;<br/>
<b>hence </b><a NAME="E6:102_2_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Maroon">c<sub>2</sub></font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Olive">b<sub>1</sub></font> </span>}</span> 
 <b>by </b><i><a class="txt" href="waybel_4.html#E3:102_2_1"><i><font color="Green">E66</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:102_2"/>
 <a href="waybel_3.html#K1">waybelow</a> <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a>  <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>2</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>1</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>1</sub></font> </span>}</span> 
 <b>by </b><i><a class="txt" href="waybel_4.html#E3:102"><i><font color="Green">E62</font></i></a>, <a class="txt" href="waybel_4.html#E4:102"><i><font color="Green">E63</font></i></a>, <a class="ref" href="setfam_1.html#T7">SETFAM_1:7</a></i>;<br/>


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


</div>
