<?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_3.html#V3">continuous</a> <a href="waybel_0.html#NM7">sup-Semilattice</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be  <a href="pre_topc.html#V2">correct</a> <a href="waybel19.html#V2">Lawson</a>  <a href="yellow_9.html#M1">TopAugmentation</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be  <a href="waybel23.html#V6">with_bottom</a>  <a href="waybel23.html#M1">CLbasis</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>





<a NAME="E1:28"/><i><font color="Green">E22</font></i>: 
 <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font> #) <a href="hidden.html#R1">=</a>  <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>2</sub></font> #)
 
<b>by </b><i><a class="ref" href="yellow_9.html#D4">YELLOW_9:def 4</a></i>;<br/>
<a NAME="E2:28"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="waybel31.html#K2" target="_self">Way_Up</a> <font color="Olive">b<sub>1</sub></font>,<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>, B is  <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM2">Subset</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> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</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>2</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</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="E1:28_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="waybel31.html#K2" target="_self">Way_Up</a> <font color="Olive">b<sub>1</sub></font>,<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>, B is  <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM2">Subset</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> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font> ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>5</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>6</sub></font> being  <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:28_1"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="waybel31.html#K2" target="_self">Way_Up</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>
 <b>and </b><br/><a NAME="E4:28_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 <b>and </b><br/><a NAME="E5:28_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>
<b>thus </b><a NAME="E6:28_1"/>
<font color="Maroon">c<sub>4</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>2</sub></font>
 <b>by </b><i><a class="txt" href="waybel31.html#E1:28"><i><font color="Green">E22</font></i></a>, <a class="txt" href="waybel31.html#E3:28_1"><i><font color="Green">E23</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>4</sub></font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="waybel31.html#K2" target="_self">Way_Up</a> <font color="Olive">b<sub>1</sub></font>,<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>, B is  <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM2">Subset</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> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font> ) </span>}</span>  as   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">c<sub>2</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="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">c<sub>2</sub></font> ;<br/>
<a NAME="E5:28"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>2</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_2"><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="E1:28_2"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>7</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>8</sub></font> being  <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:28_2"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="waybel31.html#K2" target="_self">Way_Up</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font>
 <b>and </b><br/><a NAME="E4:28_2"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 <b>and </b><br/><a NAME="E5:28_2"/>
<font color="Maroon">c<sub>8</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>
<b>reconsider </b><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>2</sub></font> <b>by </b><i><a class="txt" href="waybel31.html#E1:28"><i><font color="Green">E22</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>8</sub></font> as  <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="txt" href="waybel31.html#E1:28"><i><font color="Green">E22</font></i></a></i>;<br/>
<a NAME="E8:28_2"/><i><font color="Green">E25</font></i>: 
 <a href="waybel_3.html#K2">wayabove</a> <font color="Maroon">c<sub>9</sub></font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="ref" href="waybel19.html#T40">WAYBEL19:40</a></i>;<br/>
<a NAME="E9:28_2"/>
<span class="p1">(<span class="default"><a href="waybel_0.html#K4">uparrow</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="subset_1.html#K3">`</a>  is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="ref" href="waybel31.html#T25" target="_self">Th25</a></i>;<br/>
<a NAME="E10:28_2"/><b>then </b><i><font color="Green">E26</font></i>: 
<span class="p1">(<span class="default"><a href="waybel_3.html#K2">wayabove</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_0.html#K4">uparrow</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="txt" href="waybel31.html#E8:28_2"><i><font color="Green">E25</font></i></a>, <a class="ref" href="tops_1.html#T38">TOPS_1:38</a></i>;<br/>
<font color="Maroon">c<sub>6</sub></font> = 
<span class="p1">(<span class="default"><a href="waybel_3.html#K2">wayabove</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K4">uparrow</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="waybel31.html#E3:28_2"><i><font color="Green">E24</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="waybel_3.html#K2">wayabove</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K4">uparrow</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="waybel31.html#E1:28"><i><font color="Green">E22</font></i></a>, <a class="ref" href="yellow12.html#T13">YELLOW12:13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="waybel_3.html#K2">wayabove</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K4">uparrow</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="waybel31.html#E1:28"><i><font color="Green">E22</font></i></a>, <a class="ref" href="waybel_0.html#T13">WAYBEL_0:13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="waybel_3.html#K2">wayabove</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_0.html#K4">uparrow</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span>
<b>by </b><i><a class="ref" href="subset_1.html#T32">SUBSET_1:32</a></i>
;<br/>
<b>hence </b><a NAME="E12:28_2"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="waybel31.html#E10:28_2"><i><font color="Green">E26</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="28_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>2</sub></font>;<br/><b>assume </b><a NAME="E1:28_3"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">c<sub>6</sub></font> is <a href="pre_topc.html#V3">open</a>
 ;<br/><b>let </b><font color="Maroon">c<sub>7</sub></font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>2</sub></font>;<br/><b>assume </b><a NAME="E2:28_3"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/><b>consider </b><font color="Maroon">c<sub>8</sub></font> being  <a href="waybel11.html#V4">Scott</a>  <a href="yellow_9.html#M1">TopAugmentation</a> of <font color="Maroon">c<sub>2</sub></font>;<br/><a NAME="E4:28_3"/><i><font color="Green">E26</font></i>: 
 <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>8</sub></font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>8</sub></font> #) <a href="hidden.html#R1">=</a>  <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>2</sub></font> #)
 <b>by </b><i><a class="ref" href="yellow_9.html#D4">YELLOW_9:def 4</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>9</sub></font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="subset_1.html#K7">\</a> <span class="p3">(<span class="default"><a href="waybel_0.html#K4">uparrow</a> <font color="Olive">b<sub>2</sub></font></span>)</span></span>)</span> where B is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>2</sub></font>, B is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>2</sub></font> : ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="waybel11.html#K5">sigma</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> is <a href="finset_1.html#V1">finite</a> ) </span>}</span>  as    <a href="cantor_1.html#M1">Basis</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="ref" href="waybel19.html#T32">WAYBEL19:32</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>2</sub></font><b> such that </b><br/><a NAME="E7:28_3"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font>
 <b>and </b><br/><a NAME="E8:28_3"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font>
 <b>and </b><br/><a NAME="E9:28_3"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="waybel31.html#E1:28_3"><i><font color="Green">E24</font></i></a>, <a class="txt" href="waybel31.html#E2:28_3"><i><font color="Green">E25</font></i></a>, <a class="ref" href="yellow_9.html#T31">YELLOW_9:31</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>11</sub></font>, <font color="Maroon">c<sub>12</sub></font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E11:28_3"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K4">uparrow</a> <font color="Maroon">c<sub>12</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E12:28_3"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="waybel11.html#K5">sigma</a> <font color="Maroon">c<sub>2</sub></font>
 <b>and </b><br/><a NAME="E13:28_3"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">c<sub>12</sub></font> is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i><a class="txt" href="waybel31.html#E7:28_3"><i><font color="Green">E27</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>11</sub></font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>8</sub></font> <b>by </b><i><a class="txt" href="waybel31.html#E4:28_3"><i><font color="Green">E26</font></i></a></i>;<br/><a NAME="E15:28_3"/>
 <a href="waybel11.html#K5">sigma</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="waybel11.html#K5">sigma</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="waybel31.html#E4:28_3"><i><font color="Green">E26</font></i></a>, <a class="ref" href="yellow_9.html#T52">YELLOW_9:52</a></i>;<br/><a NAME="E16:28_3"/><b>then </b><i><font color="Green">E33</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="waybel_3.html#K2">wayabove</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>8</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>13</sub></font> </span>}</span> 
 <b>by </b><i><a class="txt" href="waybel31.html#E12:28_3"><i><font color="Green">E31</font></i></a>, <a class="ref" href="waybel14.html#T33">WAYBEL14:33</a></i>;<br/><a NAME="E17:28_3"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>11</sub></font> &amp; not <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K4">uparrow</a> <font color="Maroon">c<sub>12</sub></font> )
 <b>by </b><i><a class="txt" href="waybel31.html#E8:28_3"><i><font color="Green">E28</font></i></a>, <a class="txt" href="waybel31.html#E11:28_3"><i><font color="Green">E30</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>14</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E19:28_3"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>14</sub></font>
 <b>and </b><br/><a NAME="E20:28_3"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="waybel_3.html#K2">wayabove</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>8</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>13</sub></font> </span>}</span> 
 <b>by </b><i><a class="txt" href="waybel31.html#E16:28_3"><i><font color="Green">E33</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>15</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>8</sub></font><b> such that </b><br/><a NAME="E22:28_3"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a>  <a href="waybel_3.html#K2">wayabove</a> <font color="Maroon">c<sub>15</sub></font>
 <b>and </b><br/><a NAME="E23:28_3"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="txt" href="waybel31.html#E20:28_3"><i><font color="Green">E36</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>16</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>8</sub></font> <b>by </b><i><a class="txt" href="waybel31.html#E4:28_3"><i><font color="Green">E26</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>17</sub></font> = <font color="Maroon">c<sub>16</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="waybel31.html#E1:28"><i><font color="Green">E22</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>18</sub></font> = <font color="Maroon">c<sub>15</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="waybel31.html#E1:28"><i><font color="Green">E22</font></i></a>, <a class="txt" href="waybel31.html#E4:28_3"><i><font color="Green">E26</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>19</sub></font> = <font color="Maroon">c<sub>12</sub></font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="waybel31.html#E1:28"><i><font color="Green">E22</font></i></a></i>;<br/><a NAME="E28:28_3"/>
<font color="Maroon">c<sub>15</sub></font> <a href="waybel_3.html#NR2" target="_self">&lt;&lt;</a> <font color="Maroon">c<sub>16</sub></font>
 <b>by </b><i><a class="txt" href="waybel31.html#E19:28_3"><i><font color="Green">E35</font></i></a>, <a class="txt" href="waybel31.html#E22:28_3"><i><font color="Green">E37</font></i></a>, <a class="ref" href="waybel_3.html#T8">WAYBEL_3:8</a></i>;<br/><a NAME="E29:28_3"/><b>then </b><i><font color="Green">E39</font></i>: 
<font color="Maroon">c<sub>18</sub></font> <a href="waybel_3.html#NR2" target="_self">&lt;&lt;</a> <font color="Maroon">c<sub>17</sub></font>
 <b>by </b><i><a class="txt" href="waybel31.html#E1:28"><i><font color="Green">E22</font></i></a>, <a class="txt" href="waybel31.html#E4:28_3"><i><font color="Green">E26</font></i></a>, <a class="ref" href="waybel_8.html#T8">WAYBEL_8:8</a></i>;<br/><a NAME="E30:28_3"/>
 <a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="waybel23.html#D8">WAYBEL23:def 8</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>20</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="E32:28_3"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 <b>and </b><br/><a NAME="E33:28_3"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">c<sub>18</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>20</sub></font>
 <b>and </b><br/><a NAME="E34:28_3"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">c<sub>20</sub></font> <a href="waybel_3.html#NR2" target="_self">&lt;&lt;</a> <font color="Maroon">c<sub>17</sub></font>
 <b>by </b><i><a class="txt" href="waybel31.html#E29:28_3"><i><font color="Green">E39</font></i></a>, <a class="ref" href="waybel23.html#T47">WAYBEL23:47</a></i>;<br/><a NAME="E35:28_3"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a>  <a href="waybel_3.html#K2">wayabove</a> <font color="Maroon">c<sub>20</sub></font>
 <b>by </b><i><a class="txt" href="waybel31.html#E34:28_3"><i><font color="Green">E42</font></i></a>, <a class="ref" href="waybel_3.html#T8">WAYBEL_3:8</a></i>;<br/><a NAME="E36:28_3"/><i><font color="Green">E44</font></i>: 
 <a href="waybel_0.html#K4">uparrow</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a>  <a href="waybel_0.html#K4">uparrow</a> <font color="Maroon">c<sub>19</sub></font>
 <b>by </b><i><a class="txt" href="waybel31.html#E1:28"><i><font color="Green">E22</font></i></a>, <a class="ref" href="waybel_0.html#T13">WAYBEL_0:13</a></i>;<br/><b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] means ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>2</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; not <font color="Olive">b<sub>1</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>17</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="waybel_3.html#NR2" target="_self">&lt;&lt;</a> <font color="Olive">b<sub>2</sub></font> );<br/><a NAME="E37:28_3"/><i><font color="Green">E45</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>19</sub></font> holds <br/>ex <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>21</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:28_3_1"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>19</sub></font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>22</sub></font> = <font color="Maroon">c<sub>21</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<a NAME="E3:28_3_1"/>
not <font color="Maroon">c<sub>22</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>17</sub></font>
 
<b>by </b><i><a class="txt" href="waybel31.html#E17:28_3"><i><font color="Green">E34</font></i></a>, <a class="txt" href="waybel31.html#E36:28_3"><i><font color="Green">E44</font></i></a>, <a class="txt" href="waybel31.html#E1:28_3_1"><i><font color="Green">E46</font></i></a>, <a class="ref" href="waybel_0.html#D16">WAYBEL_0:def 16</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>23</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="E5:28_3_1"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 <b>and </b><br/><a NAME="E6:28_3_1"/><i><font color="Green">E48</font></i>: 
not <font color="Maroon">c<sub>23</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>17</sub></font>
 <b>and </b><br/><a NAME="E7:28_3_1"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">c<sub>23</sub></font> <a href="waybel_3.html#NR2" target="_self">&lt;&lt;</a> <font color="Maroon">c<sub>22</sub></font>
 <b>by </b><i><a class="ref" href="waybel23.html#T46">WAYBEL23:46</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>24</sub></font> = <font color="Maroon">c<sub>23</sub></font> as    <a href="hidden.html#M1">set</a>  ;<br/>
<b>take </b>
<font color="Maroon">c<sub>24</sub></font>
;<br/>

<b>take </b>
<font color="Maroon">c<sub>23</sub></font>
;<br/>

<b>take </b>
<font color="Maroon">c<sub>22</sub></font>
;<br/>

<b>thus </b><a NAME="E9:28_3_1"/>
( <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font> &amp; <font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>24</sub></font> &amp; <font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; not <font color="Maroon">c<sub>23</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>17</sub></font> &amp; <font color="Maroon">c<sub>23</sub></font> <a href="waybel_3.html#NR2" target="_self">&lt;&lt;</a> <font color="Maroon">c<sub>22</sub></font> )
 <b>by </b><i><a class="txt" href="waybel31.html#E5:28_3_1"><i><font color="Green">E47</font></i></a>, <a class="txt" href="waybel31.html#E6:28_3_1"><i><font color="Green">E48</font></i></a>, <a class="txt" href="waybel31.html#E7:28_3_1"><i><font color="Green">E49</font></i></a></i>;<br/>


</div><b>end;</b></div><b>consider </b><font color="Maroon">c<sub>21</sub></font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E39:28_3"/><i><font color="Green">E46</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font>
 <b>and </b><br/><a NAME="E40:28_3"/><i><font color="Green">E47</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>19</sub></font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>21</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>]
 <b>from </b><i><a class="ref" href="zfrefle1.html#S1">ZFREFLE1:sch 1</a>(<a class="txt" href="waybel31.html#E37:28_3"><i><font color="Green">E45</font></i></a>);<br/></i><a NAME="E41:28_3"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>21</sub></font> <a href="tarski.html#R1">c=</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="28_3_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>22</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="E1:28_3_2"/>
<font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>21</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>23</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:28_3_2"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>21</sub></font>
 <b>and </b><br/><a NAME="E4:28_3_2"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>23</sub></font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>24</sub></font>, <font color="Maroon">c<sub>25</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="E6:28_3_2"/>
<font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font>
 <b>and </b><br/><a NAME="E7:28_3_2"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>23</sub></font>
 <b>and </b><br/><a NAME="E8:28_3_2"/>
<font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 <b>and </b><br/><a NAME="E9:28_3_2"/>
not <font color="Maroon">c<sub>24</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>17</sub></font>
 <b>and </b><br/><a NAME="E10:28_3_2"/>
<font color="Maroon">c<sub>24</sub></font> <a href="waybel_3.html#NR2" target="_self">&lt;&lt;</a> <font color="Maroon">c<sub>25</sub></font>
 <b>by </b><i><a class="txt" href="waybel31.html#E39:28_3"><i><font color="Green">E46</font></i></a>, <a class="txt" href="waybel31.html#E40:28_3"><i><font color="Green">E47</font></i></a>, <a class="txt" href="waybel31.html#E3:28_3_2"><i><font color="Green">E48</font></i></a></i>;<br/>
<b>thus </b><a NAME="E11:28_3_2"/>
<font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R2">in</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="txt" href="waybel31.html#E4:28_3_2"><i><font color="Green">E49</font></i></a>, <a class="txt" href="waybel31.html#E7:28_3_2"><i><font color="Green">E50</font></i></a></i>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">c<sub>22</sub></font> =  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>21</sub></font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/><b>reconsider </b><font color="Maroon">c<sub>23</sub></font> = <span class="p1">(<span class="default"><a href="waybel_3.html#K2">wayabove</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K4">uparrow</a> <font color="Maroon">c<sub>22</sub></font></span>)</span> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="txt" href="waybel31.html#E1:28"><i><font color="Green">E22</font></i></a></i>;<br/><b>take </b><font color="Maroon">c<sub>24</sub></font> = <font color="Maroon">c<sub>23</sub></font>;<br/><a NAME="E44:28_3"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R1">=</a>  <a href="waybel31.html#K2" target="_self">Way_Up</a> <font color="Maroon">c<sub>20</sub></font>,<font color="Maroon">c<sub>22</sub></font>
 ;<br/><a NAME="E45:28_3"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">c<sub>22</sub></font> is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i><a class="txt" href="waybel31.html#E13:28_3"><i><font color="Green">E32</font></i></a>, <a class="txt" href="waybel31.html#E39:28_3"><i><font color="Green">E46</font></i></a>, <a class="ref" href="finset_1.html#T26">FINSET_1:26</a></i>;<br/><a NAME="E46:28_3"/>
<font color="Maroon">c<sub>22</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_3_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>25</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="E1:28_3_3"/>
<font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>22</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>26</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:28_3_3"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>21</sub></font>
 <b>and </b><br/><a NAME="E4:28_3_3"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>26</sub></font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>27</sub></font>, <font color="Maroon">c<sub>28</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="E6:28_3_3"/>
<font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>26</sub></font>
 <b>and </b><br/><a NAME="E7:28_3_3"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>26</sub></font>
 <b>and </b><br/><a NAME="E8:28_3_3"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 <b>and </b><br/><a NAME="E9:28_3_3"/>
not <font color="Maroon">c<sub>27</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>17</sub></font>
 <b>and </b><br/><a NAME="E10:28_3_3"/>
<font color="Maroon">c<sub>27</sub></font> <a href="waybel_3.html#NR2" target="_self">&lt;&lt;</a> <font color="Maroon">c<sub>28</sub></font>
 <b>by </b><i><a class="txt" href="waybel31.html#E39:28_3"><i><font color="Green">E46</font></i></a>, <a class="txt" href="waybel31.html#E40:28_3"><i><font color="Green">E47</font></i></a>, <a class="txt" href="waybel31.html#E3:28_3_3"><i><font color="Green">E50</font></i></a></i>;<br/>
<b>thus </b><a NAME="E11:28_3_3"/>
<font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="waybel31.html#E4:28_3_3"><i><font color="Green">E51</font></i></a>, <a class="txt" href="waybel31.html#E7:28_3_3"><i><font color="Green">E52</font></i></a>, <a class="txt" href="waybel31.html#E8:28_3_3"><i><font color="Green">E53</font></i></a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E47:28_3"/>
<font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="waybel31.html#E32:28_3"><i><font color="Green">E40</font></i></a>, <a class="txt" href="waybel31.html#E44:28_3"><i><font color="Green">E48</font></i></a>, <a class="txt" href="waybel31.html#E45:28_3"><i><font color="Green">E49</font></i></a></i>;<br/><a NAME="E48:28_3"/>
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 <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>22</sub></font> or  not <font color="Olive">b<sub>1</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>17</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_3_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>25</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>assume </b><a NAME="E1:28_3_4"/>
<font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>22</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>26</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:28_3_4"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>21</sub></font>
 <b>and </b><br/><a NAME="E4:28_3_4"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>26</sub></font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>27</sub></font>, <font color="Maroon">c<sub>28</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="E6:28_3_4"/>
<font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>26</sub></font>
 <b>and </b><br/><a NAME="E7:28_3_4"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>26</sub></font>
 <b>and </b><br/><a NAME="E8:28_3_4"/>
<font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 <b>and </b><br/><a NAME="E9:28_3_4"/><i><font color="Green">E53</font></i>: 
not <font color="Maroon">c<sub>27</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>17</sub></font>
 <b>and </b><br/><a NAME="E10:28_3_4"/>
<font color="Maroon">c<sub>27</sub></font> <a href="waybel_3.html#NR2" target="_self">&lt;&lt;</a> <font color="Maroon">c<sub>28</sub></font>
 <b>by </b><i><a class="txt" href="waybel31.html#E39:28_3"><i><font color="Green">E46</font></i></a>, <a class="txt" href="waybel31.html#E40:28_3"><i><font color="Green">E47</font></i></a>, <a class="txt" href="waybel31.html#E3:28_3_4"><i><font color="Green">E50</font></i></a></i>;<br/>
<b>thus </b><a NAME="E11:28_3_4"/>
not <font color="Maroon">c<sub>25</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>17</sub></font>
 <b>by </b><i><a class="txt" href="waybel31.html#E4:28_3_4"><i><font color="Green">E51</font></i></a>, <a class="txt" href="waybel31.html#E7:28_3_4"><i><font color="Green">E52</font></i></a>, <a class="txt" href="waybel31.html#E9:28_3_4"><i><font color="Green">E53</font></i></a></i>;<br/>


</div><b>end;</b></div><a NAME="E49:28_3"/><b>then </b>
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 <font color="Olive">b<sub>1</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>17</sub></font> or  not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>22</sub></font> )
 ;<br/><a NAME="E50:28_3"/><b>then </b>
not <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K4">uparrow</a> <font color="Maroon">c<sub>22</sub></font>
 <b>by </b><i><a class="ref" href="waybel_0.html#D16">WAYBEL_0:def 16</a></i>;<br/><b>hence </b><a NAME="E51:28_3"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>24</sub></font>
 <b>by </b><i><a class="txt" href="waybel31.html#E35:28_3"><i><font color="Green">E43</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E52:28_3"/><i><font color="Green">E50</font></i>: 
 <a href="waybel_3.html#K2">wayabove</a> <font color="Maroon">c<sub>20</sub></font> <a href="tarski.html#R1">c=</a>  <a href="waybel_3.html#K2">wayabove</a> <font color="Maroon">c<sub>18</sub></font>
 <b>by </b><i><a class="txt" href="waybel31.html#E33:28_3"><i><font color="Green">E41</font></i></a>, <a class="ref" href="waybel_3.html#T12">WAYBEL_3:12</a></i>;<br/><a NAME="E53:28_3"/>
 <a href="waybel_0.html#K4">uparrow</a> <font color="Maroon">c<sub>19</sub></font> <a href="tarski.html#R1">c=</a>  <a href="waybel_0.html#K4">uparrow</a> <font color="Maroon">c<sub>22</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_3_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>25</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="E1:28_3_5"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K4">uparrow</a> <font color="Maroon">c<sub>19</sub></font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>26</sub></font> = <font color="Maroon">c<sub>25</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<b>consider </b><font color="Maroon">c<sub>27</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="E4:28_3_5"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">c<sub>27</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>26</sub></font>
 <b>and </b><br/><a NAME="E5:28_3_5"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>19</sub></font>
 <b>by </b><i><a class="txt" href="waybel31.html#E1:28_3_5"><i><font color="Green">E51</font></i></a>, <a class="ref" href="waybel_0.html#D16">WAYBEL_0:def 16</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>28</sub></font>, <font color="Maroon">c<sub>29</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="E7:28_3_5"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">c<sub>29</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>27</sub></font>
 <b>and </b><br/><a NAME="E8:28_3_5"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>27</sub></font>
 <b>and </b><br/><a NAME="E9:28_3_5"/>
<font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 <b>and </b><br/><a NAME="E10:28_3_5"/>
not <font color="Maroon">c<sub>28</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>17</sub></font>
 <b>and </b><br/><a NAME="E11:28_3_5"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">c<sub>28</sub></font> <a href="waybel_3.html#NR2" target="_self">&lt;&lt;</a> <font color="Maroon">c<sub>29</sub></font>
 <b>by </b><i><a class="txt" href="waybel31.html#E40:28_3"><i><font color="Green">E47</font></i></a>, <a class="txt" href="waybel31.html#E5:28_3_5"><i><font color="Green">E53</font></i></a></i>;<br/>
<a NAME="E12:28_3_5"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>22</sub></font>
 
<b>by </b><i><a class="txt" href="waybel31.html#E39:28_3"><i><font color="Green">E46</font></i></a>, <a class="txt" href="waybel31.html#E5:28_3_5"><i><font color="Green">E53</font></i></a>, <a class="txt" href="waybel31.html#E8:28_3_5"><i><font color="Green">E55</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E13:28_3_5"/>
<font color="Maroon">c<sub>28</sub></font> <a href="waybel_3.html#NR2" target="_self">&lt;&lt;</a> <font color="Maroon">c<sub>26</sub></font>
 
<b>by </b><i><a class="txt" href="waybel31.html#E4:28_3_5"><i><font color="Green">E52</font></i></a>, <a class="txt" href="waybel31.html#E7:28_3_5"><i><font color="Green">E54</font></i></a>, <a class="txt" href="waybel31.html#E11:28_3_5"><i><font color="Green">E56</font></i></a>, <a class="ref" href="waybel_3.html#T2">WAYBEL_3:2</a></i>;<br/>
<a NAME="E14:28_3_5"/><b>then </b>
<font color="Maroon">c<sub>28</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>26</sub></font>
 
<b>by </b><i><a class="ref" href="waybel_3.html#T1">WAYBEL_3:1</a></i>;<br/>
<b>hence </b><a NAME="E15:28_3_5"/>
<font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K4">uparrow</a> <font color="Maroon">c<sub>22</sub></font>
 <b>by </b><i><a class="txt" href="waybel31.html#E12:28_3_5"><i><font color="Green">E57</font></i></a>, <a class="ref" href="waybel_0.html#D16">WAYBEL_0:def 16</a></i>;<br/>


</div><b>end;</b></div><a NAME="E54:28_3"/><b>then </b><i><font color="Green">E51</font></i>: 
<span class="p1">(<span class="default"><a href="waybel_3.html#K2">wayabove</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K4">uparrow</a> <font color="Maroon">c<sub>22</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="waybel_3.html#K2">wayabove</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K4">uparrow</a> <font color="Maroon">c<sub>19</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="waybel31.html#E52:28_3"><i><font color="Green">E50</font></i></a>, <a class="ref" href="xboole_1.html#T35">XBOOLE_1:35</a></i>;<br/><a NAME="E55:28_3"/>
 <a href="waybel_3.html#K2">wayabove</a> <font color="Maroon">c<sub>18</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>11</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_3_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>25</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="E1:28_3_6"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R2">in</a>  <a href="waybel_3.html#K2">wayabove</a> <font color="Maroon">c<sub>18</sub></font>
 ;<br/>

<a NAME="E2:28_3_6"/>
 <a href="waybel_3.html#K2">wayabove</a> <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a>  <a href="waybel_3.html#K2">wayabove</a> <font color="Maroon">c<sub>15</sub></font>
 
<b>by </b><i><a class="txt" href="waybel31.html#E1:28"><i><font color="Green">E22</font></i></a>, <a class="txt" href="waybel31.html#E4:28_3"><i><font color="Green">E26</font></i></a>, <a class="ref" href="yellow12.html#T13">YELLOW12:13</a></i>;<br/>
<a NAME="E3:28_3_6"/><b>then </b>
 <a href="waybel_3.html#K2">wayabove</a> <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="waybel_3.html#K2">wayabove</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>8</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>13</sub></font> </span>}</span> 
 
<b>by </b><i><a class="txt" href="waybel31.html#E23:28_3"><i><font color="Green">E38</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:28_3_6"/>
<font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="waybel31.html#E16:28_3"><i><font color="Green">E33</font></i></a>, <a class="txt" href="waybel31.html#E1:28_3_6"><i><font color="Green">E52</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>


</div><b>end;</b></div><a NAME="E56:28_3"/><b>then </b>
<span class="p1">(<span class="default"><a href="waybel_3.html#K2">wayabove</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K4">uparrow</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="waybel31.html#E11:28_3"><i><font color="Green">E30</font></i></a>, <a class="txt" href="waybel31.html#E36:28_3"><i><font color="Green">E44</font></i></a>, <a class="ref" href="xboole_1.html#T33">XBOOLE_1:33</a></i>;<br/><a NAME="E57:28_3"/><b>then </b>
<span class="p1">(<span class="default"><a href="waybel_3.html#K2">wayabove</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K4">uparrow</a> <font color="Maroon">c<sub>22</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="waybel31.html#E54:28_3"><i><font color="Green">E51</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><b>hence </b><a NAME="E58:28_3"/>
<font color="Maroon">c<sub>24</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="waybel31.html#E9:28_3"><i><font color="Green">E29</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:28"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="waybel31.html#K2" target="_self">Way_Up</a> <font color="Olive">b<sub>1</sub></font>,<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>, B is  <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM2">Subset</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> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font> ) </span>}</span>  is    <a href="cantor_1.html#M1">Basis</a> of <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="waybel31.html#E5:28"><i><font color="Green">E23</font></i></a>, <a class="ref" href="yellow_9.html#T32">YELLOW_9:32</a></i>;<br/>


</div>
