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<b>let </b><font color="Maroon">L1</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">T</font> be  <a href="waybel11.html#V4">Scott</a>  <a href="yellow_9.html#M1">TopAugmentation</a> of <font color="Maroon">L1</font>;<br/><b>let </b><font color="Maroon">B1</font> be    <a href="cantor_1.html#M1">Basis</a> of <font color="Maroon">T</font>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:31"/><i><font color="Green">E40</font></i>: 
not <font color="Maroon">B1</font> is <a href="finset_1.html#V1">finite</a>
 <b>and </b><br/><a NAME="E2:31"/><i><font color="Green">E41</font></i>: 
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_0.html#NK4" target="_self">inf</a> <font color="Olive">u</font></span>)</span> where <font color="Olive">u</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : <font color="Olive">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B1</font> </span>}</span>  is <a href="finset_1.html#V1">finite</a>
 ;<br/>

<b>reconsider </b><font color="Maroon">B2</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_0.html#NK4" target="_self">inf</a> <font color="Olive">u</font></span>)</span> where <font color="Olive">u</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : <font color="Olive">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B1</font> </span>}</span>  as    <a href="hidden.html#M1">set</a>  ;<br/>
<b>reconsider </b><font color="Maroon">B3</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="waybel_3.html#K2">wayabove</a> <span class="p3">(<span class="default"><a href="yellow_0.html#NK4" target="_self">inf</a> <font color="Olive">u</font></span>)</span></span>)</span> where <font color="Olive">u</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : <font color="Olive">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B1</font> </span>}</span>  as    <a href="cantor_1.html#M1">Basis</a> of <font color="Maroon">T</font> <b>by </b><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> ] <b>means </b> ex <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font> st <br/>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">y</font> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="waybel_3.html#K2">wayabove</a> <font color="Olive">y</font> );<br/>
<a NAME="E5:31"/><i><font color="Green">E43</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B2</font> holds <br/> ex <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B3</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:31_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B2</font>
 ;<br/>

<b>then consider </b><font color="Maroon">u1</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E3:31_1"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#NK4" target="_self">inf</a> <font color="Maroon">u1</font>
 <b>and </b><br/><a NAME="E4:31_1"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">u1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B1</font>
 ;<br/>
<b>reconsider </b><font color="Maroon">z</font> = <font color="Maroon">x</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font> <b>by </b><i/>;<br/>
<b>take </b><font color="Maroon">y</font> =  <a href="waybel_3.html#K2">wayabove</a> <font color="Maroon">z</font>;<br/>

<b>thus </b><a NAME="E6:31_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B3</font>
 <b>by </b><i>, </i>;<br/>

<b>take </b>
<font color="Maroon">z</font>
;<br/>

<b>thus </b><a NAME="E7:31_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="waybel_3.html#K2">wayabove</a> <font color="Maroon">z</font> )
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">f</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E7:31"/><i><font color="Green">E49</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B2</font>
 <b>and </b><br/><a NAME="E8:31"/><i><font color="Green">E50</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B3</font>
 <b>and </b><br/><a NAME="E9:31"/><i><font color="Green">E52</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B2</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font>]
 <b>from </b><i><a class="ref" href="wellord2.html#S1">WELLORD2:sch 1</a>();<br/></i>
<a NAME="E10:31"/>
<font color="Maroon">B3</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">z</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="E1:31_2"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B3</font>
 ;<br/>

<b>then consider </b><font color="Maroon">u1</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E3:31_2"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="waybel_3.html#K2">wayabove</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK4" target="_self">inf</a> <font color="Maroon">u1</font></span>)</span>
 <b>and </b><br/><a NAME="E4:31_2"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">u1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B1</font>
 ;<br/>
<a NAME="E5:31_2"/>
 <a href="yellow_0.html#NK4" target="_self">inf</a> <font color="Maroon">u1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B2</font>
 
<b>by </b><i/>;<br/>
<b>then consider </b><font color="Maroon">y</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E7:31_2"/><i><font color="Green">E55</font></i>: 
 <a href="yellow_0.html#NK4" target="_self">inf</a> <font color="Maroon">u1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 <b>and </b><br/><a NAME="E8:31_2"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK4" target="_self">inf</a> <font color="Maroon">u1</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="waybel_3.html#K2">wayabove</a> <font color="Maroon">y</font>
 <b>by </b><i/>;<br/>
<a NAME="E9:31_2"/>
( <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK4" target="_self">inf</a> <font color="Maroon">u1</font></span>)</span> &amp;  <a href="yellow_0.html#NK4" target="_self">inf</a> <font color="Maroon">u1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B2</font> )
 
<b>by </b><i>, , , </i>;<br/>
<b>hence </b><a NAME="E10:31_2"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E11:31"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B3</font>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>
<a NAME="E12:31"/><b>then </b>
<font color="Maroon">B3</font> is <a href="finset_1.html#V1">finite</a>
 
<b>by </b><i>, , <a class="ref" href="finset_1.html#T26">FINSET_1:26</a></i>;<br/>
<a NAME="E13:31"/><b>then </b>
<font color="Maroon">T</font> is <a href="group_1.html#V6">finite</a>
 
<b>by </b><i><a class="ref" href="yellow15.html#T31">YELLOW15:31</a></i>;<br/>
<a NAME="E14:31"/><b>then </b>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font> is <a href="finset_1.html#V1">finite</a>
 
<b>by </b><i><a class="ref" href="group_1.html#D14">GROUP_1:def 14</a></i>;<br/>
<b>hence </b><a NAME="E15:31"/>
contradiction
 <b>by </b><i>, <a class="ref" href="yellow15.html#T3">YELLOW15:3</a></i>;<br/>


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