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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be  <a href="waybel_3.html#V4">non-Empty</a> <a href="waybel18.html#V1">TopSpace-yielding</a>  <a href="pboole.html#M1">ManySortedSet</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="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="waybel18.html#K2">product_prebasis</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>;<br/>







<b>assume </b><b>that </b><br/><a NAME="E1:24"/><i><font color="Green">E24</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <font color="Maroon">c<sub>2</sub></font> <a href="waybel18.html#K4">.</a> <font color="Olive">b<sub>1</sub></font> is <a href="compts_1.html#V2">compact</a>
 <b>and </b><br/><a NAME="E2:24"/><i><font color="Green">E25</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>4</sub></font> holds not  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><a href="waybel18.html#K3">product</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="tarski.html#K3">union</a> <font color="Olive">b<sub>1</sub></font>
 ;<br/>

<b>assume </b><a NAME="E3:24"/><i><font color="Green">E26</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="waybel18.html#K4">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>ex <font color="Olive">b<sub>2</sub></font> being <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>4</sub></font> st <span class="p1">(<span class="default"><a href="waybel18.html#K6">proj</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">b<sub>1</sub></font></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="tarski.html#K3">union</a> <font color="Olive">b<sub>2</sub></font>
 ;<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 ( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">a<sub>2</sub></font> &amp; <span class="p1">(<span class="default"><a href="waybel18.html#K6">proj</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="relset_1.html#K11">"</a> <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="waybel18.html#K4">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>2</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> is <a href="pre_topc.html#V3">open</a> ) );<br/>
<a NAME="E4:24"/><i><font color="Green">E27</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> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="waybel18.html#K4">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> holds <br/>ex <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">b<sub>2</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 <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="waybel18.html#K4">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> &amp; <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="24_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:24_1"/>
<font color="Maroon">c<sub>5</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"><font color="Maroon">c<sub>2</sub></font> <a href="waybel18.html#K4">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>5</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="waybel18.html#K4">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> ;<br/>
<b>consider </b><font color="Maroon">c<sub>7</sub></font> being  <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>4</sub></font><b> such that </b><br/><a NAME="E4:24_1"/><i><font color="Green">E28</font></i>: 
<span class="p1">(<span class="default"><a href="waybel18.html#K6">proj</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="yellow17.html#E3:24"><i><font color="Green">E26</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>8</sub></font> being   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="waybel18.html#K4">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span><b> such that </b><br/><a NAME="E6:24_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="waybel18.html#K4">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E7:24_1"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font>
 <b>and </b><br/><a NAME="E8:24_1"/><i><font color="Green">E30</font></i>: 
<span class="p1">(<span class="default"><a href="waybel18.html#K6">proj</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font>
 <b>and </b><br/><a NAME="E9:24_1"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>8</sub></font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i><a class="txt" href="yellow17.html#E2:24"><i><font color="Green">E25</font></i></a>, <a class="txt" href="yellow17.html#E4:24_1"><i><font color="Green">E28</font></i></a>, <a class="ref" href="yellow17.html#T22" target="_self">Th22</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>8</sub></font>
;<br/>

<b>thus </b><a NAME="E10:24_1"/>
<font color="Maroon">c<sub>8</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 <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="waybel18.html#K4">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 ;<br/>

<b>thus </b><a NAME="E11:24_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="yellow17.html#E7:24_1"><i><font color="Green">E29</font></i></a></i>;<br/>

<b>thus </b><a NAME="E12:24_1"/>
<span class="p1">(<span class="default"><a href="waybel18.html#K6">proj</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="yellow17.html#E8:24_1"><i><font color="Green">E30</font></i></a></i>;<br/>

<b>let </b><font color="Maroon">c<sub>9</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="waybel18.html#K4">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>;<br/>

<b>assume </b><a NAME="E13:24_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/>

<b>hence </b><a NAME="E14:24_1"/>
<font color="Maroon">c<sub>9</sub></font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i><a class="txt" href="yellow17.html#E9:24_1"><i><font color="Green">E31</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E6:24"/><i><font color="Green">E28</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="waybel18.html#K4">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E7:24"/><i><font color="Green">E29</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>5</sub></font> <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 <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="waybel18.html#K4">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E8:24"/><i><font color="Green">E30</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> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="waybel18.html#K4">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>5</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="wellord2.html#S1">WELLORD2:sch 1</a>(<a class="txt" href="yellow17.html#E4:24"><i><font color="Green">E27</font></i></a>);<br/></i>
<b>reconsider </b><font color="Maroon">c<sub>6</sub></font> =  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>5</sub></font> as   <a href="struct_0.html#NM3">Subset-Family</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="waybel18.html#K4">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <b>by </b><i><a class="txt" href="yellow17.html#E7:24"><i><font color="Green">E29</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>6</sub></font> as   <a href="struct_0.html#NM3">Subset-Family</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="waybel18.html#K4">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> ;<br/>
<a NAME="E11:24"/><i><font color="Green">E31</font></i>: 
 <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="waybel18.html#K4">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">c<sub>7</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>8</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="E1:24_2"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="waybel18.html#K4">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 ;<br/>

<a NAME="E2:24_2"/><b>then </b><i><font color="Green">E33</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="txt" href="yellow17.html#E8:24"><i><font color="Green">E30</font></i></a></i>;<br/>
<a NAME="E3:24_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="yellow17.html#E6:24"><i><font color="Green">E28</font></i></a>, <a class="txt" href="yellow17.html#E1:24_2"><i><font color="Green">E32</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E4:24_2"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="yellow17.html#E2:24_2"><i><font color="Green">E33</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E12:24"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="waybel18.html#K4">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> is <a href="pre_topc.html#V3">open</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>8</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="waybel18.html#K4">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>;<br/>

<b>assume </b><a NAME="E1:24_3"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>9</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:24_3"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>5</sub></font>
 <b>and </b><br/><a NAME="E4:24_3"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>thus </b><a NAME="E5:24_3"/>
<font color="Maroon">c<sub>8</sub></font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i><a class="txt" href="yellow17.html#E6:24"><i><font color="Green">E28</font></i></a>, <a class="txt" href="yellow17.html#E8:24"><i><font color="Green">E30</font></i></a>, <a class="txt" href="yellow17.html#E3:24_3"><i><font color="Green">E32</font></i></a>, <a class="txt" href="yellow17.html#E4:24_3"><i><font color="Green">E33</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E13:24"/><b>then </b><i><font color="Green">E32</font></i>: 
<font color="Maroon">c<sub>7</sub></font> is <a href="tops_2.html#V1">open</a>
 
<b>by </b><i><a class="ref" href="tops_2.html#D1">TOPS_2:def 1</a></i>;<br/>
<a NAME="E14:24"/>
<font color="Maroon">c<sub>2</sub></font> <a href="waybel18.html#K4">.</a> <font color="Maroon">c<sub>3</sub></font> is <a href="compts_1.html#V2">compact</a>
 
<b>by </b><i><a class="txt" href="yellow17.html#E1:24"><i><font color="Green">E24</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>8</sub></font> being   <a href="struct_0.html#NM3">Subset-Family</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="waybel18.html#K4">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span><b> such that </b><br/><a NAME="E16:24"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>7</sub></font>
 <b>and </b><br/><a NAME="E17:24"/><i><font color="Green">E34</font></i>: 
 <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="waybel18.html#K4">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">c<sub>8</sub></font>
 <b>and </b><br/><a NAME="E18:24"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">c<sub>8</sub></font> is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i><a class="txt" href="yellow17.html#E11:24"><i><font color="Green">E31</font></i></a>, <a class="txt" href="yellow17.html#E13:24"><i><font color="Green">E32</font></i></a>, <a class="ref" href="yellow17.html#T15" target="_self">Th15</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>8</sub></font> as  non <a href="xboole_0.html#V1">empty</a> <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM3">Subset-Family</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="waybel18.html#K4">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <b>by </b><i><a class="txt" href="yellow17.html#E17:24"><i><font color="Green">E34</font></i></a>, <a class="txt" href="yellow17.html#E18:24"><i><font color="Green">E35</font></i></a>, <a class="ref" href="xboole_1.html#T3">XBOOLE_1:3</a>, <a class="ref" href="zfmisc_1.html#T2">ZFMISC_1:2</a></i>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of  <a href="zfmisc_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel18.html#K3">product</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> = <span class="p1">(<span class="default"><a href="waybel18.html#K6">proj</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="relset_1.html#K11">"</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1">set</a> ] means verum;<br/>
<b>set </b><font color="Maroon">c<sub>10</sub></font> = <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>) where B is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c<sub>9</sub></font> : <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>] </span>}</span> ;<br/>
<a NAME="E20:24"/><i><font color="Green">E36</font></i>: 
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>) where B is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c<sub>9</sub></font> : <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>] </span>}</span>  is <a href="finset_1.html#V1">finite</a>
 
<b>from </b><i><a class="ref" href="pre_circ.html#S1">PRE_CIRC:sch 1</a>();<br/></i>
<a NAME="E21:24"/>
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>) where B is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c<sub>9</sub></font> : <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>] </span>}</span>  <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_4"><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="E1:24_4"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>) where B is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c<sub>9</sub></font> : <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>] </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>12</sub></font> being    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c<sub>9</sub></font><b> such that </b><br/><a NAME="E3:24_4"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="waybel18.html#K6">proj</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>12</sub></font>
 ;<br/>
<a NAME="E4:24_4"/>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font>
 
;<br/>
<b>then consider </b><font color="Maroon">c<sub>13</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:24_4"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>5</sub></font>
 <b>and </b><br/><a NAME="E7:24_4"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="txt" href="yellow17.html#E16:24"><i><font color="Green">E33</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>thus </b><a NAME="E8:24_4"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="yellow17.html#E6:24"><i><font color="Green">E28</font></i></a>, <a class="txt" href="yellow17.html#E8:24"><i><font color="Green">E30</font></i></a>, <a class="txt" href="yellow17.html#E3:24_4"><i><font color="Green">E37</font></i></a>, <a class="txt" href="yellow17.html#E6:24_4"><i><font color="Green">E38</font></i></a>, <a class="txt" href="yellow17.html#E7:24_4"><i><font color="Green">E39</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>11</sub></font> = <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>) where B is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c<sub>9</sub></font> : <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>] </span>}</span>  as  <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>4</sub></font> <b>by </b><i><a class="txt" href="yellow17.html#E20:24"><i><font color="Green">E36</font></i></a></i>;<br/>
<a NAME="E23:24"/>
 <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><a href="waybel18.html#K3">product</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">c<sub>11</sub></font>
 
<b>by </b><i><a class="txt" href="yellow17.html#E17:24"><i><font color="Green">E34</font></i></a>, <a class="ref" href="yellow17.html#T19" target="_self">Th19</a></i>;<br/>
<b>hence </b><a NAME="E24:24"/>
contradiction
 <b>by </b><i><a class="txt" href="yellow17.html#E2:24"><i><font color="Green">E25</font></i></a></i>;<br/>


</div>
