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

<b>let </b><font color="Maroon">TS</font> be   <a href="pre_topc.html#NM1">TopSpace</a>;<br/>

<b>assume </b><a NAME="E1:12"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">TS</font> <a href="compts_1.html#NR2" target="_self">is_T2</a> 
 ;<br/>

<b>let </b><font color="Maroon">F</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">TS</font>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E2:12"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">F</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>and </b><br/><a NAME="E3:12"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">F</font> is <a href="compts_1.html#V6" target="_self">compact</a>
 ;<br/>

<b>let </b><font color="Maroon">a</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">TS</font>;<br/>

<b>assume </b><a NAME="E4:12"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> <a href="subset_1.html#K3">`</a> 
 ;<br/>

<a NAME="E5:12"/><i><font color="Green">E36</font></i>: 
not <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</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> ,   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">PS</font>, <font color="Olive">QS</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">TS</font> st <br/>( <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">PS</font> &amp; <font color="Maroon">a<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">QS</font> &amp; <font color="Olive">PS</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Olive">QS</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Olive">PS</font> &amp; <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">QS</font> &amp; <font color="Olive">PS</font> <a href="subset_1.html#K9">/\</a> <font color="Olive">QS</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  );<br/>
<a NAME="E6:12"/><i><font color="Green">E37</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">F</font> holds <br/> ex <font color="Olive">y</font>, <font color="Olive">z</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">y</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">TS</font> &amp; <font color="Olive">z</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">TS</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>,<font color="Olive">z</font>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_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="E1:12_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">p</font> = <font color="Maroon">x</font> as   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">TS</font> ;<br/>
<b>consider </b><font color="Maroon">W</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">TS</font>, <font color="Maroon">V</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">TS</font><b> such that </b><br/><a NAME="E4:12_1"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">W</font> is <a href="pre_topc.html#V3">open</a>
 <b>and </b><br/><a NAME="E5:12_1"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">V</font> is <a href="pre_topc.html#V3">open</a>
 <b>and </b><br/><a NAME="E6:12_1"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font>
 <b>and </b><br/><a NAME="E7:12_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font>
 <b>and </b><br/><a NAME="E8:12_1"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">W</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">V</font>
 <b>by </b><i>, , , </i>;<br/>
<b>reconsider </b><font color="Maroon">PS</font> = <font color="Maroon">W</font>, <font color="Maroon">QS</font> = <font color="Maroon">V</font> as    <a href="hidden.html#M1">set</a>  ;<br/>
<b>take </b>
<font color="Maroon">PS</font>
;<br/>

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

<b>thus </b><a NAME="E10:12_1"/>
( <font color="Maroon">PS</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">TS</font> &amp; <font color="Maroon">QS</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">TS</font> )
 <b>by </b><i><a class="ref" href="compts_1.html#D1" target="_self">Def1</a>, <a class="ref" href="compts_1.html#D2" target="_self">Def2</a>, <a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/>

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

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

<b>thus </b><a NAME="E11:12_1"/>
( <font color="Maroon">PS</font> <a href="hidden.html#R1">=</a> <font color="Maroon">W</font> &amp; <font color="Maroon">QS</font> <a href="hidden.html#R1">=</a> <font color="Maroon">V</font> &amp; <font color="Maroon">W</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">V</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font> &amp; <font color="Maroon">W</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">V</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 <b>by </b><i><a class="ref" href="compts_1.html#D1" target="_self">Def1</a>, <a class="ref" href="compts_1.html#D2" target="_self">Def2</a>, , <a class="ref" href="compts_1.html#D3" target="_self">Def3</a>, <a class="ref" href="compts_1.html#D4" target="_self">Def4</a>, <a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">f</font> being   <a href="funct_1.html#NM1">Function</a>, <font color="Maroon">g</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E8:12"/><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">F</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> )
 <b>and </b><br/><a NAME="E9:12"/><i><font color="Green">E50</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">F</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>,<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font>]
 <b>from </b><i><a class="ref" href="wellord2.html#S2">WELLORD2:sch 2</a>();<br/></i>
<a NAME="E10:12"/>
<font color="Maroon">g</font> <a href="relat_1.html#K9">.:</a> <font color="Maroon">F</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 <font color="Maroon">TS</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</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="E1:12_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">g</font> <a href="relat_1.html#K9">.:</a> <font color="Maroon">F</font>
 ;<br/>

<b>then consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:12_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font>
 <b>and </b><br/><a NAME="E4:12_2"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 <b>and </b><br/><a NAME="E5:12_2"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<a NAME="E6:12_2"/>
 ex <font color="Olive">PS</font>, <font color="Olive">QS</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">TS</font> st <br/>( <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">PS</font> &amp; <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">QS</font> &amp; <font color="Olive">PS</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Olive">QS</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Olive">PS</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Olive">QS</font> &amp; <font color="Olive">PS</font> <a href="subset_1.html#K9">/\</a> <font color="Olive">QS</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 
<b>by </b><i><a class="ref" href="compts_1.html#D1" target="_self">Def1</a>, <a class="ref" href="compts_1.html#D2" target="_self">Def2</a></i>;<br/>
<b>hence </b><a NAME="E7:12_2"/>
<font color="Maroon">x</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">TS</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">C</font> = <font color="Maroon">g</font> <a href="relat_1.html#K9">.:</a> <font color="Maroon">F</font> as   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">TS</font> ;<br/>
<a NAME="E12:12"/>
<font color="Maroon">F</font> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">C</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</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="E1:12_3"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 ;<br/>

<b>then consider </b><font color="Maroon">PS</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">TS</font>, <font color="Maroon">QS</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">TS</font><b> such that </b><br/><a NAME="E3:12_3"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">PS</font>
 <b>and </b><br/><a NAME="E4:12_3"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">QS</font>
 <b>and </b><br/><a NAME="E5:12_3"/>
<font color="Maroon">PS</font> is <a href="pre_topc.html#V3">open</a>
 <b>and </b><br/><a NAME="E6:12_3"/>
<font color="Maroon">QS</font> is <a href="pre_topc.html#V3">open</a>
 <b>and </b><br/><a NAME="E7:12_3"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">PS</font>
 <b>and </b><br/><a NAME="E8:12_3"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">QS</font>
 <b>and </b><br/><a NAME="E9:12_3"/>
<font color="Maroon">PS</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">QS</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="compts_1.html#D1" target="_self">Def1</a></i>;<br/>
<a NAME="E10:12_3"/>
<font color="Maroon">QS</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 
<b>by </b><i>, <a class="ref" href="compts_1.html#D2" target="_self">Def2</a>, , <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<b>hence </b><a NAME="E11:12_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">C</font>
 <b>by </b><i><a class="ref" href="compts_1.html#D3" target="_self">Def3</a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E13:12"/><b>then </b><i><font color="Green">E58</font></i>: 
<font color="Maroon">C</font> <a href="compts_1.html#R1" target="_self">is_a_cover_of</a> <font color="Maroon">F</font>
 
<b>by </b><i/>;<br/>
<a NAME="E14:12"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">C</font> is <a href="tops_2.html#V1">open</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">G</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">TS</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="tops_2.html#D1">TOPS_2:def 1</a><br/>

<b>assume </b><a NAME="E1:12_4"/>
<font color="Maroon">G</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:12_4"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font>
 <b>and </b><br/><a NAME="E4:12_4"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 <b>and </b><br/><a NAME="E5:12_4"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">G</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<a NAME="E6:12_4"/>
 ex <font color="Olive">PS</font>, <font color="Olive">QS</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">TS</font> st <br/>( <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">PS</font> &amp; <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">QS</font> &amp; <font color="Olive">PS</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Olive">QS</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Olive">PS</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">QS</font> &amp; <font color="Olive">PS</font> <a href="subset_1.html#K9">/\</a> <font color="Olive">QS</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 
<b>by </b><i><a class="ref" href="compts_1.html#D1" target="_self">Def1</a>, <a class="ref" href="compts_1.html#D3" target="_self">Def3</a></i>;<br/>
<b>hence </b><a NAME="E7:12_4"/>
<font color="Maroon">G</font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i><a class="ref" href="compts_1.html#D4" target="_self">Def4</a></i>;<br/>


</div><b>end;</b></div>
<b>then consider </b><font color="Maroon">C'</font> being   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">TS</font><b> such that </b><br/><a NAME="E16:12"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">C'</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">C</font>
 <b>and </b><br/><a NAME="E17:12"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">C'</font> <a href="compts_1.html#R1" target="_self">is_a_cover_of</a> <font color="Maroon">F</font>
 <b>and </b><br/><a NAME="E18:12"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">C'</font> is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i>, <a class="ref" href="compts_1.html#D2" target="_self">Def2</a>, </i>;<br/>
<a NAME="E19:12"/>
<font color="Maroon">C'</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font>
 
<b>by </b><i>, <a class="ref" href="compts_1.html#D3" target="_self">Def3</a>, <a class="ref" href="relat_1.html#T146">RELAT_1:146</a></i>;<br/>
<b>then consider </b><font color="Maroon">H'</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E21:12"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">H'</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font>
 <b>and </b><br/><a NAME="E22:12"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">H'</font> is <a href="finset_1.html#V1">finite</a>
 <b>and </b><br/><a NAME="E23:12"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">g</font> <a href="relat_1.html#K9">.:</a> <font color="Maroon">H'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C'</font>
 <b>by </b><i>, <a class="ref" href="orders_1.html#T195">ORDERS_1:195</a></i>;<br/>
<a NAME="E24:12"/>
<font color="Maroon">f</font> <a href="relat_1.html#K9">.:</a> <font color="Maroon">H'</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 <font color="Maroon">TS</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</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="E1:12_5"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relat_1.html#K9">.:</a> <font color="Maroon">H'</font>
 ;<br/>

<b>then consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:12_5"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font>
 <b>and </b><br/><a NAME="E4:12_5"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H'</font>
 <b>and </b><br/><a NAME="E5:12_5"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<a NAME="E6:12_5"/>
 ex <font color="Olive">PS</font>, <font color="Olive">QS</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">TS</font> st <br/>( <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">PS</font> &amp; <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">QS</font> &amp; <font color="Olive">PS</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Olive">QS</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Olive">PS</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Olive">QS</font> &amp; <font color="Olive">PS</font> <a href="subset_1.html#K9">/\</a> <font color="Olive">QS</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 
<b>by </b><i>, <a class="ref" href="compts_1.html#D1" target="_self">Def1</a>, , <a class="ref" href="compts_1.html#D7" target="_self">Def7</a></i>;<br/>
<b>hence </b><a NAME="E7:12_5"/>
<font color="Maroon">x</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">TS</font>
 <b>by </b><i><a class="ref" href="compts_1.html#T9" target="_self">Th9</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">B</font> = <font color="Maroon">f</font> <a href="relat_1.html#K9">.:</a> <font color="Maroon">H'</font> as   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">TS</font> ;<br/>
<b>take </b><font color="Maroon">G0</font> =  <a href="setfam_1.html#K6">meet</a> <font color="Maroon">B</font>;<br/>

<b>take </b><font color="Maroon">G1</font> =  <a href="setfam_1.html#K5">union</a> <font color="Maroon">C'</font>;<br/>

<a NAME="E26:12"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">B</font> is <a href="finset_1.html#V1">finite</a>
 
<b>by </b><i>, <a class="ref" href="finset_1.html#T17">FINSET_1:17</a></i>;<br/>
<a NAME="E27:12"/>
<font color="Maroon">B</font> is <a href="tops_2.html#V1">open</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">G</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">TS</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="tops_2.html#D1">TOPS_2:def 1</a><br/>

<b>assume </b><a NAME="E1:12_6"/>
<font color="Maroon">G</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:12_6"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font>
 <b>and </b><br/><a NAME="E4:12_6"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H'</font>
 <b>and </b><br/><a NAME="E5:12_6"/><i><font color="Green">E92</font></i>: 
<font color="Maroon">G</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<a NAME="E6:12_6"/>
 ex <font color="Olive">PS</font>, <font color="Olive">QS</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">TS</font> st <br/>( <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">PS</font> &amp; <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">QS</font> &amp; <font color="Olive">PS</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Olive">QS</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Olive">PS</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">QS</font> &amp; <font color="Olive">PS</font> <a href="subset_1.html#K9">/\</a> <font color="Olive">QS</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 
<b>by </b><i>, <a class="ref" href="compts_1.html#D1" target="_self">Def1</a>, <a class="ref" href="compts_1.html#T9" target="_self">Th9</a></i>;<br/>
<b>hence </b><a NAME="E7:12_6"/>
<font color="Maroon">G</font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E28:12"/>
<font color="Maroon">G0</font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i><a class="ref" href="compts_1.html#D7" target="_self">Def7</a>, <a class="ref" href="tops_2.html#T27">TOPS_2:27</a></i>;<br/>

<a NAME="E29:12"/>
<font color="Maroon">C'</font> is <a href="tops_2.html#V1">open</a>
 
<b>by </b><i>, <a class="ref" href="compts_1.html#D3" target="_self">Def3</a>, <a class="ref" href="tops_2.html#T18">TOPS_2:18</a></i>;<br/>
<b>hence </b><a NAME="E30:12"/>
<font color="Maroon">G1</font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i><a class="ref" href="tops_2.html#T26">TOPS_2:26</a></i>;<br/>

<b>consider </b><font color="Maroon">z</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F</font>;<br/>
<a NAME="E32:12"/>
<font color="Maroon">F</font> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">C'</font>
 
<b>by </b><i><a class="ref" href="compts_1.html#D4" target="_self">Def4</a>, </i>;<br/>
<a NAME="E33:12"/><b>then </b>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">C'</font>
 
<b>by </b><i>, <a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon">D</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E35:12"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font>
 <b>and </b><br/><a NAME="E36:12"/><i><font color="Green">E94</font></i>: 
<font color="Maroon">D</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C'</font>
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>reconsider </b><font color="Maroon">D'</font> = <font color="Maroon">D</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">TS</font> <b>by </b><i><a class="ref" href="compts_1.html#T9" target="_self">Th9</a></i>;<br/>
<b>consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E39:12"/><i><font color="Green">E96</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font>
 <b>and </b><br/><a NAME="E40:12"/><i><font color="Green">E97</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H'</font>
 <b>and </b><br/><a NAME="E41:12"/>
<font color="Maroon">D'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font>
 <b>by </b><i>, <a class="ref" href="compts_1.html#T9" target="_self">Th9</a>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<b>consider </b><font color="Maroon">PS</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">TS</font>, <font color="Maroon">QS</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">TS</font><b> such that </b><br/><a NAME="E43:12"/><i><font color="Green">E98</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">PS</font>
 <b>and </b><br/><a NAME="E44:12"/>
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">QS</font>
 <b>and </b><br/><a NAME="E45:12"/>
<font color="Maroon">PS</font> is <a href="pre_topc.html#V3">open</a>
 <b>and </b><br/><a NAME="E46:12"/>
<font color="Maroon">QS</font> is <a href="pre_topc.html#V3">open</a>
 <b>and </b><br/><a NAME="E47:12"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">PS</font>
 <b>and </b><br/><a NAME="E48:12"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">QS</font>
 <b>and </b><br/><a NAME="E49:12"/>
<font color="Maroon">PS</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">QS</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i>, <a class="ref" href="compts_1.html#D1" target="_self">Def1</a>, </i>;<br/>
<a NAME="E50:12"/><i><font color="Green">E99</font></i>: 
<font color="Maroon">B</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i>, , <a class="ref" href="compts_1.html#T10" target="_self">Th10</a>, <a class="txt" href="compts_1.html#E1:12"><i><font color="Green">E32</font></i></a>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<a NAME="E51:12"/>
for <font color="Olive">G</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">G</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> holds <br/><font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Olive">G</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_7"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:12_7"/><i><font color="Green">E100</font></i>: 
<font color="Maroon">G</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">G'</font> = <font color="Maroon">G</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">TS</font> ;<br/>
<b>consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:12_7"/><i><font color="Green">E101</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font>
 <b>and </b><br/><a NAME="E5:12_7"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H'</font>
 <b>and </b><br/><a NAME="E6:12_7"/><i><font color="Green">E102</font></i>: 
<font color="Maroon">G'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="txt" href="compts_1.html#E3:12"><i><font color="Green">E34</font></i></a>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<a NAME="E7:12_7"/>
 ex <font color="Olive">PS</font>, <font color="Olive">QS</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">TS</font> st <br/>( <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">PS</font> &amp; <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">QS</font> &amp; <font color="Olive">PS</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Olive">QS</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Olive">PS</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">QS</font> &amp; <font color="Olive">PS</font> <a href="subset_1.html#K9">/\</a> <font color="Olive">QS</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 
<b>by </b><i>, <a class="ref" href="compts_1.html#D1" target="_self">Def1</a>, <a class="txt" href="compts_1.html#E4:12"><i><font color="Green">E35</font></i></a></i>;<br/>
<b>hence </b><a NAME="E8:12_7"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font>
 <b>by </b><i><a class="txt" href="compts_1.html#E5:12"><i><font color="Green">E36</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E52:12"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G0</font>
 <b>by </b><i><a class="txt" href="compts_1.html#E2:12"><i><font color="Green">E33</font></i></a>, <a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/>

<b>thus </b><a NAME="E53:12"/>
<font color="Maroon">F</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">G1</font>
 <b>by </b><i><a class="ref" href="compts_1.html#D4" target="_self">Def4</a>, </i>;<br/>

<b>thus </b><a NAME="E54:12"/>
<font color="Maroon">G0</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">G1</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_8"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:12_8"/><i><font color="Green">E103</font></i>: 
<font color="Maroon">G0</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">G1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<b>consider </b><font color="Maroon">x</font> being    <a href="subset_1.html#M1">Element</a> of <span class="p1">(<span class="default"><a href="setfam_1.html#K6">meet</a> <font color="Maroon">B</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5">union</a> <font color="Maroon">C'</font></span>)</span>;<br/>
<a NAME="E3:12_8"/><i><font color="Green">E104</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">C'</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K6">meet</a> <font color="Maroon">B</font> )
 
<b>by </b><i><a class="txt" href="compts_1.html#E3:12"><i><font color="Green">E34</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon">A</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:12_8"/><i><font color="Green">E105</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>and </b><br/><a NAME="E6:12_8"/><i><font color="Green">E106</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C'</font>
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">z</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E8:12_8"/><i><font color="Green">E107</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font>
 <b>and </b><br/><a NAME="E9:12_8"/><i><font color="Green">E108</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H'</font>
 <b>and </b><br/><a NAME="E10:12_8"/><i><font color="Green">E109</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font>
 <b>by </b><i>, <a class="txt" href="compts_1.html#E6:12"><i><font color="Green">E37</font></i></a>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<b>consider </b><font color="Maroon">PS</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">TS</font>, <font color="Maroon">QS</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">TS</font><b> such that </b><br/><a NAME="E12:12_8"/><i><font color="Green">E110</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">PS</font>
 <b>and </b><br/><a NAME="E13:12_8"/><i><font color="Green">E111</font></i>: 
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">QS</font>
 <b>and </b><br/><a NAME="E14:12_8"/>
<font color="Maroon">PS</font> is <a href="pre_topc.html#V3">open</a>
 <b>and </b><br/><a NAME="E15:12_8"/>
<font color="Maroon">QS</font> is <a href="pre_topc.html#V3">open</a>
 <b>and </b><br/><a NAME="E16:12_8"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">PS</font>
 <b>and </b><br/><a NAME="E17:12_8"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">QS</font>
 <b>and </b><br/><a NAME="E18:12_8"/><i><font color="Green">E112</font></i>: 
<font color="Maroon">PS</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">QS</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i>, <a class="ref" href="compts_1.html#D1" target="_self">Def1</a>, </i>;<br/>
<a NAME="E19:12_8"/>
<font color="Maroon">PS</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 
<b>by </b><i>, , , , <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<a NAME="E20:12_8"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">QS</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">PS</font> )
 
<b>by </b><i><a class="txt" href="compts_1.html#E4:12"><i><font color="Green">E35</font></i></a>, <a class="txt" href="compts_1.html#E5:12"><i><font color="Green">E36</font></i></a>, , , <a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/>
<b>hence </b><a NAME="E21:12_8"/>
contradiction
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>


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


</div>
