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

<b>let </b><font color="Maroon">T</font> be  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a>, <font color="Maroon">S</font> be  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a>;<br/><b>let </b><font color="Maroon">f</font> be   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">T</font>,<font color="Maroon">S</font>;<br/><b>let </b><font color="Maroon">Q</font> be   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">S</font>;<br/>





<b>assume </b><a NAME="E1:13"/>
<font color="Maroon">Q</font> is <a href="finset_1.html#V1">finite</a>
 ;<br/>

<b>then consider </b><font color="Maroon">s</font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E3:13"/><i><font color="Green">E30</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Q</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T73">FINSEQ_1:73</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">S</font></span>)</span>,  <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font></span>)</span>] <b>means </b>for <font color="Olive">s</font>, <font color="Olive">t</font> being   <a href="hidden.html#M1">set</a>   st <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">s</font> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">t</font> holds <br/><font color="Olive">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Olive">s</font>;<br/>
<a NAME="E4:13"/><i><font color="Green">E32</font></i>: 
for <font color="Olive">x</font> being  <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">S</font></span>)</span>  ex <font color="Olive">y</font> being  <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font></span>)</span> st <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="13_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">S</font></span>)</span>;<br/>

<b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as    <a href="hidden.html#M1">set</a>  ;<br/>
<b>set </b><font color="Maroon">y</font> = <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">x</font>;<br/>
<b>reconsider </b><font color="Maroon">y</font> = <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">x</font> as   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font></span>)</span> ;<br/>
<b>take </b>
<font color="Maroon">y</font>
;<br/>

<b>thus </b><a NAME="E3:13_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">x</font>,<font color="Maroon">y</font>]
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">F</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="pcomps_1.html#K1">bool</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">S</font></span>)</span>, <a href="pcomps_1.html#K1">bool</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font></span>)</span><b> such that </b><br/><a NAME="E6:13"/><i><font color="Green">E33</font></i>: 
for <font color="Olive">x</font> being  <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">S</font></span>)</span> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3">FUNCT_2:sch 3</a>();<br/></i>
<a NAME="E7:13"/>
( <font color="Maroon">Q</font> <a href="tarski.html#R1">c=</a>  <a href="pcomps_1.html#K1">bool</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">S</font></span>)</span> &amp;  <a href="relset_1.html#K4">dom</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="pcomps_1.html#K1">bool</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">S</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">q</font> = <font color="Maroon">F</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">s</font> as   <a href="finseq_1.html#NM1">FinSequence</a> <b>by </b><i>, <a class="ref" href="finseq_1.html#T20">FINSEQ_1:20</a></i>;<br/>
<a NAME="E9:13"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">F</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">Q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="weierstr.html#K1" target="_self">"</a> <font color="Maroon">Q</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_2"><b>proof </b></a><div class="add">

<a NAME="E1:13_2"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">Q</font> iff <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="weierstr.html#K1" target="_self">"</a> <font color="Maroon">Q</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_2_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>thus </b><a NAME="E1:13_2_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">Q</font> implies <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="weierstr.html#K1" target="_self">"</a> <font color="Maroon">Q</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_2_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:13_2_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">Q</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="E3:13_2_1_1"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">F</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<b>reconsider </b><font color="Maroon">y</font> = <font color="Maroon">y</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">S</font> <b>by </b><i/>;<br/>
<a NAME="E5:13_2_1_1"/>
<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">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">y</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E6:13_2_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="weierstr.html#K1" target="_self">"</a> <font color="Maroon">Q</font>
 <b>by </b><i>, </i>;<br/>


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

<b>assume </b><a NAME="E2:13_2_1"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="weierstr.html#K1" target="_self">"</a> <font color="Maroon">Q</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> ;<br/>
<b>consider </b><font color="Maroon">y</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">S</font><b> such that </b><br/><a NAME="E5:13_2_1"/><i><font color="Green">E37</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">y</font> )
 <b>by </b><i>, </i>;<br/>
<a NAME="E6:13_2_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E7:13_2_1"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="pcomps_1.html#K1">bool</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">S</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<b>hence </b><a NAME="E8:13_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">Q</font>
 <b>by </b><i>, , <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:13_2"/>
<font color="Maroon">F</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">Q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="weierstr.html#K1" target="_self">"</a> <font color="Maroon">Q</font>
 <b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E10:13"/>
 ex <font color="Olive">q</font> being  <a href="finseq_1.html#NM1">FinSequence</a> st  <a href="relat_1.html#K2">rng</a> <font color="Olive">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="weierstr.html#K1" target="_self">"</a> <font color="Maroon">Q</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_3"><b>proof </b></a><div class="add">

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

<b>thus </b><a NAME="E1:13_3"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="weierstr.html#K1" target="_self">"</a> <font color="Maroon">Q</font>
 <b>by </b><i>, , <a class="ref" href="relat_1.html#T160">RELAT_1:160</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E11:13"/>
<font color="Maroon">f</font> <a href="weierstr.html#K1" target="_self">"</a> <font color="Maroon">Q</font> is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i><a class="ref" href="finseq_1.html#T73">FINSEQ_1:73</a></i>;<br/>


</div>
