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

<b>let </b><font color="Maroon" title="c1">T</font>, <font color="Maroon" title="c2">S</font> be  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>;<br/><b>let </b><font color="Maroon" title="c3">f</font> be   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">T</font>,<font color="Maroon" title="c2">S</font>;<br/><b>let </b><font color="Maroon" title="c4">P</font> be   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">T</font>;<br/>





<b>assume </b><a NAME="E1:14"/>
<font color="Maroon" title="c4">P</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c5">s</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a><b> such that </b><br/><a NAME="E3:14"/><i><font color="Green" title="E7">A1</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c5">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">P</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T73" title="FINSEQ_1:th.73">FINSEQ_1:73</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">T</font></span>)</span>,  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c2">S</font></span>)</span>] <b>means </b>for <font color="Olive" title="b1">s</font>, <font color="Olive" title="b2">t</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">s</font> &amp; $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">t</font> holds <br/><font color="Olive" title="b2">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Olive" title="b1">s</font>;<br/>
<a NAME="E4:14"/><i><font color="Green" title="E8">A2</font></i>: 
for <font color="Olive" title="b1">x</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">T</font></span>)</span>  ex <font color="Olive" title="b2">y</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c2">S</font></span>)</span> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">x</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">T</font></span>)</span>;<br/>

<b>reconsider </b><font color="Maroon" title="c7">x</font> = <font color="Maroon" title="c6">x</font> as    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ;<br/>
<b>set </b><font color="Maroon" title="c8">y</font> = <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c7">x</font>;<br/>
<b>reconsider </b><font color="Maroon" title="c9">y</font> = <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c7">x</font> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c2">S</font></span>)</span> ;<br/>
<b>take </b>
<font color="Maroon" title="c9">y</font>
;<br/>

<b>thus </b><a NAME="E3:14_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c9">y</font>]
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c6">F</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">T</font></span>)</span>, <a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c2">S</font></span>)</span><b> such that </b><br/><a NAME="E6:14"/><i><font color="Green" title="E9">A3</font></i>: 
for <font color="Olive" title="b1">x</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">T</font></span>)</span> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c6">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3" title="FUNCT_2:sch.3">FUNCT_2:sch 3</a>(<a class="txt" href="weierstr.html#E4:14"><i><font color="Green" title="E8">A2</font></i></a>);<br/></i>
<a NAME="E7:14"/>
( <font color="Maroon" title="c4">P</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">T</font></span>)</span> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">T</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c7">q</font> = <font color="Maroon" title="c6">F</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c5">s</font> as   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> <b>by </b><i><a class="txt" href="weierstr.html#E3:14"><i><font color="Green" title="E7">A1</font></i></a>, <a class="ref" href="finseq_1.html#T20" title="FINSEQ_1:th.20">FINSEQ_1:20</a></i>;<br/>
<a NAME="E9:14"/><i><font color="Green" title="E10">A4</font></i>: 
<font color="Maroon" title="c6">F</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c4">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="weierstr.html#K2" title="WEIERSTR:func.2">.:</a> <font color="Maroon" title="c4">P</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_2"><b>proof </b></a><div class="add">

<a NAME="E1:14_2"/>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">F</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c4">P</font> iff <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="weierstr.html#K2" title="WEIERSTR:func.2">.:</a> <font color="Maroon" title="c4">P</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>thus </b><a NAME="E1:14_2_1"/>
( <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">F</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c4">P</font> implies <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="weierstr.html#K2" title="WEIERSTR:func.2">.:</a> <font color="Maroon" title="c4">P</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_2_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:14_2_1_1"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">F</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c4">P</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c9">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:14_2_1_1"/><i><font color="Green" title="E11">A5</font></i>: 
( <font color="Maroon" title="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">F</font> &amp; <font color="Maroon" title="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">P</font> &amp; <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">y</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c10">y</font> = <font color="Maroon" title="c9">y</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> <b>by </b><i><a class="txt" href="weierstr.html#E3:14_2_1_1"><i><font color="Green" title="E11">A5</font></i></a></i>;<br/>
<a NAME="E5:14_2_1_1"/>
<font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c10">y</font>
 
<b>by </b><i><a class="txt" href="weierstr.html#E6:14"><i><font color="Green" title="E9">A3</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:14_2_1_1"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="weierstr.html#K2" title="WEIERSTR:func.2">.:</a> <font color="Maroon" title="c4">P</font>
 <b>by </b><i><a class="txt" href="weierstr.html#E3:14_2_1_1"><i><font color="Green" title="E11">A5</font></i></a>, <a class="ref" href="weierstr.html#D2" target="_self" title="WEIERSTR:def.2">Def2</a></i>;<br/>


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

<b>thus </b><a NAME="E2:14_2_1"/>
( <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="weierstr.html#K2" title="WEIERSTR:func.2">.:</a> <font color="Maroon" title="c4">P</font> implies <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">F</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c4">P</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_2_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:14_2_1_2"/><i><font color="Green" title="E11">A6</font></i>: 
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="weierstr.html#K2" title="WEIERSTR:func.2">.:</a> <font color="Maroon" title="c4">P</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c9">x</font> = <font color="Maroon" title="c8">x</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">S</font> ;<br/>
<b>consider </b><font color="Maroon" title="c10">y</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font><b> such that </b><br/><a NAME="E4:14_2_1_2"/><i><font color="Green" title="E12">A7</font></i>: 
( <font color="Maroon" title="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">P</font> &amp; <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c10">y</font> )
 <b>by </b><i><a class="txt" href="weierstr.html#E1:14_2_1_2"><i><font color="Green" title="E11">A6</font></i></a>, <a class="ref" href="weierstr.html#D2" target="_self" title="WEIERSTR:def.2">Def2</a></i>;<br/>
<a NAME="E5:14_2_1_2"/><i><font color="Green" title="E13">A8</font></i>: 
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">y</font>
 
<b>by </b><i><a class="txt" href="weierstr.html#E6:14"><i><font color="Green" title="E9">A3</font></i></a>, <a class="txt" href="weierstr.html#E4:14_2_1_2"><i><font color="Green" title="E12">A7</font></i></a></i>;<br/>
<a NAME="E6:14_2_1_2"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">T</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<b>hence </b><a NAME="E7:14_2_1_2"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">F</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c4">P</font>
 <b>by </b><i><a class="txt" href="weierstr.html#E4:14_2_1_2"><i><font color="Green" title="E12">A7</font></i></a>, <a class="txt" href="weierstr.html#E5:14_2_1_2"><i><font color="Green" title="E13">A8</font></i></a>, <a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/>


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


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:14_2"/>
<font color="Maroon" title="c6">F</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c4">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="weierstr.html#K2" title="WEIERSTR:func.2">.:</a> <font color="Maroon" title="c4">P</font>
 <b>by </b><i><a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E10:14"/>
 ex <font color="Olive" title="b1">q</font> being  <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> st  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b1">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="weierstr.html#K2" title="WEIERSTR:func.2">.:</a> <font color="Maroon" title="c4">P</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_3"><b>proof </b></a><div class="add">

<b>take </b>
<font color="Maroon" title="c7">q</font>
;<br/>

<b>thus </b><a NAME="E1:14_3"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="weierstr.html#K2" title="WEIERSTR:func.2">.:</a> <font color="Maroon" title="c4">P</font>
 <b>by </b><i><a class="txt" href="weierstr.html#E3:14"><i><font color="Green" title="E7">A1</font></i></a>, <a class="txt" href="weierstr.html#E9:14"><i><font color="Green" title="E10">A4</font></i></a>, <a class="ref" href="relat_1.html#T160" title="RELAT_1:th.160">RELAT_1:160</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E11:14"/>
<font color="Maroon" title="c3">f</font> <a href="weierstr.html#K2" title="WEIERSTR:func.2">.:</a> <font color="Maroon" title="c4">P</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 <b>by </b><i><a class="ref" href="finseq_1.html#T73" title="FINSEQ_1:th.73">FINSEQ_1:73</a></i>;<br/>


</div>
