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

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b><font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<a NAME="E1:6_1_1"/><i><font color="Green">E33</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">S</font></span>)</span> holds <br/> ex <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="genealg1.html#NK1" target="_self">GA-Space</a> <font color="Maroon">S</font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>,<font color="Olive">x</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:6_1_1_1"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">S</font></span>)</span>
 ;<br/>

<a NAME="E2:6_1_1_1"/><b>then </b><i><font color="Green">E35</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">S</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">k'</font> = <font color="Maroon">k</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">S</font> <b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E4:6_1_1_1"/>
<font color="Maroon">S</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
;<br/>
<a NAME="E5:6_1_1_1"/><b>then </b>
 <a href="subset_1.html#K2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <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="subset_1.html#D4">SUBSET_1:def 4</a></i>;<br/>
<b>then consider </b><font color="Maroon">x</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">S</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font><b> such that </b><br/><a NAME="E7:6_1_1_1"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="subset_1.html#K2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font></span>)</span>
 <b>by </b><i><a class="ref" href="subset_1.html#T10">SUBSET_1:10</a></i>;<br/>
<a NAME="E8:6_1_1_1"/>
<font color="Maroon">S</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">S</font>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E9:6_1_1_1"/><b>then </b>
 <a href="subset_1.html#K2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">S</font>
 
<b>by </b><i><a class="ref" href="subset_1.html#D4">SUBSET_1:def 4</a></i>;<br/>
<a NAME="E10:6_1_1_1"/><b>then </b>
 <a href="subset_1.html#K2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">S</font></span>)</span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T92">ZFMISC_1:92</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="genealg1.html#NK1" target="_self">GA-Space</a> <font color="Maroon">S</font> <b>by </b><i>, <a class="ref" href="card_3.html#D4">CARD_3:def 4</a></i>;<br/>
<b>take </b>
<font color="Maroon">x</font>
;<br/>

<b>thus </b><a NAME="E12:6_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font>,<font color="Maroon">x</font>]
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">IT</font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="genealg1.html#NK1" target="_self">GA-Space</a> <font color="Maroon">S</font><b> such that </b><br/><a NAME="E3:6_1_1"/><i><font color="Green">E37</font></i>: 
(  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">IT</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">S</font></span>)</span> &amp; ( for <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">S</font></span>)</span> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>,<font color="Maroon">IT</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font>] ) )
 <b>from </b><i><a class="ref" href="finseq_1.html#S5">FINSEQ_1:sch 5</a>();<br/></i>
<b>take </b>
<font color="Maroon">IT</font>
;<br/>

<b>thus </b><a NAME="E4:6_1_1"/>
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">IT</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">S</font> &amp; ( for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">IT</font> holds <br/><font color="Maroon">IT</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> ) )
 <b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>


</div>
