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

<b>let </b><font color="Maroon" title="c1">A</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:66"/><i><font color="Green" title="E40">A1</font></i>: 
<font color="Maroon" title="c1">A</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b> ex <font color="Olive" title="b1">p</font> being  <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> st <br/>(  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 &amp; <font color="Olive" title="b1">p</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> );<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1"><b>now </b></a><div class="add"><b>reconsider </b><font color="Maroon" title="c2">p</font> =  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  as   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> ;<br/><b>take </b><font color="Maroon" title="c3">p</font> = <font color="Maroon" title="c2">p</font>;<br/><b>thus </b><a NAME="E2:66_1"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="ref" href="finseq_1.html#T27" title="FINSEQ_1:th.27">FINSEQ_1:27</a></i>;<br/><b>thus </b><a NAME="E3:66_1"/>
<font color="Maroon" title="c3">p</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 ;<br/></div><b>end;</b></div>
<a NAME="E3:66"/><b>then </b><i><font color="Green" title="E41">A2</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ]
 
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c2">B</font>, <font color="Maroon" title="c3">C</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><b>that </b><br/><a NAME="E1:66_2"/>
<font color="Maroon" title="c2">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font>
 <b>and </b><br/><a NAME="E2:66_2"/>
<font color="Maroon" title="c3">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">A</font>
 ;<br/><b>given </b><font color="Maroon" title="c4">p</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a><b> such that </b><a NAME="E4:66_2"/><i><font color="Green" title="E42">A3</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">C</font>
 <b>and </b><br/><a NAME="E5:66_2"/><i><font color="Green" title="E43">A4</font></i>: 
<font color="Maroon" title="c4">p</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 ;<br/><div><i><font color="Green" title="E44">A5</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="66_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:66_2_1"/><i><font color="Green" title="E45">A6</font></i>: 
<font color="Maroon" title="c2">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">C</font>
 ;<br/><b>take </b><font color="Maroon" title="c5">q</font> = <font color="Maroon" title="c4">p</font>;<br/><b>thus </b><a NAME="E2:66_2_1"/>
(  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c5">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">C</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">B</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c5">q</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> )
 <b>by </b><i><a class="txt" href="finseq_4.html#E4:66_2"><i><font color="Green" title="E42">A3</font></i></a>, <a class="txt" href="finseq_4.html#E5:66_2"><i><font color="Green" title="E43">A4</font></i></a>, <a class="txt" href="finseq_4.html#E1:66_2_1"><i><font color="Green" title="E45">A6</font></i></a>, <a class="ref" href="zfmisc_1.html#T46" title="ZFMISC_1:th.46">ZFMISC_1:46</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:66_2_2"/><i><font color="Green" title="E45">A7</font></i>: 
not <font color="Maroon" title="c2">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">C</font>
 ;<br/><b>take </b><font color="Maroon" title="c5">q</font> = <font color="Maroon" title="c4">p</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c2">B</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>;<br/><b>thus </b> <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c5">q</font> = 
<span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">p</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c2">B</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T44" title="FINSEQ_1:th.44">FINSEQ_1:44</a></i>
<br/>.= 
<font color="Maroon" title="c3">C</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">B</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
<b>by </b><i><a class="txt" href="finseq_4.html#E4:66_2"><i><font color="Green" title="E42">A3</font></i></a>, <a class="ref" href="finseq_1.html#T55" title="FINSEQ_1:th.55">FINSEQ_1:55</a></i>
;<br/><b>thus </b><a NAME="E3:66_2_2"/>
<font color="Maroon" title="c5">q</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_2_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a><br/><b>let </b><font color="Maroon" title="c7">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:66_2_2_2"/><i><font color="Green" title="E46">A8</font></i>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">q</font>
 <b>and </b><br/><a NAME="E2:66_2_2_2"/><i><font color="Green" title="E47">A9</font></i>: 
<font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">q</font>
 <b>and </b><br/><a NAME="E3:66_2_2_2"/><i><font color="Green" title="E48">A10</font></i>: 
<font color="Maroon" title="c5">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">y</font>
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c8">k</font> = <font color="Maroon" title="c6">x</font>, <font color="Maroon" title="c9">l</font> = <font color="Maroon" title="c7">y</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="finseq_4.html#E1:66_2_2_2"><i><font color="Green" title="E46">A8</font></i></a>, <a class="txt" href="finseq_4.html#E2:66_2_2_2"><i><font color="Green" title="E47">A9</font></i></a></i>;<br/>
<div><i><font color="Green" title="E49">A11</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="66_2_2_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:66_2_2_2_1"/><i><font color="Green" title="E50">A12</font></i>: 
( <font color="Maroon" title="c8">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">p</font> &amp; <font color="Maroon" title="c9">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">p</font> )
 ;<br/><a NAME="E2:66_2_2_2_1"/><b>then </b>
( <font color="Maroon" title="c5">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">k</font> &amp; <font color="Maroon" title="c5">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">l</font> )
 <b>by </b><i><a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></i>;<br/><b>hence </b><a NAME="E3:66_2_2_2_1"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">y</font>
 <b>by </b><i><a class="txt" href="finseq_4.html#E5:66_2"><i><font color="Green" title="E43">A4</font></i></a>, <a class="txt" href="finseq_4.html#E3:66_2_2_2"><i><font color="Green" title="E48">A10</font></i></a>, <a class="txt" href="finseq_4.html#E1:66_2_2_2_1"><i><font color="Green" title="E50">A12</font></i></a>, <a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green" title="E50">A13</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="66_2_2_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:66_2_2_2_2"/><i><font color="Green" title="E51">A14</font></i>: 
<font color="Maroon" title="c8">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">p</font>
 ;<br/><b>given </b><font color="Maroon" title="c10">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><a NAME="E3:66_2_2_2_2"/><i><font color="Green" title="E52">A15</font></i>: 
<font color="Maroon" title="c10">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c2">B</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>
 <b>and </b><br/><a NAME="E4:66_2_2_2_2"/><i><font color="Green" title="E53">A16</font></i>: 
<font color="Maroon" title="c9">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">p</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c10">n</font>
 ;<br/><a NAME="E5:66_2_2_2_2"/>
<font color="Maroon" title="c10">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="finseq_4.html#E3:66_2_2_2_2"><i><font color="Green" title="E52">A15</font></i></a>, <a class="ref" href="finseq_1.html#T4" title="FINSEQ_1:th.4">FINSEQ_1:4</a>, <a class="ref" href="finseq_1.html#T55" title="FINSEQ_1:th.55">FINSEQ_1:55</a></i>;<br/><a NAME="E6:66_2_2_2_2"/><b>then </b><i><font color="Green" title="E54">A17</font></i>: 
<font color="Maroon" title="c10">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c2">B</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">n</font> = 
<font color="Maroon" title="c5">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">k</font>
<b>by </b><i><a class="txt" href="finseq_4.html#E3:66_2_2_2"><i><font color="Green" title="E48">A10</font></i></a>, <a class="txt" href="finseq_4.html#E3:66_2_2_2_2"><i><font color="Green" title="E52">A15</font></i></a>, <a class="txt" href="finseq_4.html#E4:66_2_2_2_2"><i><font color="Green" title="E53">A16</font></i></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></i>
<br/>.= 
<font color="Maroon" title="c4">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">k</font>
<b>by </b><i><a class="txt" href="finseq_4.html#E1:66_2_2_2_2"><i><font color="Green" title="E51">A14</font></i></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></i>
;<br/><a NAME="E8:66_2_2_2_2"/><b>then </b>
<font color="Maroon" title="c2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">k</font>
 <b>by </b><i><a class="txt" href="finseq_4.html#E6:66_2_2_2_2"><i><font color="Green" title="E54">A17</font></i></a>, <a class="ref" href="finseq_1.html#D8" title="FINSEQ_1:def.8">FINSEQ_1:def 8</a></i>;<br/><b>hence </b><a NAME="E9:66_2_2_2_2"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">y</font>
 <b>by </b><i><a class="txt" href="finseq_4.html#E4:66_2"><i><font color="Green" title="E42">A3</font></i></a>, <a class="txt" href="finseq_4.html#E1:66_2_2"><i><font color="Green" title="E45">A7</font></i></a>, <a class="txt" href="finseq_4.html#E1:66_2_2_2_2"><i><font color="Green" title="E51">A14</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green" title="E51">A18</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="66_2_2_2_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:66_2_2_2_3"/><i><font color="Green" title="E52">A19</font></i>: 
<font color="Maroon" title="c9">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">p</font>
 ;<br/><b>given </b><font color="Maroon" title="c10">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><a NAME="E3:66_2_2_2_3"/><i><font color="Green" title="E53">A20</font></i>: 
<font color="Maroon" title="c10">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c2">B</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>
 <b>and </b><br/><a NAME="E4:66_2_2_2_3"/><i><font color="Green" title="E54">A21</font></i>: 
<font color="Maroon" title="c8">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">p</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c10">n</font>
 ;<br/><a NAME="E5:66_2_2_2_3"/>
<font color="Maroon" title="c10">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="finseq_4.html#E3:66_2_2_2_3"><i><font color="Green" title="E53">A20</font></i></a>, <a class="ref" href="finseq_1.html#T4" title="FINSEQ_1:th.4">FINSEQ_1:4</a>, <a class="ref" href="finseq_1.html#T55" title="FINSEQ_1:th.55">FINSEQ_1:55</a></i>;<br/><a NAME="E6:66_2_2_2_3"/><b>then </b><i><font color="Green" title="E55">A22</font></i>: 
<font color="Maroon" title="c10">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c2">B</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">n</font> = 
<font color="Maroon" title="c5">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">l</font>
<b>by </b><i><a class="txt" href="finseq_4.html#E3:66_2_2_2"><i><font color="Green" title="E48">A10</font></i></a>, <a class="txt" href="finseq_4.html#E3:66_2_2_2_3"><i><font color="Green" title="E53">A20</font></i></a>, <a class="txt" href="finseq_4.html#E4:66_2_2_2_3"><i><font color="Green" title="E54">A21</font></i></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></i>
<br/>.= 
<font color="Maroon" title="c4">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">l</font>
<b>by </b><i><a class="txt" href="finseq_4.html#E1:66_2_2_2_3"><i><font color="Green" title="E52">A19</font></i></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></i>
;<br/><a NAME="E8:66_2_2_2_3"/><b>then </b>
<font color="Maroon" title="c2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">l</font>
 <b>by </b><i><a class="txt" href="finseq_4.html#E6:66_2_2_2_3"><i><font color="Green" title="E55">A22</font></i></a>, <a class="ref" href="finseq_1.html#D8" title="FINSEQ_1:def.8">FINSEQ_1:def 8</a></i>;<br/><b>hence </b><a NAME="E9:66_2_2_2_3"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">y</font>
 <b>by </b><i><a class="txt" href="finseq_4.html#E4:66_2"><i><font color="Green" title="E42">A3</font></i></a>, <a class="txt" href="finseq_4.html#E1:66_2_2"><i><font color="Green" title="E45">A7</font></i></a>, <a class="txt" href="finseq_4.html#E1:66_2_2_2_3"><i><font color="Green" title="E52">A19</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_2_2_2_4"><b>now </b></a><div class="add"><b>given </b><font color="Maroon" title="c10">m1</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><a NAME="E2:66_2_2_2_4"/><i><font color="Green" title="E52">A23</font></i>: 
<font color="Maroon" title="c10">m1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c2">B</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>
 <b>and </b><br/><a NAME="E3:66_2_2_2_4"/><i><font color="Green" title="E53">A24</font></i>: 
<font color="Maroon" title="c8">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">p</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c10">m1</font>
 ;<br/><b>given </b><font color="Maroon" title="c11">m2</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><a NAME="E5:66_2_2_2_4"/><i><font color="Green" title="E54">A25</font></i>: 
<font color="Maroon" title="c11">m2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c2">B</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>
 <b>and </b><br/><a NAME="E6:66_2_2_2_4"/><i><font color="Green" title="E55">A26</font></i>: 
<font color="Maroon" title="c9">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">p</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c11">m2</font>
 ;<br/><a NAME="E7:66_2_2_2_4"/>
( <font color="Maroon" title="c10">m1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c11">m2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 <b>by </b><i><a class="txt" href="finseq_4.html#E2:66_2_2_2_4"><i><font color="Green" title="E52">A23</font></i></a>, <a class="txt" href="finseq_4.html#E5:66_2_2_2_4"><i><font color="Green" title="E54">A25</font></i></a>, <a class="ref" href="finseq_1.html#T4" title="FINSEQ_1:th.4">FINSEQ_1:4</a>, <a class="ref" href="finseq_1.html#D8" title="FINSEQ_1:def.8">FINSEQ_1:def 8</a></i>;<br/><a NAME="E8:66_2_2_2_4"/><b>then </b>
( <font color="Maroon" title="c10">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c11">m2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E9:66_2_2_2_4"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">y</font>
 <b>by </b><i><a class="txt" href="finseq_4.html#E3:66_2_2_2_4"><i><font color="Green" title="E53">A24</font></i></a>, <a class="txt" href="finseq_4.html#E6:66_2_2_2_4"><i><font color="Green" title="E55">A26</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E9:66_2_2_2"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">y</font>
 <b>by </b><i><a class="txt" href="finseq_4.html#E1:66_2_2_2"><i><font color="Green" title="E46">A8</font></i></a>, <a class="txt" href="finseq_4.html#E2:66_2_2_2"><i><font color="Green" title="E47">A9</font></i></a>, <a class="txt" href="finseq_4.html#E5:66_2_2_2"><i><font color="Green" title="E49">A11</font></i></a>, <a class="txt" href="finseq_4.html#E6:66_2_2_2"><i><font color="Green" title="E50">A13</font></i></a>, <a class="txt" href="finseq_4.html#E7:66_2_2_2"><i><font color="Green" title="E51">A18</font></i></a>, <a class="ref" href="finseq_1.html#T38" title="FINSEQ_1:th.38">FINSEQ_1:38</a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E8:66_2"/>
 ex <font color="Olive" title="b1">p</font> being  <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> st <br/>(  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">C</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">B</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Olive" title="b1">p</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> )
 <b>by </b><i><a class="txt" href="finseq_4.html#E6:66_2"><i><font color="Green" title="E44">A5</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E5:66"/><b>then </b><i><font color="Green" title="E42">A27</font></i>: 
for <font color="Olive" title="b1">B</font>, <font color="Olive" title="b2">C</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Olive" title="b2">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">C</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">C</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b1">B</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>]
 
;<br/>
<b>thus </b><a NAME="E6:66"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c1">A</font>]
 <b>from </b><i><a class="ref" href="finset_1.html#S2" title="FINSET_1:sch.2">FINSET_1:sch 2</a>(<a class="txt" href="finseq_4.html#E1:66"><i><font color="Green" title="E40">A1</font></i></a>, <a class="txt" href="finseq_4.html#E3:66"><i><font color="Green" title="E41">A2</font></i></a>, <a class="txt" href="finseq_4.html#E5:66"><i><font color="Green" title="E42">A27</font></i></a>);</i><br/>


</div>
