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

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <b>means </b>for <font color="Olive" title="b1">X</font> being <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <br/> for <font color="Olive" title="b2">z</font> being   <a href="eqrel_1.html#M1" title="EQREL_1:mode.1">a_partition</a> of <font color="Olive" title="b1">X</font>  st  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b2">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 holds <br/>for <font color="Olive" title="b3">f</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Olive" title="b2">z</font>  st <font color="Olive" title="b3">f</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Olive" title="b3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">z</font> holds <br/>for <font color="Olive" title="b4">c</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b4">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b3">f</font> &amp; ( for <font color="Olive" title="b5">i</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b5">i</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="Olive" title="b4">c</font> holds <br/><font color="Olive" title="b4">c</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Olive" title="b5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b5">i</font></span>)</span> ) holds <br/> <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Olive" title="b1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Sum</a> <font color="Olive" title="b4">c</font>;<br/>
<a NAME="E1:6"/><i><font color="Green" title="E7">A1</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c1">X</font> be  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c2">z</font> be    <a href="eqrel_1.html#M1" title="EQREL_1:mode.1">a_partition</a> of <font color="Maroon" title="c1">X</font>;<br/>



<b>assume </b><a NAME="E1:6_1"/><i><font color="Green" title="E8">A2</font></i>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/>

<b>let </b><font color="Maroon" title="c3">f</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c2">z</font>;<br/>

<b>assume </b><a NAME="E2:6_1"/><i><font color="Green" title="E9">A3</font></i>: 
( <font color="Maroon" title="c3">f</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">z</font> )
 ;<br/>

<b>let </b><font color="Maroon" title="c4">c</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E3:6_1"/><i><font color="Green" title="E10">A4</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font> &amp; ( for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">i</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">c</font> holds <br/><font color="Maroon" title="c4">c</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span> ) )
 ;<br/>

<a NAME="E4:6_1"/><i><font color="Green" title="E11">A5</font></i>: 
<font color="Maroon" title="c2">z</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="txt" href="weddwitt.html#E1:6_1"><i><font color="Green" title="E8">A2</font></i></a>, <a class="ref" href="graph_5.html#T5" title="GRAPH_5:th.5">GRAPH_5:5</a></i>;<br/>
<a NAME="E5:6_1"/>
 <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c2">z</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="txt" href="weddwitt.html#E1:6_1"><i><font color="Green" title="E8">A2</font></i></a>, <a class="ref" href="graph_5.html#T5" title="GRAPH_5:th.5">GRAPH_5:5</a>, <a class="ref" href="zfmisc_1.html#T2" title="ZFMISC_1:th.2">ZFMISC_1:2</a></i>;<br/>
<a NAME="E6:6_1"/><b>then </b><i><font color="Green" title="E12">A6</font></i>: 
<font color="Maroon" title="c1">X</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="eqrel_1.html#D6" title="EQREL_1:def.6">EQREL_1:def 6</a></i>;<br/>
<a NAME="E7:6_1"/>
<font color="Maroon" title="c3">f</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="txt" href="weddwitt.html#E2:6_1"><i><font color="Green" title="E9">A3</font></i></a>, <a class="txt" href="weddwitt.html#E4:6_1"><i><font color="Green" title="E11">A5</font></i></a>, <a class="ref" href="finseq_1.html#T27" title="FINSEQ_1:th.27">FINSEQ_1:27</a></i>;<br/>
<b>hence </b><a NAME="E8:6_1"/>
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Sum</a> <font color="Maroon" title="c4">c</font>
 <b>by </b><i><a class="txt" href="weddwitt.html#E3:6_1"><i><font color="Green" title="E10">A4</font></i></a>, <a class="txt" href="weddwitt.html#E6:6_1"><i><font color="Green" title="E12">A6</font></i></a>, <a class="ref" href="card_1.html#T78" title="CARD_1:th.78">CARD_1:78</a>, <a class="ref" href="finseq_1.html#T25" title="FINSEQ_1:th.25">FINSEQ_1:25</a>, <a class="ref" href="rvsum_1.html#T102" title="RVSUM_1:th.102">RVSUM_1:102</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:6"/><i><font color="Green" title="E8">A7</font></i>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c1">k</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:6_2"/><i><font color="Green" title="E9">A8</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c1">k</font>]
 ;<br/>

<b>let </b><font color="Maroon" title="c2">X</font> be  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c3">Z</font> be    <a href="eqrel_1.html#M1" title="EQREL_1:mode.1">a_partition</a> of <font color="Maroon" title="c2">X</font>;<br/>



<b>assume </b><a NAME="E2:6_2"/><i><font color="Green" title="E10">A9</font></i>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c3">Z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ;<br/>

<b>let </b><font color="Maroon" title="c4">f</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c3">Z</font>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E3:6_2"/><i><font color="Green" title="E11">A10</font></i>: 
<font color="Maroon" title="c4">f</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 <b>and </b><br/><a NAME="E4:6_2"/><i><font color="Green" title="E12">A11</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">Z</font>
 ;<br/>

<b>let </b><font color="Maroon" title="c5">c</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E5:6_2"/><i><font color="Green" title="E13">A12</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font>
 <b>and </b><br/><a NAME="E6:6_2"/><i><font color="Green" title="E14">A13</font></i>: 
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">i</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">c</font> holds <br/><font color="Maroon" title="c5">c</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span>
 ;<br/>

<a NAME="E7:6_2"/><i><font color="Green" title="E15">A14</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<b>by </b><i><a class="txt" href="weddwitt.html#E2:6_2"><i><font color="Green" title="E10">A9</font></i></a>, <a class="txt" href="weddwitt.html#E3:6_2"><i><font color="Green" title="E11">A10</font></i></a>, <a class="txt" href="weddwitt.html#E4:6_2"><i><font color="Green" title="E12">A11</font></i></a>, <a class="ref" href="finseq_4.html#T77" title="FINSEQ_4:th.77">FINSEQ_4:77</a></i>;<br/>
<a NAME="E8:6_2"/><i><font color="Green" title="E16">A15</font></i>: 
<font color="Maroon" title="c3">Z</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="txt" href="weddwitt.html#E2:6_2"><i><font color="Green" title="E10">A9</font></i></a>, <a class="ref" href="graph_5.html#T5" title="GRAPH_5:th.5">GRAPH_5:5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c6">Z</font> = <font color="Maroon" title="c3">Z</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  <b>by </b><i><a class="txt" href="weddwitt.html#E2:6_2"><i><font color="Green" title="E10">A9</font></i></a>, <a class="ref" href="graph_5.html#T5" title="GRAPH_5:th.5">GRAPH_5:5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">f</font> = <font color="Maroon" title="c4">f</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c6">Z</font> ;<br/>
<b>reconsider </b><font color="Maroon" title="c8">X</font> = <font color="Maroon" title="c2">X</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  <b>by </b><i><a class="txt" href="weddwitt.html#E8:6_2"><i><font color="Green" title="E16">A15</font></i></a>, <a class="ref" href="eqrel_1.html#T41" title="EQREL_1:th.41">EQREL_1:41</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c9">fk</font> = <font color="Maroon" title="c7">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">k</font></span>)</span> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c6">Z</font> <b>by </b><i><a class="ref" href="finseq_1.html#T23" title="FINSEQ_1:th.23">FINSEQ_1:23</a></i>;<br/>
<a NAME="E13:6_2"/><i><font color="Green" title="E17">A17</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">fk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font>
 
<b>by </b><i><a class="txt" href="weddwitt.html#E7:6_2"><i><font color="Green" title="E15">A14</font></i></a>, <a class="ref" href="finseq_3.html#T59" title="FINSEQ_3:th.59">FINSEQ_3:59</a></i>;<br/>
<a NAME="E14:6_2"/><i><font color="Green" title="E18">A18</font></i>: 
<font color="Maroon" title="c7">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">fk</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"><span class="p2">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>
 
<b>by </b><i><a class="txt" href="weddwitt.html#E7:6_2"><i><font color="Green" title="E15">A14</font></i></a>, <a class="ref" href="finseq_3.html#T61" title="FINSEQ_3:th.61">FINSEQ_3:61</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c10">Zk</font> =  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c9">fk</font> as  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ;<br/>
<a NAME="E16:6_2"/>
 <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c10">Zk</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c6">Z</font>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T95" title="ZFMISC_1:th.95">ZFMISC_1:95</a></i>;<br/>
<a NAME="E17:6_2"/><b>then </b><i><font color="Green" title="E19">A19</font></i>: 
 <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c10">Zk</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">X</font>
 
<b>by </b><i><a class="ref" href="eqrel_1.html#D6" title="EQREL_1:def.6">EQREL_1:def 6</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c11">fk</font> = <font color="Maroon" title="c9">fk</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c10">Zk</font> <b>by </b><i><a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></i>;<br/>
<a NAME="E19:6_2"/><i><font color="Green" title="E20">A20</font></i>: 
<font color="Maroon" title="c11">fk</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="weddwitt.html#E3:6_2"><i><font color="Green" title="E11">A10</font></i></a>, <a class="ref" href="funct_1.html#T84" title="FUNCT_1:th.84">FUNCT_1:84</a></i>;<br/>
<a NAME="E20:6_2"/><b>then </b><i><font color="Green" title="E21">A21</font></i>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c10">Zk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font>
 
<b>by </b><i><a class="txt" href="weddwitt.html#E13:6_2"><i><font color="Green" title="E17">A17</font></i></a>, <a class="ref" href="finseq_4.html#T77" title="FINSEQ_4:th.77">FINSEQ_4:77</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c12">Xk</font> =  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c10">Zk</font> as  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  <b>by </b><i><a class="txt" href="weddwitt.html#E17:6_2"><i><font color="Green" title="E19">A19</font></i></a>, <a class="ref" href="finset_1.html#T13" title="FINSET_1:th.13">FINSET_1:13</a></i>;<br/>
<div><i><font color="Green" title="E22">A22</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="6_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:6_2_1_1"/>
( <font color="Maroon" title="c10">Zk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <font color="Maroon" title="c10">Zk</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_2_1_1_1"><b>suppose </b></a><a NAME="E1:6_2_1_1_1"/><i><font color="Green" title="E23">A23</font></i>: 
<font color="Maroon" title="c10">Zk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><div class="add"><b>consider </b><font color="Maroon" title="c13">B</font> being    <a href="eqrel_1.html#M1" title="EQREL_1:mode.1">a_partition</a> of <font color="Maroon" title="c12">Xk</font>;<br/><a NAME="E3:6_2_1_1_1"/>
<font color="Maroon" title="c13">B</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="txt" href="weddwitt.html#E1:6_2_1_1_1"><i><font color="Green" title="E23">A23</font></i></a>, <a class="ref" href="eqrel_1.html#T41" title="EQREL_1:th.41">EQREL_1:41</a>, <a class="ref" href="zfmisc_1.html#T2" title="ZFMISC_1:th.2">ZFMISC_1:2</a></i>;<br/><b>hence </b><a NAME="E4:6_2_1_1_1"/>
<font color="Maroon" title="c10">Zk</font> is    <a href="eqrel_1.html#M1" title="EQREL_1:mode.1">a_partition</a> of <font color="Maroon" title="c12">Xk</font>
 <b>by </b><i><a class="txt" href="weddwitt.html#E1:6_2_1_1_1"><i><font color="Green" title="E23">A23</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_2_1_1_2"><b>suppose </b></a><a NAME="E1:6_2_1_1_2"/>
<font color="Maroon" title="c10">Zk</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c13">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:6_2_1_1_2"/><i><font color="Green" title="E23">A24</font></i>: 
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">Zk</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></i>;<br/><a NAME="E4:6_2_1_1_2"/>
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">Z</font>
 <b>by </b><i><a class="txt" href="weddwitt.html#E3:6_2_1_1_2"><i><font color="Green" title="E23">A24</font></i></a></i>;<br/><a NAME="E5:6_2_1_1_2"/><b>then </b>
<font color="Maroon" title="c13">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="ref" href="eqrel_1.html#D6" title="EQREL_1:def.6">EQREL_1:def 6</a></i>;<br/><a NAME="E6:6_2_1_1_2"/><i><font color="Green" title="E24">A26</font></i>: 
for <font color="Olive" title="b1">a</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c12">Xk</font>  st <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">Zk</font> holds <br/>( <font color="Olive" title="b1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; ( for <font color="Olive" title="b2">b</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c12">Xk</font>  holds <br/>(  not <font color="Olive" title="b2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">Zk</font> or <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">b</font> or <font color="Olive" title="b1">a</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Olive" title="b2">b</font> ) ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_2_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c14">a</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c12">Xk</font>;<br/>

<b>assume </b><a NAME="E1:6_2_1_1_2_1"/><i><font color="Green" title="E25">A27</font></i>: 
<font color="Maroon" title="c14">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">Zk</font>
 ;<br/>

<a NAME="E2:6_2_1_1_2_1"/><i><font color="Green" title="E26">A28</font></i>: 
<font color="Maroon" title="c14">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">Z</font>
 
<b>by </b><i><a class="txt" href="weddwitt.html#E1:6_2_1_1_2_1"><i><font color="Green" title="E25">A27</font></i></a></i>;<br/>
<a NAME="E3:6_2_1_1_2_1"/>
for <font color="Olive" title="b1">b</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c12">Xk</font>  holds <br/>(  not <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">Zk</font> or <font color="Maroon" title="c14">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">b</font> or <font color="Maroon" title="c14">a</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Olive" title="b1">b</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_2_1_1_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c15">b</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c12">Xk</font>;<br/>

<b>assume </b><a NAME="E1:6_2_1_1_2_1_1"/><i><font color="Green" title="E27">A29</font></i>: 
<font color="Maroon" title="c15">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">Zk</font>
 ;<br/>

<a NAME="E2:6_2_1_1_2_1_1"/>
( <font color="Maroon" title="c14">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">Z</font> &amp; <font color="Maroon" title="c15">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">Z</font> )
 
<b>by </b><i><a class="txt" href="weddwitt.html#E1:6_2_1_1_2_1"><i><font color="Green" title="E25">A27</font></i></a>, <a class="txt" href="weddwitt.html#E1:6_2_1_1_2_1_1"><i><font color="Green" title="E27">A29</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:6_2_1_1_2_1_1"/>
( <font color="Maroon" title="c14">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">b</font> or <font color="Maroon" title="c14">a</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c15">b</font> )
 <b>by </b><i><a class="ref" href="eqrel_1.html#D6" title="EQREL_1:def.6">EQREL_1:def 6</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:6_2_1_1_2_1"/>
( <font color="Maroon" title="c14">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; ( for <font color="Olive" title="b1">b</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c12">Xk</font>  holds <br/>(  not <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">Zk</font> or <font color="Maroon" title="c14">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">b</font> or <font color="Maroon" title="c14">a</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Olive" title="b1">b</font> ) ) )
 <b>by </b><i><a class="txt" href="weddwitt.html#E2:6_2_1_1_2_1"><i><font color="Green" title="E26">A28</font></i></a>, <a class="ref" href="eqrel_1.html#D6" title="EQREL_1:def.6">EQREL_1:def 6</a></i>;<br/>


</div><b>end;</b></div><a NAME="E7:6_2_1_1_2"/>
<font color="Maroon" title="c10">Zk</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c10">Zk</font></span>)</span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T100" title="ZFMISC_1:th.100">ZFMISC_1:100</a></i>;<br/><b>hence </b><a NAME="E8:6_2_1_1_2"/>
<font color="Maroon" title="c10">Zk</font> is    <a href="eqrel_1.html#M1" title="EQREL_1:mode.1">a_partition</a> of <font color="Maroon" title="c12">Xk</font>
 <b>by </b><i><a class="txt" href="weddwitt.html#E6:6_2_1_1_2"><i><font color="Green" title="E24">A26</font></i></a>, <a class="ref" href="eqrel_1.html#D6" title="EQREL_1:def.6">EQREL_1:def 6</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>reconsider </b><font color="Maroon" title="c13">ck</font> = <font color="Maroon" title="c5">c</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">k</font></span>)</span> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="finseq_1.html#T23" title="FINSEQ_1:th.23">FINSEQ_1:23</a></i>;<br/>
<a NAME="E24:6_2"/><i><font color="Green" title="E23">A30</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">ck</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">fk</font>
 
<b>by </b><i><a class="txt" href="weddwitt.html#E5:6_2"><i><font color="Green" title="E13">A12</font></i></a>, <a class="txt" href="weddwitt.html#E7:6_2"><i><font color="Green" title="E15">A14</font></i></a>, <a class="txt" href="weddwitt.html#E13:6_2"><i><font color="Green" title="E17">A17</font></i></a>, <a class="ref" href="finseq_3.html#T59" title="FINSEQ_3:th.59">FINSEQ_3:59</a></i>;<br/>
<a NAME="E25:6_2"/>
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">i</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="c13">ck</font> holds <br/><font color="Maroon" title="c13">ck</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">fk</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c14">i</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:6_2_2"/><i><font color="Green" title="E24">A31</font></i>: 
<font color="Maroon" title="c14">i</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="c13">ck</font>
 ;<br/>

<a NAME="E2:6_2_2"/><i><font color="Green" title="E25">A32</font></i>: 
<font color="Maroon" title="c1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/>
<a NAME="E3:6_2_2"/><b>then </b>
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c13">ck</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">k</font>
 
<b>by </b><i><a class="txt" href="weddwitt.html#E5:6_2"><i><font color="Green" title="E13">A12</font></i></a>, <a class="txt" href="weddwitt.html#E7:6_2"><i><font color="Green" title="E15">A14</font></i></a>, <a class="ref" href="finseq_1.html#T21" title="FINSEQ_1:th.21">FINSEQ_1:21</a></i>;<br/>
<a NAME="E4:6_2_2"/><b>then </b><i><font color="Green" title="E26">A33</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c13">ck</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c11">fk</font>
 
<b>by </b><i><a class="txt" href="weddwitt.html#E7:6_2"><i><font color="Green" title="E15">A14</font></i></a>, <a class="txt" href="weddwitt.html#E2:6_2_2"><i><font color="Green" title="E25">A32</font></i></a>, <a class="ref" href="finseq_1.html#T21" title="FINSEQ_1:th.21">FINSEQ_1:21</a></i>;<br/>
<a NAME="E5:6_2_2"/><i><font color="Green" title="E27">A34</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c13">ck</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">c</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T89" title="RELAT_1:th.89">RELAT_1:89</a></i>;<br/>
<a NAME="E6:6_2_2"/>
<font color="Maroon" title="c13">ck</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Maroon" title="c14">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">c</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Maroon" title="c14">i</font>
 
<b>by </b><i><a class="txt" href="weddwitt.html#E1:6_2_2"><i><font color="Green" title="E24">A31</font></i></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></i>;<br/>
<a NAME="E7:6_2_2"/><b>then </b>
<font color="Maroon" title="c13">ck</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Maroon" title="c14">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">i</font></span>)</span>
 
<b>by </b><i><a class="txt" href="weddwitt.html#E6:6_2"><i><font color="Green" title="E14">A13</font></i></a>, <a class="txt" href="weddwitt.html#E1:6_2_2"><i><font color="Green" title="E24">A31</font></i></a>, <a class="txt" href="weddwitt.html#E5:6_2_2"><i><font color="Green" title="E27">A34</font></i></a></i>;<br/>
<b>hence </b><a NAME="E8:6_2_2"/>
<font color="Maroon" title="c13">ck</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Maroon" title="c14">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">fk</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">i</font></span>)</span>
 <b>by </b><i><a class="txt" href="weddwitt.html#E1:6_2_2"><i><font color="Green" title="E24">A31</font></i></a>, <a class="txt" href="weddwitt.html#E4:6_2_2"><i><font color="Green" title="E26">A33</font></i></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E26:6_2"/><b>then </b><i><font color="Green" title="E24">A35</font></i>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c12">Xk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Sum</a> <font color="Maroon" title="c13">ck</font>
 
<b>by </b><i><a class="txt" href="weddwitt.html#E1:6_2"><i><font color="Green" title="E9">A8</font></i></a>, <a class="txt" href="weddwitt.html#E19:6_2"><i><font color="Green" title="E20">A20</font></i></a>, <a class="txt" href="weddwitt.html#E20:6_2"><i><font color="Green" title="E21">A21</font></i></a>, <a class="txt" href="weddwitt.html#E22:6_2"><i><font color="Green" title="E22">A22</font></i></a>, <a class="txt" href="weddwitt.html#E24:6_2"><i><font color="Green" title="E23">A30</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c14">fk1</font> = <font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> as    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ;<br/>
<a NAME="E28:6_2"/>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">Zk</font> holds <br/><font color="Olive" title="b1">x</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c14">fk1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_2_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c15">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:6_2_3"/><i><font color="Green" title="E25">A36</font></i>: 
<font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">Zk</font>
 ;<br/>

<a NAME="E2:6_2_3"/><i><font color="Green" title="E26">A37</font></i>: 
<font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">Z</font>
 
<b>by </b><i><a class="txt" href="weddwitt.html#E1:6_2_3"><i><font color="Green" title="E25">A36</font></i></a></i>;<br/>
<a NAME="E3:6_2_3"/><i><font color="Green" title="E27">A38</font></i>: 
<font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T6" title="FINSEQ_1:th.6">FINSEQ_1:6</a></i>;<br/>
<a NAME="E4:6_2_3"/>
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c7">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">f</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E5:6_2_3"/><b>then </b><i><font color="Green" title="E28">A39</font></i>: 
<font color="Maroon" title="c14">fk1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c7">f</font>
 
<b>by </b><i><a class="txt" href="weddwitt.html#E7:6_2"><i><font color="Green" title="E15">A14</font></i></a>, <a class="txt" href="weddwitt.html#E3:6_2_3"><i><font color="Green" title="E27">A38</font></i></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></i>;<br/>
<a NAME="E6:6_2_3"/><b>then </b><i><font color="Green" title="E29">A40</font></i>: 
<font color="Maroon" title="c14">fk1</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c8">X</font>
 
<b>by </b><i><a class="txt" href="weddwitt.html#E4:6_2"><i><font color="Green" title="E12">A11</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c16">i</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E8:6_2_3"/><i><font color="Green" title="E30">A41</font></i>: 
( <font color="Maroon" title="c16">i</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="c11">fk</font> &amp; <font color="Maroon" title="c11">fk</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">x</font> )
 <b>by </b><i><a class="txt" href="weddwitt.html#E1:6_2_3"><i><font color="Green" title="E25">A36</font></i></a>, <a class="ref" href="finseq_2.html#T11" title="FINSEQ_2:th.11">FINSEQ_2:11</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_2_3_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:6_2_3_1"/><i><font color="Green" title="E31">A42</font></i>: 
<font color="Maroon" title="c11">fk</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">fk1</font>
 ;<br/><a NAME="E2:6_2_3_1"/><i><font color="Green" title="E32">A43</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c11">fk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">k</font>
 <b>by </b><i><a class="txt" href="weddwitt.html#E13:6_2"><i><font color="Green" title="E17">A17</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E3:6_2_3_1"/><i><font color="Green" title="E33">A44</font></i>: 
<font color="Maroon" title="c16">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">k</font>
 <b>by </b><i><a class="txt" href="weddwitt.html#E13:6_2"><i><font color="Green" title="E17">A17</font></i></a>, <a class="txt" href="weddwitt.html#E8:6_2_3"><i><font color="Green" title="E30">A41</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E4:6_2_3_1"/>
<font color="Maroon" title="c16">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">k</font>
 <b>by </b><i><a class="txt" href="weddwitt.html#E8:6_2_3"><i><font color="Green" title="E30">A41</font></i></a>, <a class="txt" href="weddwitt.html#E2:6_2_3_1"><i><font color="Green" title="E32">A43</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><a NAME="E5:6_2_3_1"/><b>then </b><i><font color="Green" title="E34">A45</font></i>: 
<font color="Maroon" title="c16">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E6:6_2_3_1"/><i><font color="Green" title="E35">A46</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c7">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="weddwitt.html#E7:6_2"><i><font color="Green" title="E15">A14</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E7:6_2_3_1"/>
<font color="Maroon" title="c1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></i>;<br/><a NAME="E8:6_2_3_1"/><b>then </b><i><font color="Green" title="E36">A47</font></i>: 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">k</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T7" title="FINSEQ_1:th.7">FINSEQ_1:7</a></i>;<br/><a NAME="E9:6_2_3_1"/><i><font color="Green" title="E37">A48</font></i>: 
<font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <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="c7">f</font>
 <b>by </b><i><a class="txt" href="weddwitt.html#E6:6_2_3_1"><i><font color="Green" title="E35">A46</font></i></a>, <a class="ref" href="finseq_1.html#T6" title="FINSEQ_1:th.6">FINSEQ_1:6</a></i>;<br/><a NAME="E10:6_2_3_1"/>
<font color="Maroon" title="c11">fk</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">i</font>
 <b>by </b><i><a class="txt" href="weddwitt.html#E8:6_2_3"><i><font color="Green" title="E30">A41</font></i></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></i>;<br/><b>hence </b><a NAME="E11:6_2_3_1"/>
contradiction
 <b>by </b><i><a class="txt" href="weddwitt.html#E3:6_2"><i><font color="Green" title="E11">A10</font></i></a>, <a class="txt" href="weddwitt.html#E1:6_2_3_1"><i><font color="Green" title="E31">A42</font></i></a>, <a class="txt" href="weddwitt.html#E3:6_2_3_1"><i><font color="Green" title="E33">A44</font></i></a>, <a class="txt" href="weddwitt.html#E5:6_2_3_1"><i><font color="Green" title="E34">A45</font></i></a>, <a class="txt" href="weddwitt.html#E6:6_2_3_1"><i><font color="Green" title="E35">A46</font></i></a>, <a class="txt" href="weddwitt.html#E8:6_2_3_1"><i><font color="Green" title="E36">A47</font></i></a>, <a class="txt" href="weddwitt.html#E9:6_2_3_1"><i><font color="Green" title="E37">A48</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>
<b>hence </b><a NAME="E10:6_2_3"/>
<font color="Maroon" title="c15">x</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c14">fk1</font>
 <b>by </b><i><a class="txt" href="weddwitt.html#E2:6_2_3"><i><font color="Green" title="E26">A37</font></i></a>, <a class="txt" href="weddwitt.html#E5:6_2_3"><i><font color="Green" title="E28">A39</font></i></a>, <a class="txt" href="weddwitt.html#E6:6_2_3"><i><font color="Green" title="E29">A40</font></i></a>, <a class="txt" href="weddwitt.html#E8:6_2_3"><i><font color="Green" title="E30">A41</font></i></a>, <a class="ref" href="eqrel_1.html#D6" title="EQREL_1:def.6">EQREL_1:def 6</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E29:6_2"/><b>then </b><i><font color="Green" title="E25">A49</font></i>: 
<font color="Maroon" title="c14">fk1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c12">Xk</font>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T98" title="ZFMISC_1:th.98">ZFMISC_1:98</a></i>;<br/>
<a NAME="E30:6_2"/><i><font color="Green" title="E26">A50</font></i>: 
 <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c6">Z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">X</font>
 
<b>by </b><i><a class="ref" href="eqrel_1.html#D6" title="EQREL_1:def.6">EQREL_1:def 6</a></i>;<br/>
<a NAME="E31:6_2"/><i><font color="Green" title="E27">A51</font></i>: 
<font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T6" title="FINSEQ_1:th.6">FINSEQ_1:6</a></i>;<br/>
<a NAME="E32:6_2"/>
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c7">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">f</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E33:6_2"/><b>then </b>
<font color="Maroon" title="c14">fk1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c7">f</font>
 
<b>by </b><i><a class="txt" href="weddwitt.html#E7:6_2"><i><font color="Green" title="E15">A14</font></i></a>, <a class="txt" href="weddwitt.html#E31:6_2"><i><font color="Green" title="E27">A51</font></i></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c15">fk1</font> = <font color="Maroon" title="c14">fk1</font> as  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  <b>by </b><i><a class="txt" href="weddwitt.html#E30:6_2"><i><font color="Green" title="E26">A50</font></i></a>, <a class="ref" href="finset_1.html#T25" title="FINSET_1:th.25">FINSET_1:25</a></i>;<br/>
<i><font color="Green" title="E28">A52</font></i>: <font color="Maroon" title="c6">Z</font> = 
<span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c11">fk</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="c15">fk1</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span>
<b>by </b><i><a class="txt" href="weddwitt.html#E4:6_2"><i><font color="Green" title="E12">A11</font></i></a>, <a class="txt" href="weddwitt.html#E14:6_2"><i><font color="Green" title="E18">A18</font></i></a>, <a class="ref" href="finseq_1.html#T44" title="FINSEQ_1:th.44">FINSEQ_1:44</a></i>
<br/>.= 
<font color="Maroon" title="c10">Zk</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="c15">fk1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
<b>by </b><i><a class="ref" href="finseq_1.html#T56" title="FINSEQ_1:th.56">FINSEQ_1:56</a></i>
;<br/>
<i><font color="Green" title="E29">A53</font></i>: <font color="Maroon" title="c8">X</font> = 
 <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c6">Z</font>
<b>by </b><i><a class="ref" href="eqrel_1.html#D6" title="EQREL_1:def.6">EQREL_1:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c10">Zk</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c15">fk1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>
<b>by </b><i><a class="txt" href="weddwitt.html#E35:6_2"><i><font color="Green" title="E28">A52</font></i></a>, <a class="ref" href="zfmisc_1.html#T96" title="ZFMISC_1:th.96">ZFMISC_1:96</a></i>
<br/>.= 
<font color="Maroon" title="c12">Xk</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c15">fk1</font>
<b>by </b><i><a class="ref" href="zfmisc_1.html#T31" title="ZFMISC_1:th.31">ZFMISC_1:31</a></i>
;<br/>
<a NAME="E37:6_2"/>
<font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T6" title="FINSEQ_1:th.6">FINSEQ_1:6</a></i>;<br/>
<a NAME="E38:6_2"/><b>then </b><i><font color="Green" title="E30">A54</font></i>: 
<font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <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">c</font>
 
<b>by </b><i><a class="txt" href="weddwitt.html#E5:6_2"><i><font color="Green" title="E13">A12</font></i></a>, <a class="txt" href="weddwitt.html#E7:6_2"><i><font color="Green" title="E15">A14</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E39:6_2"/>
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c13">ck</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 
;<br/>
<b>then reconsider </b><font color="Maroon" title="c16">ckc</font> = <font color="Maroon" title="c13">ck</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  <b>by </b><i><a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></i>;<br/>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c8">X</font> = 
<span class="p1">(<span class="default"><a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Sum</a> <font color="Maroon" title="c13">ck</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c15">fk1</font></span>)</span>
<b>by </b><i><a class="txt" href="weddwitt.html#E26:6_2"><i><font color="Green" title="E24">A35</font></i></a>, <a class="txt" href="weddwitt.html#E29:6_2"><i><font color="Green" title="E25">A49</font></i></a>, <a class="txt" href="weddwitt.html#E36:6_2"><i><font color="Green" title="E29">A53</font></i></a>, <a class="ref" href="card_2.html#T53" title="CARD_2:th.53">CARD_2:53</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Sum</a> <font color="Maroon" title="c13">ck</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">c</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="weddwitt.html#E6:6_2"><i><font color="Green" title="E14">A13</font></i></a>, <a class="txt" href="weddwitt.html#E38:6_2"><i><font color="Green" title="E30">A54</font></i></a></i>
<br/>.= 
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">ckc</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">c</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span>
<b>by </b><i><a class="ref" href="rvsum_1.html#T104" title="RVSUM_1:th.104">RVSUM_1:104</a></i>
<br/>.= 
 <a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Sum</a> <font color="Maroon" title="c5">c</font>
<b>by </b><i><a class="txt" href="weddwitt.html#E5:6_2"><i><font color="Green" title="E13">A12</font></i></a>, <a class="txt" href="weddwitt.html#E7:6_2"><i><font color="Green" title="E15">A14</font></i></a>, <a class="ref" href="finseq_3.html#T61" title="FINSEQ_3:th.61">FINSEQ_3:61</a></i>
;<br/>
<b>hence </b><a NAME="E42:6_2"/>
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c2">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Sum</a> <font color="Maroon" title="c5">c</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E3:6"/><i><font color="Green" title="E9">A55</font></i>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="weddwitt.html#E1:6"><i><font color="Green" title="E7">A1</font></i></a>, <a class="txt" href="weddwitt.html#E2:6"><i><font color="Green" title="E8">A7</font></i></a>);<br/></i>
<b>let </b><font color="Maroon" title="c1">X</font> be  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c2">Y</font> be    <a href="eqrel_1.html#M1" title="EQREL_1:mode.1">a_partition</a> of <font color="Maroon" title="c1">X</font>;<br/>



<a NAME="E4:6"/>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">Y</font>
 
;<br/>
<b>hence </b><a NAME="E5:6"/>
for <font color="Olive" title="b1">f</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c2">Y</font>  st <font color="Olive" title="b1">f</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">Y</font> holds <br/>for <font color="Olive" title="b2">c</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">f</font> &amp; ( for <font color="Olive" title="b3">i</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b3">i</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="Olive" title="b2">c</font> holds <br/><font color="Olive" title="b2">c</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Olive" title="b3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">i</font></span>)</span> ) holds <br/> <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Sum</a> <font color="Olive" title="b2">c</font>
 <b>by </b><i><a class="txt" href="weddwitt.html#E3:6"><i><font color="Green" title="E9">A55</font></i></a></i>;<br/>


</div>
