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

<b>let </b><font color="Maroon">Y</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">PA</font> be    <a href="eqrel_1.html#M1">a_partition</a> of <font color="Maroon">Y</font>, <font color="Maroon">PB</font> be    <a href="eqrel_1.html#M1">a_partition</a> of <font color="Maroon">Y</font>;<br/><b>let </b><font color="Maroon">p0</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">f</font> be    <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">Y</font>;<br/>







<b>assume </b><a NAME="E1:31"/><i><font color="Green">E42</font></i>: 
( <font color="Maroon">p0</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">p0</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</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 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> holds <br/> ex <font color="Olive">p2</font>, <font color="Olive">p3</font>, <font color="Olive">u</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">p2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">PA</font> &amp; <font color="Olive">p3</font> <a href="hidden.html#R2">in</a> <font color="Maroon">PB</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Olive">p2</font> &amp; <font color="Olive">u</font> <a href="hidden.html#R2">in</a> <font color="Olive">p2</font> &amp; <font color="Olive">u</font> <a href="hidden.html#R2">in</a> <font color="Olive">p3</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a> <font color="Olive">p3</font> ) ) &amp; <font color="Maroon">p0</font> <a href="partit1.html#R1" target="_self">is_a_dependent_set_of</a> <font color="Maroon">PA</font> &amp; <font color="Maroon">p0</font> <a href="partit1.html#R1" target="_self">is_a_dependent_set_of</a> <font color="Maroon">PB</font> )
 ;<br/>

<b>then consider </b><font color="Maroon">A1</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:31"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">A1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">PA</font> &amp; <font color="Maroon">A1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">p0</font> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">A1</font> )
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">B1</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:31"/><i><font color="Green">E45</font></i>: 
( <font color="Maroon">B1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">PB</font> &amp; <font color="Maroon">B1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">p0</font> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">B1</font> )
 <b>by </b><i>, </i>;<br/>
<a NAME="E6:31"/><i><font color="Green">E48</font></i>: 
(  <a href="setfam_1.html#K5">union</a> <font color="Maroon">PA</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Y</font> &amp; ( for <font color="Olive">A</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">PA</font> holds <br/>( <font color="Olive">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; ( for <font color="Olive">B</font> being   <a href="hidden.html#M1">set</a>   holds <br/>(  not <font color="Olive">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">PA</font> or <font color="Olive">A</font> <a href="hidden.html#R1">=</a> <font color="Olive">B</font> or <font color="Olive">A</font> <a href="xboole_0.html#R1">misses</a> <font color="Olive">B</font> ) ) ) ) )
 
<b>by </b><i><a class="ref" href="eqrel_1.html#D6">EQREL_1:def 6</a></i>;<br/>
<a NAME="E7:31"/><i><font color="Green">E52</font></i>: 
(  <a href="setfam_1.html#K5">union</a> <font color="Maroon">PB</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Y</font> &amp; ( for <font color="Olive">A</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">PB</font> holds <br/>( <font color="Olive">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; ( for <font color="Olive">B</font> being   <a href="hidden.html#M1">set</a>   holds <br/>(  not <font color="Olive">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">PB</font> or <font color="Olive">A</font> <a href="hidden.html#R1">=</a> <font color="Olive">B</font> or <font color="Olive">A</font> <a href="xboole_0.html#R1">misses</a> <font color="Olive">B</font> ) ) ) ) )
 
<b>by </b><i><a class="ref" href="eqrel_1.html#D6">EQREL_1:def 6</a></i>;<br/>
<a NAME="E8:31"/>
for <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> holds <br/><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font> <a href="hidden.html#R2">in</a> <font color="Maroon">p0</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1"><b>proof </b></a><div class="add">

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Maroon">a<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> implies <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">p0</font> );<br/>
<a NAME="E1:31_1"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
;<br/>
<a NAME="E2:31_1"/><i><font color="Green">E60</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="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_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="E1:31_1_1"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font>]
 ;<br/>

<b>assume </b><a NAME="E2:31_1_1"/><i><font color="Green">E62</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> )
 ;<br/>

<a NAME="E3:31_1_1"/><b>then </b><i><font color="Green">E63</font></i>: 
<font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E4:31_1_1"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> or 1 <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 )
 
<b>by </b><i>, <a class="ref" href="nat_1.html#T26">NAT_1:26</a></i>;<br/>
<a NAME="E5:31_1_1"/><b>then </b><i><font color="Green">E64</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> or 1 <a href="xcmplx_0.html#K6">-</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> )
 
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:31_1_1_1_1"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> or 0 <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:31_1_1_1_1_1"/><i><font color="Green">E65</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">p2</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">p3</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">u</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:31_1_1_1_1_1"/><i><font color="Green">E75</font></i>: 
( <font color="Maroon">p2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">PA</font> &amp; <font color="Maroon">p3</font> <a href="hidden.html#R2">in</a> <font color="Maroon">PB</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <font color="Maroon">p2</font> &amp; <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">p2</font> &amp; <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">p3</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">p3</font> )
 <b>by </b><i>, </i>;<br/><b>consider </b><font color="Maroon">A</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:31_1_1_1_1_1"/><i><font color="Green">E76</font></i>: 
( <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">PA</font> )
 <b>by </b><i>, , , , , <a class="ref" href="nat_1.html#T38">NAT_1:38</a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><a NAME="E6:31_1_1_1_1_1"/><i><font color="Green">E77</font></i>: 
( <font color="Maroon">p2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> or <font color="Maroon">p2</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">A</font> )
 <b>by </b><i>, , <a class="ref" href="eqrel_1.html#D6">EQREL_1:def 6</a></i>;<br/><b>consider </b><font color="Maroon">a</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E8:31_1_1_1_1_1"/><i><font color="Green">E79</font></i>: 
( <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A1</font> )
 <b>by </b><i>, , , , <a class="ref" href="nat_1.html#T38">NAT_1:38</a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><a NAME="E9:31_1_1_1_1_1"/>
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p2</font> or <font color="Maroon">a</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">p2</font> )
 <b>by </b><i>, , , </i>;<br/><a NAME="E10:31_1_1_1_1_1"/><b>then </b><i><font color="Green">E80</font></i>: 
<font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">p0</font>
 <b>by </b><i>, , , , , <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a>, <a class="ref" href="zfmisc_1.html#T92">ZFMISC_1:92</a></i>;<br/><b>consider </b><font color="Maroon">B</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E12:31_1_1_1_1_1"/><i><font color="Green">E81</font></i>: 
( <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Maroon">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">PB</font> )
 <b>by </b><i/>;<br/><a NAME="E13:31_1_1_1_1_1"/><i><font color="Green">E82</font></i>: 
( <font color="Maroon">p3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font> or <font color="Maroon">p3</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">B</font> )
 <b>by </b><i>, , <a class="ref" href="eqrel_1.html#D6">EQREL_1:def 6</a></i>;<br/><b>consider </b><font color="Maroon">b</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E15:31_1_1_1_1_1"/><i><font color="Green">E83</font></i>: 
( <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">b</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B1</font> )
 <b>by </b><i>, , , , , <a class="ref" href="tarski.html#D4">TARSKI:def 4</a>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/><a NAME="E16:31_1_1_1_1_1"/>
( <font color="Maroon">p3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> or <font color="Maroon">p3</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">b</font> )
 <b>by </b><i>, , , <a class="txt" href="partit1.html#E1:31"><i><font color="Green">E42</font></i></a></i>;<br/><a NAME="E17:31_1_1_1_1_1"/><b>then </b>
<font color="Maroon">B</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">p0</font>
 <b>by </b><i>, , , <a class="ref" href="partit1.html#T1" target="_self">Th1</a>, <a class="txt" href="partit1.html#E1:31"><i><font color="Green">E42</font></i></a>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a>, <a class="ref" href="zfmisc_1.html#T92">ZFMISC_1:92</a></i>;<br/><b>hence </b><a NAME="E18:31_1_1_1_1_1"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">p0</font>
 <b>by </b><i>, , <a class="ref" href="partit1.html#T1" target="_self">Th1</a>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:31_1_1_1_1_2"/>
0 <a href="hidden.html#R1">=</a> <font color="Maroon">k</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:31_1_1_1_1_2"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">p0</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E7:31_1_1"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">p0</font>
 ;<br/>


</div><b>end;</b></div>
<b>thus </b><a NAME="E3:31_1"/>
for <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(, );</i><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E9:31"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">p0</font>
 <b>by </b><i/>;<br/>


</div>
