<?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/>



<a NAME="E1:33"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">PA</font> <a href="setfam_1.html#R1">is_finer_than</a> <font color="Maroon">PA</font> <a href="partit1.html#K3" target="_self">'\/'</a> <font color="Maroon">PB</font>
 
<b>by </b><i/>;<br/>
<a NAME="E2:33"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">PB</font> <a href="setfam_1.html#R1">is_finer_than</a> <font color="Maroon">PA</font> <a href="partit1.html#K3" target="_self">'\/'</a> <font color="Maroon">PB</font>
 
<b>by </b><i/>;<br/>
<a NAME="E3:33"/><i><font color="Green">E45</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="E4:33"/><i><font color="Green">E48</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="E5:33"/><i><font color="Green">E52</font></i>: 
 <a href="partit1.html#K4" target="_self">ERl</a> <span class="p1">(<span class="default"><font color="Maroon">PA</font> <a href="partit1.html#K3" target="_self">'\/'</a> <font color="Maroon">PB</font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="partit1.html#K4" target="_self">ERl</a> <font color="Maroon">PA</font></span>)</span> <a href="eqrel_1.html#K5">"\/"</a> <span class="p1">(<span class="default"><a href="partit1.html#K4" target="_self">ERl</a> <font color="Maroon">PB</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1"><b>proof </b></a><div class="add">

<b>let </b><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> ; <i><font color="Red">:: according to </font></i><a class="ref" href="relat_1.html#D3">RELAT_1:def 3</a><br/>

<b>assume </b><a NAME="E1:33_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="partit1.html#K4" target="_self">ERl</a> <span class="p1">(<span class="default"><font color="Maroon">PA</font> <a href="partit1.html#K3" target="_self">'\/'</a> <font color="Maroon">PB</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">p0</font> being   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">Y</font><b> such that </b><br/><a NAME="E3:33_1"/><i><font color="Green">E53</font></i>: 
( <font color="Maroon">p0</font> <a href="hidden.html#R2">in</a> <font color="Maroon">PA</font> <a href="partit1.html#K3" target="_self">'\/'</a> <font color="Maroon">PB</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">p0</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">p0</font> )
 <b>by </b><i/>;<br/>
<a NAME="E4:33_1"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">p0</font> <a href="partit1.html#R2" target="_self">is_min_depend</a> <font color="Maroon">PA</font>,<font color="Maroon">PB</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E5:33_1"/><b>then </b><i><font color="Green">E61</font></i>: 
( <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> &amp; ( for <font color="Olive">d</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">d</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">p0</font> &amp; <font color="Olive">d</font> <a href="partit1.html#R1" target="_self">is_a_dependent_set_of</a> <font color="Maroon">PA</font> &amp; <font color="Olive">d</font> <a href="partit1.html#R1" target="_self">is_a_dependent_set_of</a> <font color="Maroon">PB</font> holds <br/><font color="Olive">d</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p0</font> ) )
 
<b>by </b><i/>;<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="E7:33_1"/><i><font color="Green">E62</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">a</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E9:33_1"/><i><font color="Green">E63</font></i>: 
( <font color="Maroon">x</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="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>reconsider </b><font color="Maroon">Ca</font> = <span class="p1">{<span class="default"> <font color="Olive">p</font> where <font color="Olive">p</font> is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">PA</font> :  ex <font color="Olive">f</font> being   <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">Y</font> st <br/>( 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Olive">f</font> &amp; <font color="Olive">f</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Olive">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="Olive">f</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Olive">p</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="Olive">f</font> holds <br/> ex <font color="Olive">p1</font>, <font color="Olive">p2</font>, <font color="Olive">x0</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">p1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">PA</font> &amp; <font color="Olive">p2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">PB</font> &amp; <font color="Olive">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Olive">p1</font> &amp; <font color="Olive">x0</font> <a href="hidden.html#R2">in</a> <font color="Olive">p1</font> <a href="xboole_0.html#K3">/\</a> <font color="Olive">p2</font> &amp; <font color="Olive">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">p2</font> ) ) ) </span>}</span>  as    <a href="hidden.html#M1">set</a>  ;<br/>
<b>reconsider </b><font color="Maroon">pb</font> =  <a href="tarski.html#K3">union</a> <font color="Maroon">Ca</font> as    <a href="hidden.html#M1">set</a>  ;<br/>
<b>reconsider </b><font color="Maroon">Cb</font> = <span class="p1">{<span class="default"> <font color="Olive">p</font> where <font color="Olive">p</font> is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">PB</font> :  ex <font color="Olive">q</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Ca</font> &amp; <font color="Olive">p</font> <a href="subset_1.html#K8">/\</a> <font color="Olive">q</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  ) </span>}</span>  as    <a href="hidden.html#M1">set</a>  ;<br/>
<b>reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font> <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">fx</font> = <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">x'</font></span><a href="finseq_1.html#K12">*&gt;</a></span> as    <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">Y</font> ;<br/>
<a NAME="E15:33_1"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">fx</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D8">FINSEQ_1:def 8</a></i>;<br/>
<a NAME="E16:33_1"/><b>then </b><i><font color="Green">E65</font></i>: 
<font color="Maroon">fx</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">fx</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">a</font>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a></i>;<br/>
<a NAME="E17:33_1"/><i><font color="Green">E75</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">fx</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a></i>;<br/>
<a NAME="E18:33_1"/>
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">fx</font> holds <br/> ex <font color="Olive">p1</font>, <font color="Olive">p2</font>, <font color="Olive">x0</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">p1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">PA</font> &amp; <font color="Olive">p2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">PB</font> &amp; <font color="Maroon">fx</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Olive">p1</font> &amp; <font color="Olive">x0</font> <a href="hidden.html#R2">in</a> <font color="Olive">p1</font> <a href="xboole_0.html#K3">/\</a> <font color="Olive">p2</font> &amp; <font color="Maroon">fx</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">p2</font> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a></i>;<br/>
<a NAME="E19:33_1"/><b>then </b><i><font color="Green">E76</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Ca</font>
 
<b>by </b><i>, , , , </i>;<br/>
<b>then consider </b><font color="Maroon">y5</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E21:33_1"/><i><font color="Green">E77</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">y5</font> &amp; <font color="Maroon">y5</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Ca</font> )
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">z5</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E23:33_1"/><i><font color="Green">E79</font></i>: 
( <font color="Maroon">x'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">z5</font> &amp; <font color="Maroon">z5</font> <a href="hidden.html#R2">in</a> <font color="Maroon">PB</font> )
 <b>by </b><i>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<a NAME="E24:33_1"/>
<font color="Maroon">y5</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">z5</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i>, , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E25:33_1"/><b>then </b><i><font color="Green">E80</font></i>: 
<font color="Maroon">z5</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Cb</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E26:33_1"/><i><font color="Green">E81</font></i>: 
<font color="Maroon">pb</font> <a href="partit1.html#R1" target="_self">is_a_dependent_set_of</a> <font color="Maroon">PA</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:33_1_1"/>
<font color="Maroon">Ca</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">PA</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">z</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:33_1_1_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Ca</font>
 ;<br/>

<a NAME="E2:33_1_1_1"/><b>then </b>
 ex <font color="Olive">p</font> being   <a href="subset_1.html#M2">Element</a> of <font color="Maroon">PA</font> st <br/>( <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Olive">p</font> &amp;  ex <font color="Olive">f</font> being   <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">Y</font> st <br/>( 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Olive">f</font> &amp; <font color="Olive">f</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Olive">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="Olive">f</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Olive">p</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="Olive">f</font> holds <br/> ex <font color="Olive">p1</font>, <font color="Olive">p2</font>, <font color="Olive">x0</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">p1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">PA</font> &amp; <font color="Olive">p2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">PB</font> &amp; <font color="Olive">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Olive">p1</font> &amp; <font color="Olive">x0</font> <a href="hidden.html#R2">in</a> <font color="Olive">p1</font> <a href="xboole_0.html#K3">/\</a> <font color="Olive">p2</font> &amp; <font color="Olive">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">p2</font> ) ) ) )
 
;<br/>
<b>hence </b><a NAME="E3:33_1_1_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">PA</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:33_1_1"/>
<font color="Maroon">pb</font> <a href="partit1.html#R1" target="_self">is_a_dependent_set_of</a> <font color="Maroon">PA</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E27:33_1"/><i><font color="Green">E82</font></i>: 
<font color="Maroon">pb</font> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">Cb</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_2"><b>proof </b></a><div class="add">

<a NAME="E1:33_1_2"/><i><font color="Green">E83</font></i>: 
<font color="Maroon">pb</font> <a href="tarski.html#R1">c=</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">Cb</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x1</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:33_1_2_1"/>
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">pb</font>
 ;<br/>

<b>then consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:33_1_2_1"/><i><font color="Green">E84</font></i>: 
( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">y</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Ca</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">p</font> being    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">PA</font><b> such that </b><br/><a NAME="E5:33_1_2_1"/><i><font color="Green">E85</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> &amp;  ex <font color="Olive">f</font> being   <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">Y</font> st <br/>( 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Olive">f</font> &amp; <font color="Olive">f</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Olive">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="Olive">f</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">p</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="Olive">f</font> holds <br/> ex <font color="Olive">p1</font>, <font color="Olive">p2</font>, <font color="Olive">x0</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">p1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">PA</font> &amp; <font color="Olive">p2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">PB</font> &amp; <font color="Olive">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Olive">p1</font> &amp; <font color="Olive">x0</font> <a href="hidden.html#R2">in</a> <font color="Olive">p1</font> <a href="xboole_0.html#K3">/\</a> <font color="Olive">p2</font> &amp; <font color="Olive">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">p2</font> ) ) ) )
 <b>by </b><i><a class="txt" href="partit1.html#E3:33"><i><font color="Green">E45</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">z</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E7:33_1_2_1"/><i><font color="Green">E86</font></i>: 
( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">z</font> &amp; <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">PB</font> )
 <b>by </b><i>, <a class="txt" href="partit1.html#E3:33"><i><font color="Green">E45</font></i></a>, <a class="ref" href="partit1.html#T4" target="_self">Th4</a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<a NAME="E8:33_1_2_1"/>
<font color="Maroon">y</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="txt" href="partit1.html#E3:33"><i><font color="Green">E45</font></i></a>, , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E9:33_1_2_1"/><b>then </b>
( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">z</font> &amp; <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Cb</font> )
 
<b>by </b><i><a class="txt" href="partit1.html#E3:33"><i><font color="Green">E45</font></i></a>, </i>;<br/>
<b>hence </b><a NAME="E10:33_1_2_1"/>
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">Cb</font>
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:33_1_2"/>
 <a href="tarski.html#K3">union</a> <font color="Maroon">Cb</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">pb</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x1</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:33_1_2_2"/>
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">Cb</font>
 ;<br/>

<b>then consider </b><font color="Maroon">y1</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:33_1_2_2"/><i><font color="Green">E87</font></i>: 
( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">y1</font> &amp; <font color="Maroon">y1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Cb</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">p</font> being    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">PB</font><b> such that </b><br/><a NAME="E5:33_1_2_2"/><i><font color="Green">E88</font></i>: 
( <font color="Maroon">y1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> &amp;  ex <font color="Olive">q</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Ca</font> &amp; <font color="Maroon">p</font> <a href="subset_1.html#K8">/\</a> <font color="Olive">q</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  ) )
 <b>by </b><i><a class="txt" href="partit1.html#E3:33"><i><font color="Green">E45</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">q</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E7:33_1_2_2"/><i><font color="Green">E89</font></i>: 
( <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Ca</font> &amp; <font color="Maroon">y1</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">q</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 <b>by </b><i><a class="ref" href="partit1.html#T4" target="_self">Th4</a></i>;<br/>
<b>consider </b><font color="Maroon">pd</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E9:33_1_2_2"/><i><font color="Green">E90</font></i>: 
( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">pd</font> &amp; <font color="Maroon">pd</font> <a href="hidden.html#R2">in</a> <font color="Maroon">PA</font> )
 <b>by </b><i>, <a class="txt" href="partit1.html#E3:33"><i><font color="Green">E45</font></i></a>, <a class="ref" href="partit1.html#T4" target="_self">Th4</a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">pp</font> being    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">PA</font><b> such that </b><br/><a NAME="E11:33_1_2_2"/><i><font color="Green">E91</font></i>: 
( <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">pp</font> &amp;  ex <font color="Olive">f</font> being   <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">Y</font> st <br/>( 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Olive">f</font> &amp; <font color="Olive">f</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Olive">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="Olive">f</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">pp</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="Olive">f</font> holds <br/> ex <font color="Olive">p1</font>, <font color="Olive">p2</font>, <font color="Olive">x0</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">p1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">PA</font> &amp; <font color="Olive">p2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">PB</font> &amp; <font color="Olive">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Olive">p1</font> &amp; <font color="Olive">x0</font> <a href="hidden.html#R2">in</a> <font color="Olive">p1</font> <a href="xboole_0.html#K3">/\</a> <font color="Olive">p2</font> &amp; <font color="Olive">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">p2</font> ) ) ) )
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">fd</font> being    <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">Y</font><b> such that </b><br/><a NAME="E13:33_1_2_2"/><i><font color="Green">E92</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">fd</font> &amp; <font color="Maroon">fd</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">fd</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">fd</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">q</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">fd</font> holds <br/> ex <font color="Olive">p1</font>, <font color="Olive">p2</font>, <font color="Olive">x0</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">p1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">PA</font> &amp; <font color="Olive">p2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">PB</font> &amp; <font color="Maroon">fd</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Olive">p1</font> &amp; <font color="Olive">x0</font> <a href="hidden.html#R2">in</a> <font color="Olive">p1</font> <a href="xboole_0.html#K3">/\</a> <font color="Olive">p2</font> &amp; <font color="Maroon">fd</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">p2</font> ) ) )
 <b>by </b><i><a class="ref" href="partit1.html#T5" target="_self">Th5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">x1</font> = <font color="Maroon">x1</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font> <b>by </b><i><a class="txt" href="partit1.html#E3:33"><i><font color="Green">E45</font></i></a>, <a class="ref" href="partit1.html#T4" target="_self">Th4</a></i>;<br/>
<b>reconsider </b><font color="Maroon">f</font> = <font color="Maroon">fd</font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">x1</font></span><a href="finseq_1.html#K12">*&gt;</a></span> as    <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">Y</font> ;<br/>
<a NAME="E16:33_1_2_2"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">fd</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">x1</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>;<br/>
<a NAME="E17:33_1_2_2"/><b>then </b><i><font color="Green">E93</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">fd</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a></i>;<br/>
<a NAME="E18:33_1_2_2"/>
1 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">fd</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="partit1.html#T7" target="_self">Th7</a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E19:33_1_2_2"/><b>then </b><i><font color="Green">E129</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font>
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/>
<a NAME="E20:33_1_2_2"/><i><font color="Green">E162</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">fd</font> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">x1</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">fd</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">x1</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T59">FINSEQ_1:59</a></i>;<br/>
<a NAME="E21:33_1_2_2"/><i><font color="Green">E163</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">fd</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">y1</font>
 
<b>by </b><i><a class="txt" href="partit1.html#E3:33"><i><font color="Green">E45</font></i></a>, <a class="ref" href="finseq_1.html#T59">FINSEQ_1:59</a></i>;<br/>
<a NAME="E22:33_1_2_2"/>
<font color="Maroon">y1</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">q</font>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>
<b>then consider </b><font color="Maroon">z0</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E24:33_1_2_2"/><i><font color="Green">E164</font></i>: 
( <font color="Maroon">z0</font> <a href="hidden.html#R2">in</a> <font color="Maroon">y1</font> &amp; <font color="Maroon">z0</font> <a href="hidden.html#R2">in</a> <font color="Maroon">q</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>
<a NAME="E25:33_1_2_2"/><i><font color="Green">E165</font></i>: 
<font color="Maroon">z0</font> <a href="hidden.html#R2">in</a> <font color="Maroon">y1</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">q</font>
 
<b>by </b><i><a class="ref" href="partit1.html#D2" target="_self">Def2</a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E26:33_1_2_2"/><i><font color="Green">E166</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">fd</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">fd</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E27:33_1_2_2"/><i><font color="Green">E167</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 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">fd</font> holds <br/><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">fd</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_2_2_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="E1:33_1_2_2_1"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">fd</font> )
 ;<br/>

<a NAME="E2:33_1_2_2_1"/><b>then </b>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">fd</font>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<b>hence </b><a NAME="E3:33_1_2_2_1"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">fd</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E28:33_1_2_2"/><b>then </b><i><font color="Green">E168</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 
<b>by </b><i><a class="ref" href="partit1.html#T7" target="_self">Th7</a></i>;<br/>
<a NAME="E29:33_1_2_2"/><i><font color="Green">E169</font></i>: 
<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">fd</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">q</font>
 
<b>by </b><i><a class="ref" href="partit1.html#T7" target="_self">Th7</a>, </i>;<br/>
<a NAME="E30:33_1_2_2"/><i><font color="Green">E170</font></i>: 
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">fd</font> holds <br/> ex <font color="Olive">p1</font>, <font color="Olive">p2</font>, <font color="Olive">x0</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">p1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">PA</font> &amp; <font color="Olive">p2</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">p1</font> &amp; <font color="Olive">x0</font> <a href="hidden.html#R2">in</a> <font color="Olive">p1</font> <a href="xboole_0.html#K3">/\</a> <font color="Olive">p2</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">p2</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_2_2_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:33_1_2_2_2"/><i><font color="Green">E171</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">fd</font> )
 ;<br/>

<a NAME="E2:33_1_2_2_2"/><b>then </b><i><font color="Green">E172</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">fd</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 
<b>by </b><i/>;<br/>
<a NAME="E3:33_1_2_2_2"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">i</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">fd</font> )
 
<b>by </b><i><a class="txt" href="partit1.html#E7:33_1"><i><font color="Green">E62</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E4:33_1_2_2_2"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">fd</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E5:33_1_2_2_2"/>
 ex <font color="Olive">p1</font>, <font color="Olive">p2</font>, <font color="Olive">x0</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">p1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">PA</font> &amp; <font color="Olive">p2</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">i</font> <a href="hidden.html#R2">in</a> <font color="Olive">p1</font> &amp; <font color="Olive">x0</font> <a href="hidden.html#R2">in</a> <font color="Olive">p1</font> <a href="xboole_0.html#K3">/\</a> <font color="Olive">p2</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a> <font color="Olive">p2</font> )
 <b>by </b><i><a class="ref" href="partit1.html#T7" target="_self">Th7</a>, <a class="txt" href="partit1.html#E7:33_1"><i><font color="Green">E62</font></i></a>, <a class="txt" href="partit1.html#E9:33_1"><i><font color="Green">E63</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E31:33_1_2_2"/>
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">p1</font>, <font color="Olive">p2</font>, <font color="Olive">x0</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">p1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">PA</font> &amp; <font color="Olive">p2</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">p1</font> &amp; <font color="Olive">x0</font> <a href="hidden.html#R2">in</a> <font color="Olive">p1</font> <a href="xboole_0.html#K3">/\</a> <font color="Olive">p2</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">p2</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_2_2_3"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:33_1_2_2_3"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">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> )
 ;<br/>

<a NAME="E2:33_1_2_2_3"/><b>then </b><i><font color="Green">E173</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">fd</font> )
 
<b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_2_2_3_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_2_2_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:33_1_2_2_3_1_1"/>
( ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">fd</font> ) or ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">fd</font> ) )
 </span><b>by </b><i><a class="txt" href="partit1.html#E7:33_1"><i><font color="Green">E62</font></i></a>, <a class="ref" href="xreal_1.html#T1">XREAL_1:1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_2_2_3_1_1_1"><b>case </b></a><a NAME="E1:33_1_2_2_3_1_1_1"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">fd</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:33_1_2_2_3_1_1_1"/>
 ex <font color="Olive">p1</font>, <font color="Olive">p2</font>, <font color="Olive">x0</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">p1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">PA</font> &amp; <font color="Olive">p2</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">i</font> <a href="hidden.html#R2">in</a> <font color="Olive">p1</font> &amp; <font color="Olive">x0</font> <a href="hidden.html#R2">in</a> <font color="Olive">p1</font> <a href="xboole_0.html#K3">/\</a> <font color="Olive">p2</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a> <font color="Olive">p2</font> )
 <b>by </b><i><a class="txt" href="partit1.html#E5:33_1"><i><font color="Green">E61</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_2_2_3_1_1_2"><b>case </b></a><a NAME="E1:33_1_2_2_3_1_1_2"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">fd</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:33_1_2_2_3_1_1_2"/>
 ex <font color="Olive">p1</font>, <font color="Olive">p2</font>, <font color="Olive">x0</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">p1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">PA</font> &amp; <font color="Olive">p2</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">i</font> <a href="hidden.html#R2">in</a> <font color="Olive">p1</font> &amp; <font color="Olive">x0</font> <a href="hidden.html#R2">in</a> <font color="Olive">p1</font> <a href="xboole_0.html#K3">/\</a> <font color="Olive">p2</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a> <font color="Olive">p2</font> )
 <b>by </b><i><a class="ref" href="partit1.html#T4" target="_self">Th4</a>, <a class="ref" href="partit1.html#T5" target="_self">Th5</a>, <a class="ref" href="partit1.html#D1" target="_self">Def1</a>, <a class="ref" href="partit1.html#T8" target="_self">Th8</a>, <a class="txt" href="partit1.html#E4:33_1"><i><font color="Green">E60</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E4:33_1_2_2_3"/>
 ex <font color="Olive">p1</font>, <font color="Olive">p2</font>, <font color="Olive">x0</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">p1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">PA</font> &amp; <font color="Olive">p2</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">i</font> <a href="hidden.html#R2">in</a> <font color="Olive">p1</font> &amp; <font color="Olive">x0</font> <a href="hidden.html#R2">in</a> <font color="Olive">p1</font> <a href="xboole_0.html#K3">/\</a> <font color="Olive">p2</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a> <font color="Olive">p2</font> )
 ;<br/>


</div><b>end;</b></div>
<a NAME="E32:33_1_2_2"/><b>then </b>
<font color="Maroon">pd</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Ca</font>
 
<b>by </b><i><a class="txt" href="partit1.html#E4:33"><i><font color="Green">E48</font></i></a>, , <a class="txt" href="partit1.html#E5:33"><i><font color="Green">E52</font></i></a>, <a class="txt" href="partit1.html#E3:33_1"><i><font color="Green">E53</font></i></a>, </i>;<br/>
<b>hence </b><a NAME="E33:33_1_2_2"/>
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">pb</font>
 <b>by </b><i><a class="txt" href="partit1.html#E4:33"><i><font color="Green">E48</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:33_1_2"/>
<font color="Maroon">pb</font> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">Cb</font>
 <b>by </b><i><a class="ref" href="partit1.html#T3" target="_self">Th3</a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E28:33_1"/><i><font color="Green">E174</font></i>: 
<font color="Maroon">pb</font> <a href="partit1.html#R1" target="_self">is_a_dependent_set_of</a> <font color="Maroon">PB</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_3"><b>proof </b></a><div class="add">

<a NAME="E1:33_1_3"/>
<font color="Maroon">Cb</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">PB</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">z</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:33_1_3_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Cb</font>
 ;<br/>

<a NAME="E2:33_1_3_1"/><b>then </b>
 ex <font color="Olive">p</font> being   <a href="subset_1.html#M2">Element</a> of <font color="Maroon">PB</font> st <br/>( <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Olive">p</font> &amp;  ex <font color="Olive">q</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Ca</font> &amp; <font color="Olive">p</font> <a href="subset_1.html#K8">/\</a> <font color="Olive">q</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  ) )
 
;<br/>
<b>hence </b><a NAME="E3:33_1_3_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">PB</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:33_1_3"/>
<font color="Maroon">pb</font> <a href="partit1.html#R1" target="_self">is_a_dependent_set_of</a> <font color="Maroon">PB</font>
 <b>by </b><i><a class="ref" href="partit1.html#T1" target="_self">Th1</a>, <a class="txt" href="partit1.html#E2:33"><i><font color="Green">E43</font></i></a>, </i>;<br/>


</div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_4"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:33_1_4"/>
not <font color="Maroon">pb</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">p0</font>
 ;<br/><a NAME="E2:33_1_4"/><b>then </b>
<font color="Maroon">pb</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">p0</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="xboole_1.html#T37">XBOOLE_1:37</a></i>;<br/><b>then consider </b><font color="Maroon">x1</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:33_1_4"/><i><font color="Green">E175</font></i>: 
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">pb</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">p0</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/><a NAME="E5:33_1_4"/><i><font color="Green">E176</font></i>: 
( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">pb</font> &amp; not <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">p0</font> )
 <b>by </b><i><a class="txt" href="partit1.html#E3:33"><i><font color="Green">E45</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><b>then consider </b><font color="Maroon">y1</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E7:33_1_4"/><i><font color="Green">E177</font></i>: 
( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">y1</font> &amp; <font color="Maroon">y1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Cb</font> )
 <b>by </b><i><a class="txt" href="partit1.html#E2:33"><i><font color="Green">E43</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><b>consider </b><font color="Maroon">p</font> being    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">PB</font><b> such that </b><br/><a NAME="E9:33_1_4"/><i><font color="Green">E178</font></i>: 
( <font color="Maroon">y1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> &amp;  ex <font color="Olive">q</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Ca</font> &amp; <font color="Maroon">p</font> <a href="subset_1.html#K8">/\</a> <font color="Olive">q</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  ) )
 <b>by </b><i/>;<br/><b>consider </b><font color="Maroon">q</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E11:33_1_4"/><i><font color="Green">E179</font></i>: 
( <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Ca</font> &amp; <font color="Maroon">y1</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">q</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 <b>by </b><i><a class="txt" href="partit1.html#E4:33"><i><font color="Green">E48</font></i></a></i>;<br/><b>consider </b><font color="Maroon">pp</font> being    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">PA</font><b> such that </b><br/><a NAME="E13:33_1_4"/><i><font color="Green">E180</font></i>: 
( <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">pp</font> &amp;  ex <font color="Olive">f</font> being   <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">Y</font> st <br/>( 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Olive">f</font> &amp; <font color="Olive">f</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Olive">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="Olive">f</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">pp</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="Olive">f</font> holds <br/> ex <font color="Olive">p1</font>, <font color="Olive">p2</font>, <font color="Olive">x0</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">p1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">PA</font> &amp; <font color="Olive">p2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">PB</font> &amp; <font color="Olive">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Olive">p1</font> &amp; <font color="Olive">x0</font> <a href="hidden.html#R2">in</a> <font color="Olive">p1</font> <a href="xboole_0.html#K3">/\</a> <font color="Olive">p2</font> &amp; <font color="Olive">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">p2</font> ) ) ) )
 <b>by </b><i><a class="ref" href="partit1.html#T5" target="_self">Th5</a></i>;<br/><b>consider </b><font color="Maroon">fd</font> being    <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">Y</font><b> such that </b><br/><a NAME="E15:33_1_4"/><i><font color="Green">E181</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">fd</font> &amp; <font color="Maroon">fd</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">fd</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">fd</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">q</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">fd</font> holds <br/> ex <font color="Olive">p1</font>, <font color="Olive">p2</font>, <font color="Olive">x0</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">p1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">PA</font> &amp; <font color="Olive">p2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">PB</font> &amp; <font color="Maroon">fd</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Olive">p1</font> &amp; <font color="Olive">x0</font> <a href="hidden.html#R2">in</a> <font color="Olive">p1</font> <a href="xboole_0.html#K3">/\</a> <font color="Olive">p2</font> &amp; <font color="Maroon">fd</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">p2</font> ) ) )
 <b>by </b><i><a class="ref" href="partit1.html#T7" target="_self">Th7</a></i>;<br/><b>reconsider </b><font color="Maroon">x1</font> = <font color="Maroon">x1</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font> <b>by </b><i>, <a class="txt" href="partit1.html#E4:33"><i><font color="Green">E48</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon">f</font> = <font color="Maroon">fd</font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">x1</font></span><a href="finseq_1.html#K12">*&gt;</a></span> as    <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">Y</font> ;<br/><a NAME="E18:33_1_4"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">fd</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">x1</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>;<br/><a NAME="E19:33_1_4"/><b>then </b><i><font color="Green">E182</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">fd</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a></i>;<br/><a NAME="E20:33_1_4"/>
1 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">fd</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E21:33_1_4"/><b>then </b><i><font color="Green">E183</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="txt" href="partit1.html#E5:33"><i><font color="Green">E52</font></i></a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><a NAME="E22:33_1_4"/><i><font color="Green">E184</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">fd</font> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">x1</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">fd</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">x1</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T59">FINSEQ_1:59</a></i>;<br/><a NAME="E23:33_1_4"/><i><font color="Green">E185</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">fd</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">y1</font>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T59">FINSEQ_1:59</a></i>;<br/><a NAME="E24:33_1_4"/>
<font color="Maroon">y1</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">q</font>
 <b>by </b><i><a class="ref" href="partit1.html#T5" target="_self">Th5</a>, <a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/><b>then consider </b><font color="Maroon">z0</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E26:33_1_4"/><i><font color="Green">E186</font></i>: 
( <font color="Maroon">z0</font> <a href="hidden.html#R2">in</a> <font color="Maroon">y1</font> &amp; <font color="Maroon">z0</font> <a href="hidden.html#R2">in</a> <font color="Maroon">q</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/><a NAME="E27:33_1_4"/><i><font color="Green">E187</font></i>: 
<font color="Maroon">z0</font> <a href="hidden.html#R2">in</a> <font color="Maroon">y1</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">q</font>
 <b>by </b><i><a class="ref" href="partit1.html#T8" target="_self">Th8</a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E28:33_1_4"/><i><font color="Green">E188</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">fd</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">fd</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E29:33_1_4"/><i><font color="Green">E189</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 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">fd</font> holds <br/><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">fd</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_4_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="E1:33_1_4_1"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">fd</font> )
 ;<br/>

<a NAME="E2:33_1_4_1"/><b>then </b>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">fd</font>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<b>hence </b><a NAME="E3:33_1_4_1"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">fd</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>;<br/>


</div><b>end;</b></div><a NAME="E30:33_1_4"/><b>then </b><i><font color="Green">E190</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 <b>by </b><i/>;<br/><a NAME="E31:33_1_4"/><i><font color="Green">E191</font></i>: 
<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">fd</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">q</font>
 <b>by </b><i>, </i>;<br/><a NAME="E32:33_1_4"/><i><font color="Green">E192</font></i>: 
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">fd</font> holds <br/> ex <font color="Olive">p1</font>, <font color="Olive">p2</font>, <font color="Olive">x0</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">p1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">PA</font> &amp; <font color="Olive">p2</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">p1</font> &amp; <font color="Olive">x0</font> <a href="hidden.html#R2">in</a> <font color="Olive">p1</font> <a href="xboole_0.html#K3">/\</a> <font color="Olive">p2</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">p2</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_4_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:33_1_4_2"/><i><font color="Green">E193</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">fd</font> )
 ;<br/>

<a NAME="E2:33_1_4_2"/><b>then </b><i><font color="Green">E194</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">fd</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 
<b>by </b><i/>;<br/>
<a NAME="E3:33_1_4_2"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">i</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">fd</font> )
 
<b>by </b><i><a class="txt" href="partit1.html#E9:33_1"><i><font color="Green">E63</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E4:33_1_4_2"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">fd</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E5:33_1_4_2"/>
 ex <font color="Olive">p1</font>, <font color="Olive">p2</font>, <font color="Olive">x0</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">p1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">PA</font> &amp; <font color="Olive">p2</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">i</font> <a href="hidden.html#R2">in</a> <font color="Olive">p1</font> &amp; <font color="Olive">x0</font> <a href="hidden.html#R2">in</a> <font color="Olive">p1</font> <a href="xboole_0.html#K3">/\</a> <font color="Olive">p2</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a> <font color="Olive">p2</font> )
 <b>by </b><i>, <a class="txt" href="partit1.html#E9:33_1"><i><font color="Green">E63</font></i></a>, <a class="txt" href="partit1.html#E15:33_1"><i><font color="Green">E64</font></i></a></i>;<br/>


</div><b>end;</b></div><a NAME="E33:33_1_4"/><i><font color="Green">E195</font></i>: 
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">p1</font>, <font color="Olive">p2</font>, <font color="Olive">x0</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">p1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">PA</font> &amp; <font color="Olive">p2</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">p1</font> &amp; <font color="Olive">x0</font> <a href="hidden.html#R2">in</a> <font color="Olive">p1</font> <a href="xboole_0.html#K3">/\</a> <font color="Olive">p2</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">p2</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_4_3"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:33_1_4_3"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">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> )
 ;<br/>

<a NAME="E2:33_1_4_3"/><b>then </b><i><font color="Green">E196</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">fd</font> )
 
<b>by </b><i><a class="txt" href="partit1.html#E5:33"><i><font color="Green">E52</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_4_3_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_4_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:33_1_4_3_1_1"/>
( ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">fd</font> ) or ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">fd</font> ) )
 </span><b>by </b><i><a class="txt" href="partit1.html#E15:33_1"><i><font color="Green">E64</font></i></a>, <a class="ref" href="xreal_1.html#T1">XREAL_1:1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_4_3_1_1_1"><b>case </b></a><a NAME="E1:33_1_4_3_1_1_1"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">fd</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:33_1_4_3_1_1_1"/>
 ex <font color="Olive">p1</font>, <font color="Olive">p2</font>, <font color="Olive">x0</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">p1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">PA</font> &amp; <font color="Olive">p2</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">i</font> <a href="hidden.html#R2">in</a> <font color="Olive">p1</font> &amp; <font color="Olive">x0</font> <a href="hidden.html#R2">in</a> <font color="Olive">p1</font> <a href="xboole_0.html#K3">/\</a> <font color="Olive">p2</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a> <font color="Olive">p2</font> )
 <b>by </b><i><a class="txt" href="partit1.html#E7:33_1"><i><font color="Green">E62</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_4_3_1_1_2"><b>case </b></a><a NAME="E1:33_1_4_3_1_1_2"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">fd</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:33_1_4_3_1_1_2"/>
 ex <font color="Olive">p1</font>, <font color="Olive">p2</font>, <font color="Olive">x0</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">p1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">PA</font> &amp; <font color="Olive">p2</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">i</font> <a href="hidden.html#R2">in</a> <font color="Olive">p1</font> &amp; <font color="Olive">x0</font> <a href="hidden.html#R2">in</a> <font color="Olive">p1</font> <a href="xboole_0.html#K3">/\</a> <font color="Olive">p2</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a> <font color="Olive">p2</font> )
 <b>by </b><i><a class="txt" href="partit1.html#E4:33"><i><font color="Green">E48</font></i></a>, <a class="ref" href="partit1.html#T7" target="_self">Th7</a>, <a class="ref" href="partit1.html#D2" target="_self">Def2</a>, , <a class="txt" href="partit1.html#E5:33_1"><i><font color="Green">E61</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E4:33_1_4_3"/>
 ex <font color="Olive">p1</font>, <font color="Olive">p2</font>, <font color="Olive">x0</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">p1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">PA</font> &amp; <font color="Olive">p2</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">i</font> <a href="hidden.html#R2">in</a> <font color="Olive">p1</font> &amp; <font color="Olive">x0</font> <a href="hidden.html#R2">in</a> <font color="Olive">p1</font> <a href="xboole_0.html#K3">/\</a> <font color="Olive">p2</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a> <font color="Olive">p2</font> )
 ;<br/>


</div><b>end;</b></div><a NAME="E34:33_1_4"/>
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">p1</font>, <font color="Olive">p2</font>, <font color="Olive">x0</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">p1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">PA</font> &amp; <font color="Olive">p2</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">p1</font> &amp; <font color="Olive">x0</font> <a href="hidden.html#R2">in</a> <font color="Olive">p1</font> &amp; <font color="Olive">x0</font> <a href="hidden.html#R2">in</a> <font color="Olive">p2</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">p2</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_4_4"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:33_1_4_4"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">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> )
 ;<br/>

<b>then consider </b><font color="Maroon">p1</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">p2</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">x0</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:33_1_4_4"/><i><font color="Green">E197</font></i>: 
( <font color="Maroon">p1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">PA</font> &amp; <font color="Maroon">p2</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">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">p1</font> &amp; <font color="Maroon">x0</font> <a href="hidden.html#R2">in</a> <font color="Maroon">p1</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">p2</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">p2</font> )
 <b>by </b><i><a class="txt" href="partit1.html#E9:33_1"><i><font color="Green">E63</font></i></a></i>;<br/>
<a NAME="E4:33_1_4_4"/>
( <font color="Maroon">x0</font> <a href="hidden.html#R2">in</a> <font color="Maroon">p1</font> &amp; <font color="Maroon">x0</font> <a href="hidden.html#R2">in</a> <font color="Maroon">p2</font> )
 
<b>by </b><i><a class="txt" href="partit1.html#E15:33_1"><i><font color="Green">E64</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>hence </b><a NAME="E5:33_1_4_4"/>
 ex <font color="Olive">p1</font>, <font color="Olive">p2</font>, <font color="Olive">x0</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">p1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">PA</font> &amp; <font color="Olive">p2</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">i</font> <a href="hidden.html#R2">in</a> <font color="Olive">p1</font> &amp; <font color="Olive">x0</font> <a href="hidden.html#R2">in</a> <font color="Olive">p1</font> &amp; <font color="Olive">x0</font> <a href="hidden.html#R2">in</a> <font color="Olive">p2</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a> <font color="Olive">p2</font> )
 <b>by </b><i><a class="txt" href="partit1.html#E15:33_1"><i><font color="Green">E64</font></i></a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E35:33_1_4"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="partit1.html#T4" target="_self">Th4</a>, <a class="txt" href="partit1.html#E5:33"><i><font color="Green">E52</font></i></a>, <a class="txt" href="partit1.html#E3:33_1"><i><font color="Green">E53</font></i></a>, <a class="ref" href="partit1.html#D1" target="_self">Def1</a>, <a class="txt" href="partit1.html#E4:33_1"><i><font color="Green">E60</font></i></a>, </i>;<br/></div><b>end;</b></div>
<a NAME="E30:33_1"/><b>then </b>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">pb</font>
 
<b>by </b><i>, , <a class="txt" href="partit1.html#E1:33"><i><font color="Green">E42</font></i></a>, <a class="ref" href="partit1.html#T3" target="_self">Th3</a>, </i>;<br/>
<b>then consider </b><font color="Maroon">y1</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E32:33_1"/><i><font color="Green">E198</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">y1</font> &amp; <font color="Maroon">y1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Cb</font> )
 <b>by </b><i><a class="txt" href="partit1.html#E2:33"><i><font color="Green">E43</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">p</font> being    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">PB</font><b> such that </b><br/><a NAME="E34:33_1"/><i><font color="Green">E199</font></i>: 
( <font color="Maroon">y1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> &amp;  ex <font color="Olive">q</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Ca</font> &amp; <font color="Maroon">p</font> <a href="subset_1.html#K8">/\</a> <font color="Olive">q</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  ) )
 <b>by </b><i><a class="txt" href="partit1.html#E3:33"><i><font color="Green">E45</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">q</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E36:33_1"/><i><font color="Green">E200</font></i>: 
( <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Ca</font> &amp; <font color="Maroon">y1</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">q</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 <b>by </b><i><a class="ref" href="partit1.html#T4" target="_self">Th4</a></i>;<br/>
<b>consider </b><font color="Maroon">pd</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E38:33_1"/><i><font color="Green">E201</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">pd</font> &amp; <font color="Maroon">pd</font> <a href="hidden.html#R2">in</a> <font color="Maroon">PA</font> )
 <b>by </b><i>, <a class="txt" href="partit1.html#E3:33"><i><font color="Green">E45</font></i></a>, <a class="ref" href="partit1.html#T4" target="_self">Th4</a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">pp</font> being    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">PA</font><b> such that </b><br/><a NAME="E40:33_1"/><i><font color="Green">E202</font></i>: 
( <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">pp</font> &amp;  ex <font color="Olive">f</font> being   <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">Y</font> st <br/>( 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Olive">f</font> &amp; <font color="Olive">f</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Olive">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="Olive">f</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">pp</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="Olive">f</font> holds <br/> ex <font color="Olive">p1</font>, <font color="Olive">p2</font>, <font color="Olive">x0</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">p1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">PA</font> &amp; <font color="Olive">p2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">PB</font> &amp; <font color="Olive">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Olive">p1</font> &amp; <font color="Olive">x0</font> <a href="hidden.html#R2">in</a> <font color="Olive">p1</font> <a href="xboole_0.html#K3">/\</a> <font color="Olive">p2</font> &amp; <font color="Olive">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">p2</font> ) ) ) )
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">fd</font> being    <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">Y</font><b> such that </b><br/><a NAME="E42:33_1"/><i><font color="Green">E203</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">fd</font> &amp; <font color="Maroon">fd</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">fd</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">fd</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">q</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">fd</font> holds <br/> ex <font color="Olive">p1</font>, <font color="Olive">p2</font>, <font color="Olive">x0</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">p1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">PA</font> &amp; <font color="Olive">p2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">PB</font> &amp; <font color="Maroon">fd</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Olive">p1</font> &amp; <font color="Olive">x0</font> <a href="hidden.html#R2">in</a> <font color="Olive">p1</font> <a href="xboole_0.html#K3">/\</a> <font color="Olive">p2</font> &amp; <font color="Maroon">fd</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">p2</font> ) ) )
 <b>by </b><i><a class="ref" href="partit1.html#T5" target="_self">Th5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">y'</font> = <font color="Maroon">y</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font> <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">f</font> = <font color="Maroon">fd</font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">y'</font></span><a href="finseq_1.html#K12">*&gt;</a></span> as    <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">Y</font> ;<br/>
<a NAME="E45:33_1"/><i><font color="Green">E205</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">fd</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">y'</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>;<br/>
<a NAME="E46:33_1"/><b>then </b><i><font color="Green">E206</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">fd</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a></i>;<br/>
<a NAME="E47:33_1"/><b>then </b><i><font color="Green">E207</font></i>: 
1 <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>
 
<b>by </b><i><a class="ref" href="partit1.html#T7" target="_self">Th7</a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E48:33_1"/><i><font color="Green">E208</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">fd</font> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">y'</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">fd</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T59">FINSEQ_1:59</a></i>;<br/>
<a NAME="E49:33_1"/><i><font color="Green">E209</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">fd</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">y1</font>
 
<b>by </b><i><a class="txt" href="partit1.html#E3:33"><i><font color="Green">E45</font></i></a>, <a class="ref" href="finseq_1.html#T59">FINSEQ_1:59</a></i>;<br/>
<a NAME="E50:33_1"/>
<font color="Maroon">y1</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">q</font>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>
<b>then consider </b><font color="Maroon">z0</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E52:33_1"/><i><font color="Green">E210</font></i>: 
( <font color="Maroon">z0</font> <a href="hidden.html#R2">in</a> <font color="Maroon">y1</font> &amp; <font color="Maroon">z0</font> <a href="hidden.html#R2">in</a> <font color="Maroon">q</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>
<a NAME="E53:33_1"/><i><font color="Green">E211</font></i>: 
<font color="Maroon">z0</font> <a href="hidden.html#R2">in</a> <font color="Maroon">y1</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">q</font>
 
<b>by </b><i><a class="ref" href="partit1.html#T8" target="_self">Th8</a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E54:33_1"/><i><font color="Green">E212</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">fd</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">fd</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E55:33_1"/><i><font color="Green">E213</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 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">fd</font> holds <br/><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">fd</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_5"><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="E1:33_1_5"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">fd</font> )
 ;<br/>

<a NAME="E2:33_1_5"/><b>then </b>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">fd</font>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<b>hence </b><a NAME="E3:33_1_5"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">fd</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E56:33_1"/><b>then </b><i><font color="Green">E214</font></i>: 
<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">fd</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">q</font>
 
<b>by </b><i><a class="ref" href="partit1.html#T7" target="_self">Th7</a></i>;<br/>
<a NAME="E57:33_1"/><i><font color="Green">E215</font></i>: 
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">fd</font> holds <br/> ex <font color="Olive">p1</font>, <font color="Olive">p2</font>, <font color="Olive">x0</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">p1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">PA</font> &amp; <font color="Olive">p2</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">p1</font> &amp; <font color="Olive">x0</font> <a href="hidden.html#R2">in</a> <font color="Olive">p1</font> <a href="xboole_0.html#K3">/\</a> <font color="Olive">p2</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">p2</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_6"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:33_1_6"/><i><font color="Green">E216</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">fd</font> )
 ;<br/>

<a NAME="E2:33_1_6"/><b>then </b><i><font color="Green">E217</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">fd</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 
<b>by </b><i/>;<br/>
<a NAME="E3:33_1_6"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">i</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">fd</font> )
 
<b>by </b><i><a class="txt" href="partit1.html#E7:33_1"><i><font color="Green">E62</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E4:33_1_6"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">fd</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E5:33_1_6"/>
 ex <font color="Olive">p1</font>, <font color="Olive">p2</font>, <font color="Olive">x0</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">p1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">PA</font> &amp; <font color="Olive">p2</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">i</font> <a href="hidden.html#R2">in</a> <font color="Olive">p1</font> &amp; <font color="Olive">x0</font> <a href="hidden.html#R2">in</a> <font color="Olive">p1</font> <a href="xboole_0.html#K3">/\</a> <font color="Olive">p2</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a> <font color="Olive">p2</font> )
 <b>by </b><i><a class="ref" href="partit1.html#T7" target="_self">Th7</a>, <a class="txt" href="partit1.html#E7:33_1"><i><font color="Green">E62</font></i></a>, <a class="txt" href="partit1.html#E9:33_1"><i><font color="Green">E63</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E58:33_1"/><i><font color="Green">E218</font></i>: 
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">p1</font>, <font color="Olive">p2</font>, <font color="Olive">x0</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">p1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">PA</font> &amp; <font color="Olive">p2</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">p1</font> &amp; <font color="Olive">x0</font> <a href="hidden.html#R2">in</a> <font color="Olive">p1</font> <a href="xboole_0.html#K3">/\</a> <font color="Olive">p2</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">p2</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_7"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:33_1_7"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">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> )
 ;<br/>

<a NAME="E2:33_1_7"/><b>then </b><i><font color="Green">E219</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">fd</font> )
 
<b>by </b><i><a class="txt" href="partit1.html#E5:33"><i><font color="Green">E52</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_7_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_7_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:33_1_7_1_1"/>
( ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">fd</font> ) or ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">fd</font> ) )
 </span><b>by </b><i><a class="txt" href="partit1.html#E9:33_1"><i><font color="Green">E63</font></i></a>, <a class="ref" href="xreal_1.html#T1">XREAL_1:1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_7_1_1_1"><b>case </b></a><a NAME="E1:33_1_7_1_1_1"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">fd</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:33_1_7_1_1_1"/>
 ex <font color="Olive">p1</font>, <font color="Olive">p2</font>, <font color="Olive">x0</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">p1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">PA</font> &amp; <font color="Olive">p2</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">i</font> <a href="hidden.html#R2">in</a> <font color="Olive">p1</font> &amp; <font color="Olive">x0</font> <a href="hidden.html#R2">in</a> <font color="Olive">p1</font> <a href="xboole_0.html#K3">/\</a> <font color="Olive">p2</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a> <font color="Olive">p2</font> )
 <b>by </b><i><a class="txt" href="partit1.html#E5:33_1"><i><font color="Green">E61</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_7_1_1_2"><b>case </b></a><a NAME="E1:33_1_7_1_1_2"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">fd</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:33_1_7_1_1_2"/>
 ex <font color="Olive">p1</font>, <font color="Olive">p2</font>, <font color="Olive">x0</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">p1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">PA</font> &amp; <font color="Olive">p2</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">i</font> <a href="hidden.html#R2">in</a> <font color="Olive">p1</font> &amp; <font color="Olive">x0</font> <a href="hidden.html#R2">in</a> <font color="Olive">p1</font> <a href="xboole_0.html#K3">/\</a> <font color="Olive">p2</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a> <font color="Olive">p2</font> )
 <b>by </b><i><a class="ref" href="partit1.html#T4" target="_self">Th4</a>, <a class="ref" href="partit1.html#T5" target="_self">Th5</a>, <a class="ref" href="partit1.html#D2" target="_self">Def2</a>, , <a class="txt" href="partit1.html#E4:33_1"><i><font color="Green">E60</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E4:33_1_7"/>
 ex <font color="Olive">p1</font>, <font color="Olive">p2</font>, <font color="Olive">x0</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">p1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">PA</font> &amp; <font color="Olive">p2</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">i</font> <a href="hidden.html#R2">in</a> <font color="Olive">p1</font> &amp; <font color="Olive">x0</font> <a href="hidden.html#R2">in</a> <font color="Olive">p1</font> <a href="xboole_0.html#K3">/\</a> <font color="Olive">p2</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a> <font color="Olive">p2</font> )
 ;<br/>


</div><b>end;</b></div>
<a NAME="E59:33_1"/><b>then </b><i><font color="Green">E220</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</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#R2">in</a> <font color="Maroon">pd</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">p1</font>, <font color="Olive">p2</font>, <font color="Olive">x0</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">p1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">PA</font> &amp; <font color="Olive">p2</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">p1</font> &amp; <font color="Olive">x0</font> <a href="hidden.html#R2">in</a> <font color="Olive">p1</font> <a href="xboole_0.html#K3">/\</a> <font color="Olive">p2</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">p2</font> ) ) )
 
<b>by </b><i><a class="txt" href="partit1.html#E4:33"><i><font color="Green">E48</font></i></a>, <a class="ref" href="partit1.html#T7" target="_self">Th7</a>, , <a class="txt" href="partit1.html#E3:33_1"><i><font color="Green">E53</font></i></a>, <a class="ref" href="partit1.html#D1" target="_self">Def1</a>, , <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a>, <a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a></i>;<br/>
<a NAME="E60:33_1"/>
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">u</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font></span>)</span>,<font color="Olive">u</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="partit1.html#K4" target="_self">ERl</a> <font color="Maroon">PA</font></span>)</span> <a href="eqrel_1.html#K3">\/</a> <span class="p1">(<span class="default"><a href="partit1.html#K4" target="_self">ERl</a> <font color="Maroon">PB</font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">u</font>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Olive">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="partit1.html#K4" target="_self">ERl</a> <font color="Maroon">PA</font></span>)</span> <a href="eqrel_1.html#K3">\/</a> <span class="p1">(<span class="default"><a href="partit1.html#K4" target="_self">ERl</a> <font color="Maroon">PB</font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_8"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:33_1_8"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">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> )
 ;<br/>

<b>then consider </b><font color="Maroon">p1</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">p2</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="E3:33_1_8"/><i><font color="Green">E221</font></i>: 
( <font color="Maroon">p1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">PA</font> &amp; <font color="Maroon">p2</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">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">p1</font> &amp; <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">p1</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">p2</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">p2</font> )
 <b>by </b><i><a class="txt" href="partit1.html#E7:33_1"><i><font color="Green">E62</font></i></a></i>;<br/>
<a NAME="E4:33_1_8"/><i><font color="Green">E222</font></i>: 
( <font color="Maroon">p1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">PA</font> &amp; <font color="Maroon">p2</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">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">p1</font> &amp; <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">p1</font> &amp; <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">p2</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">p2</font> )
 
<b>by </b><i><a class="txt" href="partit1.html#E15:33_1"><i><font color="Green">E64</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">x2</font> = <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> as    <a href="hidden.html#M1">set</a>  ;<br/>
<b>reconsider </b><font color="Maroon">y2</font> = <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> as    <a href="hidden.html#M1">set</a>  ;<br/>
<a NAME="E7:33_1_8"/><i><font color="Green">E224</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x2</font>,<font color="Maroon">u</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="partit1.html#K4" target="_self">ERl</a> <font color="Maroon">PA</font>
 
<b>by </b><i><a class="txt" href="partit1.html#E16:33_1"><i><font color="Green">E65</font></i></a>, </i>;<br/>
<a NAME="E8:33_1_8"/><i><font color="Green">E225</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">u</font>,<font color="Maroon">y2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="partit1.html#K4" target="_self">ERl</a> <font color="Maroon">PB</font>
 
<b>by </b><i><a class="txt" href="partit1.html#E16:33_1"><i><font color="Green">E65</font></i></a>, </i>;<br/>
<a NAME="E9:33_1_8"/><i><font color="Green">E226</font></i>: 
 <a href="partit1.html#K4" target="_self">ERl</a> <font color="Maroon">PA</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="partit1.html#K4" target="_self">ERl</a> <font color="Maroon">PA</font></span>)</span> <a href="eqrel_1.html#K3">\/</a> <span class="p1">(<span class="default"><a href="partit1.html#K4" target="_self">ERl</a> <font color="Maroon">PB</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E10:33_1_8"/>
 <a href="partit1.html#K4" target="_self">ERl</a> <font color="Maroon">PB</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="partit1.html#K4" target="_self">ERl</a> <font color="Maroon">PA</font></span>)</span> <a href="eqrel_1.html#K3">\/</a> <span class="p1">(<span class="default"><a href="partit1.html#K4" target="_self">ERl</a> <font color="Maroon">PB</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/>
<b>hence </b><a NAME="E11:33_1_8"/>
 ex <font color="Olive">u</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>,<font color="Olive">u</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="partit1.html#K4" target="_self">ERl</a> <font color="Maroon">PA</font></span>)</span> <a href="eqrel_1.html#K3">\/</a> <span class="p1">(<span class="default"><a href="partit1.html#K4" target="_self">ERl</a> <font color="Maroon">PB</font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">u</font>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="partit1.html#K4" target="_self">ERl</a> <font color="Maroon">PA</font></span>)</span> <a href="eqrel_1.html#K3">\/</a> <span class="p1">(<span class="default"><a href="partit1.html#K4" target="_self">ERl</a> <font color="Maroon">PB</font></span>)</span> )
 <b>by </b><i><a class="txt" href="partit1.html#E16:33_1"><i><font color="Green">E65</font></i></a>, <a class="ref" href="partit1.html#T9" target="_self">Th9</a>, <a class="ref" href="partit1.html#T10" target="_self">Th10</a>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E61:33_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x'</font>,<font color="Maroon">y'</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="partit1.html#K4" target="_self">ERl</a> <font color="Maroon">PA</font></span>)</span> <a href="eqrel_1.html#K5">"\/"</a> <span class="p1">(<span class="default"><a href="partit1.html#K4" target="_self">ERl</a> <font color="Maroon">PB</font></span>)</span>
 
<b>by </b><i><a class="txt" href="partit1.html#E5:33"><i><font color="Green">E52</font></i></a>, <a class="ref" href="partit1.html#D1" target="_self">Def1</a>, <a class="txt" href="partit1.html#E9:33_1"><i><font color="Green">E63</font></i></a>, </i>;<br/>
<b>hence </b><a NAME="E62:33_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="partit1.html#K4" target="_self">ERl</a> <font color="Maroon">PA</font></span>)</span> <a href="eqrel_1.html#K5">"\/"</a> <span class="p1">(<span class="default"><a href="partit1.html#K4" target="_self">ERl</a> <font color="Maroon">PB</font></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E6:33"/>
for <font color="Olive">x1</font>, <font color="Olive">x2</font> being   <a href="hidden.html#M1">set</a>   st <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x1</font>,<font color="Olive">x2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="partit1.html#K4" target="_self">ERl</a> <font color="Maroon">PA</font></span>)</span> <a href="eqrel_1.html#K3">\/</a> <span class="p1">(<span class="default"><a href="partit1.html#K4" target="_self">ERl</a> <font color="Maroon">PB</font></span>)</span> holds <br/><span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x1</font>,<font color="Olive">x2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="partit1.html#K4" target="_self">ERl</a> <span class="p1">(<span class="default"><font color="Maroon">PA</font> <a href="partit1.html#K3" target="_self">'\/'</a> <font color="Maroon">PB</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x1</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">x2</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:33_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="partit1.html#K4" target="_self">ERl</a> <font color="Maroon">PA</font></span>)</span> <a href="eqrel_1.html#K3">\/</a> <span class="p1">(<span class="default"><a href="partit1.html#K4" target="_self">ERl</a> <font color="Maroon">PB</font></span>)</span>
 ;<br/>

<a NAME="E2:33_2"/><b>then </b>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="partit1.html#K4" target="_self">ERl</a> <font color="Maroon">PA</font> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="partit1.html#K4" target="_self">ERl</a> <font color="Maroon">PB</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E3:33_2"/><b>then </b><i><font color="Green">E227</font></i>: 
(  ex <font color="Olive">A</font> being  <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">Y</font> st <br/>( <font color="Olive">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">PA</font> &amp; <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Olive">A</font> &amp; <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <font color="Olive">A</font> ) or  ex <font color="Olive">B</font> being  <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">Y</font> st <br/>( <font color="Olive">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">PB</font> &amp; <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Olive">B</font> &amp; <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <font color="Olive">B</font> ) )
 
<b>by </b><i/>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:33_2_1_1"/>
( ( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp;  ex <font color="Olive">A</font> being  <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">Y</font> st <br/>( <font color="Olive">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">PA</font> &amp; <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Olive">A</font> &amp; <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <font color="Olive">A</font> ) ) or ( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp;  ex <font color="Olive">B</font> being  <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">Y</font> st <br/>( <font color="Olive">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">PB</font> &amp; <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Olive">B</font> &amp; <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <font color="Olive">B</font> ) ) )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_2_1_1_1"><b>case </b></a><a NAME="E1:33_2_1_1_1"/>
( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp;  ex <font color="Olive">A</font> being  <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">Y</font> st <br/>( <font color="Olive">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">PA</font> &amp; <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Olive">A</font> &amp; <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <font color="Olive">A</font> ) )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">A</font> being   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">Y</font><b> such that </b><br/><a NAME="E3:33_2_1_1_1"/><i><font color="Green">E228</font></i>: 
( <font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">PA</font> &amp; <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> )
 ;<br/><a NAME="E4:33_2_1_1_1"/>
 ex <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">PA</font> <a href="partit1.html#K3" target="_self">'\/'</a> <font color="Maroon">PB</font> &amp; <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Olive">y</font> )
 <b>by </b><i>, , <a class="ref" href="setfam_1.html#D2">SETFAM_1:def 2</a></i>;<br/><b>hence </b><a NAME="E5:33_2_1_1_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="partit1.html#K4" target="_self">ERl</a> <span class="p1">(<span class="default"><font color="Maroon">PA</font> <a href="partit1.html#K3" target="_self">'\/'</a> <font color="Maroon">PB</font></span>)</span>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_2_1_1_2"><b>case </b></a><a NAME="E1:33_2_1_1_2"/>
( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp;  ex <font color="Olive">B</font> being  <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">Y</font> st <br/>( <font color="Olive">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">PB</font> &amp; <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Olive">B</font> &amp; <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <font color="Olive">B</font> ) )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">B</font> being   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">Y</font><b> such that </b><br/><a NAME="E3:33_2_1_1_2"/><i><font color="Green">E229</font></i>: 
( <font color="Maroon">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">PB</font> &amp; <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> )
 ;<br/><a NAME="E4:33_2_1_1_2"/>
 ex <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">PA</font> <a href="partit1.html#K3" target="_self">'\/'</a> <font color="Maroon">PB</font> &amp; <font color="Maroon">B</font> <a href="tarski.html#R1">c=</a> <font color="Olive">y</font> )
 <b>by </b><i>, , <a class="ref" href="setfam_1.html#D2">SETFAM_1:def 2</a></i>;<br/><b>hence </b><a NAME="E5:33_2_1_1_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="partit1.html#K4" target="_self">ERl</a> <span class="p1">(<span class="default"><font color="Maroon">PA</font> <a href="partit1.html#K3" target="_self">'\/'</a> <font color="Maroon">PB</font></span>)</span>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E5:33_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="partit1.html#K4" target="_self">ERl</a> <span class="p1">(<span class="default"><font color="Maroon">PA</font> <a href="partit1.html#K3" target="_self">'\/'</a> <font color="Maroon">PB</font></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E7:33"/><b>then </b>
<span class="p1">(<span class="default"><a href="partit1.html#K4" target="_self">ERl</a> <font color="Maroon">PA</font></span>)</span> <a href="eqrel_1.html#K3">\/</a> <span class="p1">(<span class="default"><a href="partit1.html#K4" target="_self">ERl</a> <font color="Maroon">PB</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="partit1.html#K4" target="_self">ERl</a> <span class="p1">(<span class="default"><font color="Maroon">PA</font> <a href="partit1.html#K3" target="_self">'\/'</a> <font color="Maroon">PB</font></span>)</span>
 
<b>by </b><i><a class="ref" href="relat_1.html#D3">RELAT_1:def 3</a></i>;<br/>
<a NAME="E8:33"/><b>then </b>
<span class="p1">(<span class="default"><a href="partit1.html#K4" target="_self">ERl</a> <font color="Maroon">PA</font></span>)</span> <a href="eqrel_1.html#K5">"\/"</a> <span class="p1">(<span class="default"><a href="partit1.html#K4" target="_self">ERl</a> <font color="Maroon">PB</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="partit1.html#K4" target="_self">ERl</a> <span class="p1">(<span class="default"><font color="Maroon">PA</font> <a href="partit1.html#K3" target="_self">'\/'</a> <font color="Maroon">PB</font></span>)</span>
 
<b>by </b><i><a class="ref" href="eqrel_1.html#D3">EQREL_1:def 3</a></i>;<br/>
<b>hence </b><a NAME="E9:33"/>
 <a href="partit1.html#K4" target="_self">ERl</a> <span class="p1">(<span class="default"><font color="Maroon">PA</font> <a href="partit1.html#K3" target="_self">'\/'</a> <font color="Maroon">PB</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="partit1.html#K4" target="_self">ERl</a> <font color="Maroon">PA</font></span>)</span> <a href="eqrel_1.html#K5">"\/"</a> <span class="p1">(<span class="default"><a href="partit1.html#K4" target="_self">ERl</a> <font color="Maroon">PB</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


</div>
