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

<b>let </b><font color="Maroon">X</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">S</font> be   <a href="prob_1.html#NM3">SigmaField</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">f</font> be   <a href="partfun1.html#NM1">PartFunc</a> of <font color="Maroon">X</font>, <a href="supinf_1.html#K3">REAL</a> ;<br/><b>let </b><font color="Maroon">A</font> be    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">S</font>;<br/>







<b>assume </b><a NAME="E1:77"/>
<font color="Maroon">f</font> <a href="mesfunc6.html#R2" target="_self">is_simple_func_in</a> <font color="Maroon">S</font>
 ;<br/>

<b>then consider </b><font color="Maroon">F</font> being   <a href="mesfunc2.html#NM1">Finite_Sep_Sequence</a> of <font color="Maroon">S</font><b> such that </b><br/><a NAME="E3:77"/><i><font color="Green">E37</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">F</font></span>)</span>
 <b>and </b><br/><a NAME="E4:77"/><i><font color="Green">E39</font></i>: 
for <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a><br/> for <font color="Olive">x</font>, <font color="Olive">y</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font>  st <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F</font> &amp; <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> holds <br/><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Olive">y</font>
 <b>by </b><i/>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="nat_1.html#NM1">Nat</a>) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">X</font> = <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font>;<br/>
<b>consider </b><font color="Maroon">G</font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E6:77"/><i><font color="Green">E43</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font> &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font></span>)</span> holds <br/><font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">n</font>) ) )
 <b>from </b><i><a class="ref" href="finseq_1.html#S2">FINSEQ_1:sch 2</a>();<br/></i>
<a NAME="E7:77"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">G</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">S</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="77_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">P</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="E1:77_1"/>
<font color="Maroon">P</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">G</font>
 ;<br/>

<b>then consider </b><font color="Maroon">k</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:77_1"/><i><font color="Green">E45</font></i>: 
( <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">G</font> &amp; <font color="Maroon">P</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">k</font> = <font color="Maroon">k</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i/>;<br/>
<a NAME="E5:77_1"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E6:77_1"/><b>then </b><i><font color="Green">E49</font></i>: 
<font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font>
 
<b>by </b><i/>;<br/>
<a NAME="E7:77_1"/>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F</font>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E8:77_1"/><b>then </b>
( <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">F</font> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">F</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">S</font> )
 
<b>by </b><i><a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/>
<b>hence </b><a NAME="E9:77_1"/>
<font color="Maroon">P</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 <b>by </b><i>, , <a class="ref" href="finsub_1.html#D2">FINSUB_1:def 2</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">G</font> = <font color="Maroon">G</font> as    <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">S</font> <b>by </b><i><a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/>
<a NAME="E9:77"/>
for <font color="Olive">i</font>, <font color="Olive">j</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">G</font> &amp; <font color="Olive">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">G</font> &amp; <font color="Olive">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">j</font> holds <br/><font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Olive">j</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="77_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be   <a href="nat_1.html#NM1">Nat</a>, <font color="Maroon">j</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:77_2"/><i><font color="Green">E50</font></i>: 
( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">G</font> &amp; <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">G</font> &amp; <font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">j</font> )
 ;<br/>

<a NAME="E2:77_2"/><b>then </b><i><font color="Green">E77</font></i>: 
( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font></span>)</span> &amp; <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font></span>)</span> )
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E3:77_2"/><b>then </b>
( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F</font> &amp; <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F</font> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E4:77_2"/><b>then </b><i><font color="Green">E78</font></i>: 
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
 
<b>by </b><i>, <a class="ref" href="mesfunc3.html#T4">MESFUNC3:4</a></i>;<br/>
<a NAME="E5:77_2"/>
( <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font> &amp; <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font> )
 
<b>by </b><i>, </i>;<br/>
<b>then </b><span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font></span>)</span> <a href="subset_1.html#K8">/\</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font>
<b>by </b><i><a class="ref" href="xboole_1.html#T16">XBOOLE_1:16</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font>
<b>by </b><i><a class="ref" href="xboole_1.html#T16">XBOOLE_1:16</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="xboole_0.html#K1">{}</a>  <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font>
<b>by </b><i>, <a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>
;<br/>
<b>hence </b><a NAME="E7:77_2"/>
<font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">G</font> = <font color="Maroon">G</font> as   <a href="mesfunc2.html#NM1">Finite_Sep_Sequence</a> of <font color="Maroon">S</font> <b>by </b><i><a class="ref" href="mesfunc3.html#T4">MESFUNC3:4</a></i>;<br/>
<a NAME="E11:77"/>
for <font color="Olive">v</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">v</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">A</font></span>)</span> holds <br/><font color="Olive">v</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">G</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="77_3"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:77_3"/>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">A</font></span>)</span>
 ;<br/>

<a NAME="E2:77_3"/><b>then </b>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/>
<a NAME="E3:77_3"/><b>then </b><i><font color="Green">E79</font></i>: 
( <font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon">W</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:77_3"/><i><font color="Green">E80</font></i>: 
( <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font> &amp; <font color="Maroon">W</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">F</font> )
 <b>by </b><i>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">k</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E7:77_3"/><i><font color="Green">E81</font></i>: 
( <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F</font> &amp; <font color="Maroon">W</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> )
 <b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">k</font> = <font color="Maroon">k</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i/>;<br/>
<a NAME="E9:77_3"/><i><font color="Green">E82</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E10:77_3"/><b>then </b>
<font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font>
 
<b>by </b><i/>;<br/>
<a NAME="E11:77_3"/><b>then </b><i><font color="Green">E83</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>
 
<b>by </b><i>, , , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E12:77_3"/>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">G</font>
 
<b>by </b><i>, , <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E13:77_3"/><b>then </b>
<font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">G</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/>
<b>hence </b><a NAME="E14:77_3"/>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">G</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E12:77"/><b>then </b><i><font color="Green">E84</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">A</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">G</font></span>)</span>
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<a NAME="E13:77"/>
for <font color="Olive">v</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">v</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">G</font></span>)</span> holds <br/><font color="Olive">v</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">A</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="77_4"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:77_4"/>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">G</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">W</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:77_4"/><i><font color="Green">E85</font></i>: 
( <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font> &amp; <font color="Maroon">W</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">G</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">k</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:77_4"/><i><font color="Green">E86</font></i>: 
( <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">G</font> &amp; <font color="Maroon">W</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> )
 <b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">k</font> = <font color="Maroon">k</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i/>;<br/>
<a NAME="E7:77_4"/><i><font color="Green">E87</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E8:77_4"/><b>then </b>
<font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font>
 
<b>by </b><i/>;<br/>
<a NAME="E9:77_4"/><b>then </b><i><font color="Green">E88</font></i>: 
( <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> &amp; <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> )
 
<b>by </b><i>, , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E10:77_4"/>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F</font>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E11:77_4"/><b>then </b>
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">F</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/>
<a NAME="E12:77_4"/><b>then </b>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">F</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<a NAME="E13:77_4"/><b>then </b>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font>
 
<b>by </b><i>, , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>hence </b><a NAME="E14:77_4"/>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">A</font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E14:77"/><b>then </b>
 <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">G</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">A</font></span>)</span>
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<a NAME="E15:77"/><b>then </b><i><font color="Green">E89</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">G</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>
<a NAME="E16:77"/>
for <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a><br/> for <font color="Olive">x</font>, <font color="Olive">y</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font>  st <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">G</font> &amp; <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">A</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">A</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="77_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font>, <font color="Maroon">y</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font>;<br/>



<b>assume </b><a NAME="E1:77_5"/><i><font color="Green">E90</font></i>: 
( <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">G</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> )
 ;<br/>

<a NAME="E2:77_5"/><b>then </b>
<font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">G</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/>
<a NAME="E3:77_5"/><b>then </b><i><font color="Green">E91</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">A</font></span>)</span> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">A</font></span>)</span> )
 
<b>by </b><i>, , <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<a NAME="E4:77_5"/><i><font color="Green">E92</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E5:77_5"/><b>then </b>
<font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font>
 
<b>by </b><i/>;<br/>
<a NAME="E6:77_5"/><b>then </b><i><font color="Green">E93</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> )
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E7:77_5"/>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F</font>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E8:77_5"/><b>then </b>
<font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">y</font>
 
<b>by </b><i><a class="ref" href="mesfunc6.html#T1" target="_self">Th1</a>, </i>;<br/>
<a NAME="E9:77_5"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">A</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">y</font>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>;<br/>
<b>hence </b><a NAME="E10:77_5"/>
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">A</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">A</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">y</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E17:77"/>
<font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">A</font> <a href="mesfunc6.html#R2" target="_self">is_simple_func_in</a> <font color="Maroon">S</font>
 <b>by </b><i>, </i>;<br/>


</div>
