<?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="numbers.html#K6">ExtREAL</a> ;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:16"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">f</font> <a href="mesfunc2.html#R2">is_simple_func_in</a> <font color="Maroon">S</font>
 <b>and </b><br/><a NAME="E2:16"/><i><font color="Green">E36</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> holds <br/> <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">f</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">x</font>
 <b>and </b><br/><a NAME="E3:16"/><i><font color="Green">E37</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> holds <br/> <a href="supinf_2.html#K1">0.</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">f</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">x</font>
 ;<br/>

<b>consider </b><font color="Maroon">F1</font> being   <a href="mesfunc2.html#NM1">Finite_Sep_Sequence</a> of <font color="Maroon">S</font><b> such that </b><br/><a NAME="E5:16"/><i><font color="Green">E38</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">F1</font></span>)</span>
 <b>and </b><br/><a NAME="E6:16"/><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="finsop_1.html#K5">dom</a> <font color="Maroon">F1</font> &amp; <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F1</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">F1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> holds <br/><font color="Maroon">f</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">y</font>
 <b>by </b><i>, <a class="ref" href="mesfunc2.html#D5">MESFUNC2:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon">G</font> being   <a href="mesfunc2.html#NM1">Finite_Sep_Sequence</a> of <font color="Maroon">S</font><b> such that </b><br/><a NAME="E8:16"/><i><font color="Green">E40</font></i>: 
 <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">F1</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>and </b><br/><a NAME="E9:16"/><i><font color="Green">E42</font></i>: 
for <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">G</font> holds <br/>( <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp;  ex <font color="Olive">m</font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( <font color="Olive">m</font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">F1</font> &amp; <font color="Maroon">F1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> ) )
 <b>by </b><i/>;<br/>
<b>set </b><font color="Maroon">F</font> = <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="finseq_1.html#K9">*&gt;</a></span> <a href="finseq_1.html#K7">^</a> <font color="Maroon">G</font>;<br/>
<a NAME="E10:16"/>
(  <a href="relat_1.html#K2">rng</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="finseq_1.html#K9">*&gt;</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span> &amp;  <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T55">FINSEQ_1:55</a>, <a class="ref" href="measure1.html#T21">MEASURE1:21</a></i>;<br/>
<a NAME="E11:16"/><b>then </b>
(  <a href="relat_1.html#K2">rng</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="finseq_1.html#K9">*&gt;</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">S</font> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">G</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">S</font> )
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/>
<a NAME="E12:16"/><b>then </b>
<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="finseq_1.html#K9">*&gt;</a></span></span>)</span> <a href="xboole_0.html#K2">\/</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> <font color="Maroon">S</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T8">XBOOLE_1:8</a></i>;<br/>
<a NAME="E13:16"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="finseq_1.html#K9">*&gt;</a></span> <a href="finseq_1.html#K7">^</a> <font color="Maroon">G</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">S</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T44">FINSEQ_1:44</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">F</font> = <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="finseq_1.html#K9">*&gt;</a></span> <a href="finseq_1.html#K7">^</a> <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="E15:16"/>
for <font color="Olive">x</font>, <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">y</font> holds <br/><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Olive">y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_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> ;<br/>

<b>assume </b><a NAME="E1:16_1"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">y</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:16_1_1_1"/>
( ( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">F</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">F</font> ) or  not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">F</font> or  not <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">F</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_1_1_1"><b>case </b></a><a NAME="E1:16_1_1_1_1"/><i><font color="Green">E45</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">F</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">F</font> )
 ;<br/><div class="add"><a NAME="E2:16_1_1_1_1"/>
 <a href="finsop_1.html#K5">dom</a> <font color="Maroon">F</font> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K5">NAT</a> 
 <b>by </b><i><a class="ref" href="relset_1.html#T12">RELSET_1:12</a></i>;<br/><b>then reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font>, <font color="Maroon">y</font> = <font color="Maroon">y</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="mesfunc3.html#T1" target="_self">Th1</a></i>;<br/><a NAME="E4:16_1_1_1_1"/>
( <font color="Maroon">x</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> <span class="p2">(<span class="default"><span class="p3"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="finseq_1.html#K9">*&gt;</a></span> <a href="finseq_1.html#K7">^</a> <font color="Maroon">G</font></span>)</span></span>)</span> &amp; <font color="Maroon">y</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> <span class="p2">(<span class="default"><span class="p3"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="finseq_1.html#K9">*&gt;</a></span> <a href="finseq_1.html#K7">^</a> <font color="Maroon">G</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="mesfunc3.html#T1" target="_self">Th1</a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E5:16_1_1_1_1"/><b>then </b><i><font color="Green">E46</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="finseq_1.html#K9">*&gt;</a></span> <a href="finseq_1.html#K7">^</a> <font color="Maroon">G</font></span>)</span> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">y</font> &amp; <font color="Maroon">y</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="finseq_1.html#K9">*&gt;</a></span> <a href="finseq_1.html#K7">^</a> <font color="Maroon">G</font></span>)</span> )
 <b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:16_1_1_1_1_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> 1 or ( <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1 &amp; <font color="Maroon">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1 ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_1_1_1_1_1_1"><b>case </b></a><a NAME="E1:16_1_1_1_1_1_1_1"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> 1 )
 ;<br/><div class="add"><a NAME="E2:16_1_1_1_1_1_1_1"/><i><font color="Green">E48</font></i>: 
 <a href="finsop_1.html#K5">dom</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="finseq_1.html#K9">*&gt;</a></span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> 1
 <b>by </b><i><a class="ref" href="finseq_1.html#T55">FINSEQ_1:55</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_1_1_1_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_1_1_1_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:16_1_1_1_1_1_1_1_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> 1 )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_1_1_1_1_1_1_1_1_1"><b>case </b></a><a NAME="E1:16_1_1_1_1_1_1_1_1_1_1"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><a NAME="E2:16_1_1_1_1_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="finseq_1.html#K9">*&gt;</a></span>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E3:16_1_1_1_1_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="finseq_1.html#K9">*&gt;</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>;<br/><a NAME="E4:16_1_1_1_1_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i>, <a class="ref" href="finseq_1.html#D8">FINSEQ_1:def 8</a></i>;<br/><b>hence </b><a NAME="E5:16_1_1_1_1_1_1_1_1_1_1"/>
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T65">XBOOLE_1:65</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_1_1_1_1_1_1_1_1_2"><b>case </b></a><a NAME="E1:16_1_1_1_1_1_1_1_1_1_2"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><a NAME="E2:16_1_1_1_1_1_1_1_1_1_2"/><b>then </b>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="finseq_1.html#K9">*&gt;</a></span>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E3:16_1_1_1_1_1_1_1_1_1_2"/><b>then </b>
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="finseq_1.html#K9">*&gt;</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>;<br/><a NAME="E4:16_1_1_1_1_1_1_1_1_1_2"/><b>then </b>
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i>, <a class="ref" href="finseq_1.html#D8">FINSEQ_1:def 8</a></i>;<br/><b>hence </b><a NAME="E5:16_1_1_1_1_1_1_1_1_1_2"/>
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T65">XBOOLE_1:65</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:16_1_1_1_1_1_1_1"/>
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_1_1_1_1_1_2"><b>case </b></a><a NAME="E1:16_1_1_1_1_1_1_2"/>
( <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1 &amp; <font color="Maroon">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1 )
 ;<br/><div class="add"><a NAME="E2:16_1_1_1_1_1_1_2"/><b>then </b>
( 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x</font> &amp; 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">y</font> )
 <b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E3:16_1_1_1_1_1_1_2"/><b>then </b>
(  <a href="finseq_1.html#K3">len</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="finseq_1.html#K9">*&gt;</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x</font> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="finseq_1.html#K9">*&gt;</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">y</font> )
 <b>by </b><i><a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a></i>;<br/><a NAME="E4:16_1_1_1_1_1_1_2"/><b>then </b><i><font color="Green">E53</font></i>: 
( <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="finseq_1.html#K9">*&gt;</a></span></span>)</span></span>)</span> &amp; <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="finseq_1.html#K9">*&gt;</a></span></span>)</span></span>)</span> )
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T37">FINSEQ_1:37</a></i>;<br/><a NAME="E5:16_1_1_1_1_1_1_2"/>
<font color="Maroon">x</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="finseq_1.html#K9">*&gt;</a></span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">y</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="finseq_1.html#K9">*&gt;</a></span></span>)</span>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E6:16_1_1_1_1_1_1_2"/>
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font>
 <b>by </b><i>, <a class="ref" href="prob_2.html#D3">PROB_2:def 3</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E7:16_1_1_1_1"/>
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_1_1_2"><b>case </b></a><a NAME="E1:16_1_1_1_2"/>
(  not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">F</font> or  not <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">F</font> )
 ;<br/><div class="add"><a NAME="E2:16_1_1_1_2"/><b>then </b>
( <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 <b>by </b><i><a class="ref" href="funct_1.html#D4">FUNCT_1:def 4</a></i>;<br/><b>hence </b><a NAME="E3:16_1_1_1_2"/>
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T65">XBOOLE_1:65</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:16_1"/>
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font>
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">F</font> = <font color="Maroon">F</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="prob_2.html#D3">PROB_2:def 3</a></i>;<br/>
<i><font color="Green">E54</font></i>:  <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> = 
 <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p3"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="finseq_1.html#K9">*&gt;</a></span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">G</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T44">FINSEQ_1:44</a></i>
<br/>.= 
 <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">G</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T55">FINSEQ_1:55</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">G</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="zfmisc_1.html#T96">ZFMISC_1:96</a></i>
<br/>.= 
<a href="xboole_0.html#K1">{}</a>  <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">G</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="zfmisc_1.html#T31">ZFMISC_1:31</a></i>
<br/>.= 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
<b>by </b><i>, </i>
;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>,   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K6">ExtREAL</a> ] <b>means </b>( ( for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font>  st <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">F</font> &amp; <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> 1 holds <br/><font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a>  ) &amp; ( for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font>  st <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">F</font> &amp; <font color="Maroon">a<sub>1</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2 &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="Maroon">a<sub>1</sub></font> holds <br/><font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">x</font> ) );<br/>
<a NAME="E18:16"/><i><font color="Green">E55</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">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> holds <br/> ex <font color="Olive">y</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K6">ExtREAL</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>,<font color="Olive">y</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:16_3"/><i><font color="Green">E56</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>
 ;<br/>

<a NAME="E2:16_3"/>
 ex <font color="Olive">y</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K6">ExtREAL</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font>,<font color="Olive">y</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_3_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_3_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_3_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:16_3_1_1_1"/>
( <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_3_1_1_1_1"><b>case </b></a><a NAME="E1:16_3_1_1_1_1"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><b>take </b><font color="Maroon">y</font> =  <a href="supinf_2.html#K1">0.</a> ;<br/><a NAME="E2:16_3_1_1_1_1"/><i><font color="Green">E58</font></i>: 
for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font>  st <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">F</font> &amp; <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 1 holds <br/><font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a> 
 ;<br/><a NAME="E3:16_3_1_1_1_1"/>
for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font>  st <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">F</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2 &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="Maroon">k</font> holds <br/><font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">x</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E4:16_3_1_1_1_1"/>
 ex <font color="Olive">y</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K6">ExtREAL</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font>,<font color="Olive">y</font>]
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_3_1_1_1_2"><b>case </b></a><a NAME="E1:16_3_1_1_1_2"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1
 ;<br/><div class="add"><a NAME="E2:16_3_1_1_1_2"/>
( 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">F</font> )
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E3:16_3_1_1_1_2"/><b>then </b><i><font color="Green">E60</font></i>: 
( 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="finseq_1.html#K9">*&gt;</a></span></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font></span>)</span> )
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E4:16_3_1_1_1_2"/><b>then </b>
( 1 <a href="binop_2.html#K23">+</a> 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> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="finseq_1.html#K9">*&gt;</a></span></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font></span>)</span> )
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E5:16_3_1_1_1_2"/><b>then </b><i><font color="Green">E61</font></i>: 
( <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="finseq_1.html#K9">*&gt;</a></span></span>)</span> <a href="binop_2.html#K23">+</a> 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> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="finseq_1.html#K9">*&gt;</a></span></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font></span>)</span> )
 <b>by </b><i><a class="ref" href="finseq_1.html#T56">FINSEQ_1:56</a></i>;<br/><b>then consider </b><font color="Maroon">k2</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E7:16_3_1_1_1_2"/><i><font color="Green">E62</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="finseq_1.html#K9">*&gt;</a></span></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">k2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T28">NAT_1:28</a></i>;<br/><b>reconsider </b><font color="Maroon">k2</font> = <font color="Maroon">k2</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/><a NAME="E9:16_3_1_1_1_2"/>
1 <a href="binop_2.html#K23">+</a> <font color="Maroon">k2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="finseq_1.html#K9">*&gt;</a></span></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E10:16_3_1_1_1_2"/><b>then </b><i><font color="Green">E65</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="binop_2.html#K23">+</a> <font color="Maroon">k2</font>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/><a NAME="E11:16_3_1_1_1_2"/>
<font color="Maroon">k</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="finseq_1.html#K9">*&gt;</a></span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T22">XREAL_1:22</a></i>;<br/><a NAME="E12:16_3_1_1_1_2"/><b>then </b>
1 <a href="binop_2.html#K23">+</a> <font color="Maroon">k2</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">G</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E13:16_3_1_1_1_2"/><b>then </b><i><font color="Green">E66</font></i>: 
1 <a href="binop_2.html#K23">+</a> <font color="Maroon">k2</font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">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/><i><font color="Green">E67</font></i>: <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> = 
<font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="finseq_1.html#K9">*&gt;</a></span></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="finseq_1.html#T36">FINSEQ_1:36</a></i>
<br/>.= 
<font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K23">+</a> <font color="Maroon">k2</font></span>)</span>
<b>by </b><i/>
;<br/><a NAME="E15:16_3_1_1_1_2"/><b>then </b>
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i>, </i>;<br/><b>then consider </b><font color="Maroon">z</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E17:16_3_1_1_1_2"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">z</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>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/><b>take </b><font color="Maroon">y</font> = <font color="Maroon">f</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">z</font>;<br/><b>consider </b><font color="Maroon">m1</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E19:16_3_1_1_1_2"/><i><font color="Green">E69</font></i>: 
( <font color="Maroon">m1</font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">F1</font> &amp; <font color="Maroon">F1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K23">+</a> <font color="Maroon">k2</font></span>)</span> )
 <b>by </b><i>, </i>;<br/><a NAME="E20:16_3_1_1_1_2"/>
( <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">F1</font> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">F1</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/><a NAME="E21:16_3_1_1_1_2"/><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> <font color="Maroon">S</font> &amp; <font color="Maroon">S</font> <a href="tarski.html#R1">c=</a>  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">X</font> )
 ;<br/><b>then reconsider </b><font color="Maroon">z</font> = <font color="Maroon">z</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font> <b>by </b><i/>;<br/><a NAME="E23:16_3_1_1_1_2"/><i><font color="Green">E70</font></i>: 
for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font>  st <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">F</font> &amp; <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 1 holds <br/><font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a> 
 <b>by </b><i/>;<br/><a NAME="E24:16_3_1_1_1_2"/>
for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font>  st <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">F</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2 &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="Maroon">k</font> holds <br/><font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">x</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_3_1_1_1_2_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:16_3_1_1_1_2_2"/>
( <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">F</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2 &amp; <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">k</font> )
 ;<br/>

<a NAME="E2:16_3_1_1_1_2_2"/><b>then </b>
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m1</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m1</font> )
 
<b>by </b><i>, , </i>;<br/>
<b>hence </b><a NAME="E3:16_3_1_1_1_2_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E25:16_3_1_1_1_2"/>
 ex <font color="Olive">y</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K6">ExtREAL</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font>,<font color="Olive">y</font>]
 <b>by </b><i><a class="txt" href="mesfunc3.html#E1:16"><i><font color="Green">E35</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="E2:16_3_1"/>
 ex <font color="Olive">y</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K6">ExtREAL</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font>,<font color="Olive">y</font>]
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:16_3"/>
 ex <font color="Olive">y</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K6">ExtREAL</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font>,<font color="Olive">y</font>]
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">a</font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K6">ExtREAL</a> <b> such that </b><br/><a NAME="E20:16"/><i><font color="Green">E71</font></i>: 
(  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">a</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">F</font></span>)</span> &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">S<sub>1</sub></font>[<font color="Olive">n</font>,<font color="Maroon">a</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">n</font>] ) )
 <b>from </b><i><a class="ref" href="finseq_1.html#S5">FINSEQ_1:sch 5</a>(<a class="ref" href="mesfunc3.html#T1" target="_self">Th1</a>);<br/></i>
<a NAME="E21:16"/><i><font color="Green">E73</font></i>: 
(  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">a</font> &amp; <font color="Maroon">a</font> <a href="mesfunc1.html#K4">.</a> 1 <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a>  &amp; ( for <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a><br/> for <font color="Olive">x</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="finsop_1.html#K5">dom</a> <font color="Maroon">F</font> &amp; <font color="Olive">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2 &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> holds <br/><font color="Maroon">a</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">x</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_4"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:16_4"/>
 <a href="finsop_1.html#K5">dom</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="finsop_1.html#K5">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 href="finseq_1.html#K3">len</a> <font color="Maroon">F</font> = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="finseq_1.html#K9">*&gt;</a></span></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>
<br/>.= 
1 <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T56">FINSEQ_1:56</a></i>
;<br/>
<a NAME="E3:16_4"/><b>then </b>
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="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<a NAME="E4:16_4"/><b>then </b><i><font color="Green">E74</font></i>: 
1 <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#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E5:16_4"/><b>then </b>
1 <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">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/>
<b>hence </b><a NAME="E6:16_4"/>
<font color="Maroon">a</font> <a href="mesfunc1.html#K4">.</a> 1 <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a> 
 <b>by </b><i>, </i>;<br/>

<a NAME="E7:16_4"/>
for <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a><br/> for <font color="Olive">x</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="finsop_1.html#K5">dom</a> <font color="Maroon">F</font> &amp; <font color="Olive">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2 &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> holds <br/><font color="Maroon">a</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_4_2"><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>;<br/>



<b>assume </b><a NAME="E1:16_4_2"/><i><font color="Green">E75</font></i>: 
( <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">F</font> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2 &amp; <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> )
 ;<br/>

<a NAME="E2:16_4_2"/><b>then </b>
<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/>
<b>hence </b><a NAME="E3:16_4_2"/>
<font color="Maroon">a</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E8:16_4"/>
for <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a><br/> for <font color="Olive">x</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="finsop_1.html#K5">dom</a> <font color="Maroon">F</font> &amp; <font color="Olive">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2 &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> holds <br/><font color="Maroon">a</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">x</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E22:16"/><i><font color="Green">E76</font></i>: 
for <font color="Olive">n'</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">n'</font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">F</font> holds <br/>for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <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> holds <br/><font color="Maroon">f</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">n'</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_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>assume </b><a NAME="E1:16_5"/><i><font color="Green">E77</font></i>: 
<font color="Maroon">n'</font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">F</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_5_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_5_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:16_5_1_1"/>
( <font color="Maroon">n'</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">n'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_5_1_1_1"><b>case </b></a><a NAME="E1:16_5_1_1_1"/><i><font color="Green">E78</font></i>: 
<font color="Maroon">n'</font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><b>thus </b><a NAME="E2:16_5_1_1_1"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <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="Maroon">n'</font> holds <br/><font color="Maroon">f</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">n'</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_5_1_1_1_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:16_5_1_1_1_1"/><i><font color="Green">E79</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>
 ;<br/>

<a NAME="E2:16_5_1_1_1_1"/>
 <a href="finsop_1.html#K5">dom</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="finseq_1.html#K9">*&gt;</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a>, <a class="ref" href="finseq_1.html#T55">FINSEQ_1:55</a></i>;<br/>
<a NAME="E3:16_5_1_1_1_1"/><b>then </b>
1 <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="finseq_1.html#K9">*&gt;</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>then </b><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> 1 = 
<span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="finseq_1.html#K9">*&gt;</a></span> <a href="funct_1.html#K1">.</a> 1
<b>by </b><i><a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>
<br/>.= 
 <a href="xboole_0.html#K1">{}</a> 
<b>by </b><i><a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a></i>
;<br/>
<b>hence </b><a NAME="E5:16_5_1_1_1_1"/>
<font color="Maroon">f</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">n'</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_5_1_1_2"><b>case </b></a><a NAME="E1:16_5_1_1_2"/><i><font color="Green">E80</font></i>: 
<font color="Maroon">n'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1
 ;<br/><div class="add"><a NAME="E2:16_5_1_1_2"/>
<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="E3:16_5_1_1_2"/><b>then </b>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n'</font> &amp; <font color="Maroon">n'</font> <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="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E4:16_5_1_1_2"/><b>then </b>
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n'</font>
 <b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E5:16_5_1_1_2"/><b>then </b><i><font color="Green">E81</font></i>: 
1 <a href="binop_2.html#K23">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n'</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>thus </b><a NAME="E6:16_5_1_1_2"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <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="Maroon">n'</font> holds <br/><font color="Maroon">f</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">n'</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_5_1_1_2_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:16_5_1_1_2_1"/><i><font color="Green">E82</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>
 ;<br/>

<a NAME="E2:16_5_1_1_2_1"/>
( <font color="Maroon">F</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">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/>
<a NAME="E3:16_5_1_1_2_1"/><b>then </b>
( <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> &amp; <font color="Maroon">S</font> <a href="tarski.html#R1">c=</a>  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">X</font> )
 
;<br/>
<b>then 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">X</font> <b>by </b><i/>;<br/>
<a NAME="E5:16_5_1_1_2_1"/>
<font color="Maroon">a</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">n'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font>
 
<b>by </b><i>, , , </i>;<br/>
<b>hence </b><a NAME="E6:16_5_1_1_2_1"/>
<font color="Maroon">f</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">n'</font>
 ;<br/>


</div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:16_5"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <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="Maroon">n'</font> holds <br/><font color="Maroon">f</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">n'</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E23:16"/><i><font color="Green">E84</font></i>: 
for <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a>  st 2 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">n</font> &amp; <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">a</font> holds <br/>(  <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">n</font> &amp; <font color="Maroon">a</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_1.html#K1">+infty</a>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_6"><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>assume </b><b>that </b><br/><a NAME="E1:16_6"/><i><font color="Green">E85</font></i>: 
2 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 <b>and </b><br/><a NAME="E2:16_6"/><i><font color="Green">E86</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">a</font>
 ;<br/>

<a NAME="E3:16_6"/>
<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="E4:16_6"/><b>then </b>
<font color="Maroon">n</font> <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="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E5:16_6"/><b>then </b><i><font color="Green">E87</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="finseq_1.html#K9">*&gt;</a></span></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>;<br/>
<a NAME="E6:16_6"/>
1 <a href="binop_2.html#K23">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 
<b>by </b><i/>;<br/>
<a NAME="E7:16_6"/><b>then </b>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="finseq_1.html#K9">*&gt;</a></span></span>)</span> <a href="binop_2.html#K23">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T56">FINSEQ_1:56</a></i>;<br/>
<b>then </b><i><font color="Green">E88</font></i>: <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> = 
<font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="finseq_1.html#K9">*&gt;</a></span></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="finseq_1.html#T36">FINSEQ_1:36</a></i>
<br/>.= 
<font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T56">FINSEQ_1:56</a></i>
;<br/>
<a NAME="E9:16_6"/><i><font color="Green">E89</font></i>: 
 <a href="finsop_1.html#K5">dom</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="finseq_1.html#K9">*&gt;</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a>, <a class="ref" href="finseq_1.html#T55">FINSEQ_1:55</a></i>;<br/>
<a NAME="E10:16_6"/><i><font color="Green">E90</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E11:16_6"/>
1 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">n</font>
 
<b>by </b><i/>;<br/>
<a NAME="E12:16_6"/><b>then </b>
not <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="finseq_1.html#K9">*&gt;</a></span>
 
<b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon">k</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E14:16_6"/><i><font color="Green">E91</font></i>: 
( <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">G</font> &amp; <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="finseq_1.html#K9">*&gt;</a></span></span>)</span> <a href="binop_2.html#K23">+</a> <font color="Maroon">k</font> )
 <b>by </b><i>, , , <a class="ref" href="finseq_1.html#T38">FINSEQ_1:38</a></i>;<br/>
<a NAME="E15:16_6"/>
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 1 <a href="binop_2.html#K23">+</a> <font color="Maroon">k</font>
 
<b>by </b><i><a class="txt" href="mesfunc3.html#E1:16"><i><font color="Green">E35</font></i></a>, <a class="ref" href="finseq_1.html#T56">FINSEQ_1:56</a></i>;<br/>
<a NAME="E16:16_6"/><b>then </b>
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> 1
 
;<br/>
<a NAME="E17:16_6"/><b>then </b>
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</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="mesfunc3.html#E1:16"><i><font color="Green">E35</font></i></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="E19:16_6"/><i><font color="Green">E92</font></i>: 
<font color="Maroon">x1</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#D1">XBOOLE_0:def 1</a></i>;<br/>
<a NAME="E20:16_6"/>
<font color="Maroon">F</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">F</font>
 
<b>by </b><i>, , <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/>
<a NAME="E21:16_6"/><b>then </b><i><font color="Green">E93</font></i>: 
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <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">F</font></span>)</span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T92">ZFMISC_1:92</a></i>;<br/>
<a NAME="E22:16_6"/><b>then </b>
( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp;  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font> )
 
<b>by </b><i>, <a class="txt" href="mesfunc3.html#E2:16"><i><font color="Green">E36</font></i></a></i>;<br/>
<b>then 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">X</font> ;<br/>
<a NAME="E24:16_6"/><i><font color="Green">E94</font></i>: 
<font color="Maroon">a</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x1</font>
 
<b>by </b><i>, , , <a class="txt" href="mesfunc3.html#E2:16"><i><font color="Green">E36</font></i></a></i>;<br/>
<a NAME="E25:16_6"/>
<font color="Maroon">f</font> <a href="mesfunc2.html#R1">is_finite</a> 
 
<b>by </b><i>, <a class="ref" href="mesfunc2.html#D5">MESFUNC2:def 5</a></i>;<br/>
<a NAME="E26:16_6"/><b>then </b><i><font color="Green">E95</font></i>: 
<span class="p1"><a href="extreal1.html#K4">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x1</font></span>)</span></span><a href="extreal1.html#K4">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_1.html#K1">+infty</a> 
 
<b>by </b><i>, <a class="txt" href="mesfunc3.html#E2:16"><i><font color="Green">E36</font></i></a>, <a class="txt" href="mesfunc3.html#E3:16"><i><font color="Green">E37</font></i></a>, <a class="ref" href="mesfunc2.html#D1">MESFUNC2:def 1</a></i>;<br/>
<a NAME="E27:16_6"/>
(  <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">f</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x1</font> &amp;  <a href="supinf_2.html#K1">0.</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">f</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x1</font> )
 
<b>by </b><i>, , , <a class="txt" href="mesfunc3.html#E2:16"><i><font color="Green">E36</font></i></a>, <a class="txt" href="mesfunc3.html#E3:16"><i><font color="Green">E37</font></i></a></i>;<br/>
<b>hence </b><a NAME="E28:16_6"/>
(  <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">n</font> &amp; <font color="Maroon">a</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_1.html#K1">+infty</a>  )
 <b>by </b><i><a class="txt" href="mesfunc3.html#E5:16"><i><font color="Green">E38</font></i></a>, <a class="txt" href="mesfunc3.html#E6:16"><i><font color="Green">E39</font></i></a>, <a class="ref" href="extreal2.html#T58">EXTREAL2:58</a></i>;<br/>


</div><b>end;</b></div>
<b>take </b>
<font color="Maroon">F</font>
;<br/>

<b>take </b>
<font color="Maroon">a</font>
;<br/>

<b>thus </b><a NAME="E24:16"/>
( <font color="Maroon">F</font>,<font color="Maroon">a</font> <a href="mesfunc3.html#R1" target="_self">are_Re-presentation_of</a> <font color="Maroon">f</font> &amp; <font color="Maroon">a</font> <a href="mesfunc1.html#K4">.</a> 1 <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a>  &amp; ( for <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a>  st 2 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">n</font> &amp; <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">a</font> holds <br/>(  <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">n</font> &amp; <font color="Maroon">a</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_1.html#K1">+infty</a>  ) ) )
 <b>by </b><i>, , , , </i>;<br/>


</div>
