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<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">M</font> be    <a href="measure1.html#M3">sigma_Measure</a> of <font color="Maroon">S</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> , <font color="Maroon">g</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="E1:99"/><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="E2:99"/><i><font color="Green">E37</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>and </b><br/><a NAME="E3:99"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">f</font> is <a href="supinf_2.html#V6">nonnegative</a>
 <b>and </b><br/><a NAME="E4:99"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">g</font> <a href="mesfunc2.html#R2">is_simple_func_in</a> <font color="Maroon">S</font>
 <b>and </b><br/><a NAME="E5:99"/><i><font color="Green">E40</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 <b>and </b><br/><a NAME="E6:99"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">g</font> is <a href="supinf_2.html#V6">nonnegative</a>
 <b>and </b><br/><a NAME="E7:99"/><i><font color="Green">E42</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/><font color="Maroon">g</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">f</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">x</font>
 ;<br/>

<a NAME="E8:99"/><i><font color="Green">E43</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/>0 <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>by </b><i><a class="txt" href="mesfunc5.html#E20:99"><i><font color="Green">E95</font></i></a>, <a class="ref" href="supinf_2.html#T70">SUPINF_2:70</a></i>;<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>, <font color="Maroon">a</font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K6">ExtREAL</a> , <font color="Maroon">x</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="E10:99"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">F</font>,<font color="Maroon">a</font> <a href="mesfunc3.html#R1">are_Re-presentation_of</a> <font color="Maroon">f</font>
 <b>and </b><br/><a NAME="E11:99"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">F</font>
 <b>and </b><br/><a NAME="E12:99"/>
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="relset_1.html#K4">dom</a> <font color="Maroon">x</font> holds <br/><font color="Maroon">x</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">n</font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="finseqop.html#K5">*</a> <font color="Maroon">F</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Olive">n</font></span>)</span>
 <b>and </b><br/><a NAME="E13:99"/>
 <a href="mesfunc3.html#K1">integral</a> <font color="Maroon">X</font>,<font color="Maroon">S</font>,<font color="Maroon">M</font>,<font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="convfun1.html#K5">Sum</a> <font color="Maroon">x</font>
 <b>by </b><i>, <a class="txt" href="mesfunc5.html#E16:99"><i><font color="Green">E94</font></i></a>, <a class="ref" href="mesfunc4.html#T4">MESFUNC4:4</a></i>;<br/>
<a NAME="E14:99"/><i><font color="Green">E47</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">g</font> holds <br/>0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">g</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">x</font>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E22:99"><i><font color="Green">E98</font></i></a>, <a class="ref" href="supinf_2.html#T70">SUPINF_2:70</a></i>;<br/>
<b>then consider </b><font color="Maroon">G</font> being   <a href="mesfunc2.html#NM1">Finite_Sep_Sequence</a> of <font color="Maroon">S</font>, <font color="Maroon">b</font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K6">ExtREAL</a> , <font color="Maroon">y</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="E16:99"/><i><font color="Green">E94</font></i>: 
<font color="Maroon">G</font>,<font color="Maroon">b</font> <a href="mesfunc3.html#R1">are_Re-presentation_of</a> <font color="Maroon">g</font>
 <b>and </b><br/><a NAME="E17:99"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">G</font>
 <b>and </b><br/><a NAME="E18:99"/>
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="relset_1.html#K4">dom</a> <font color="Maroon">y</font> holds <br/><font color="Maroon">y</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">n</font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="finseqop.html#K5">*</a> <font color="Maroon">G</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Olive">n</font></span>)</span>
 <b>and </b><br/><a NAME="E19:99"/>
 <a href="mesfunc3.html#K1">integral</a> <font color="Maroon">X</font>,<font color="Maroon">S</font>,<font color="Maroon">M</font>,<font color="Maroon">g</font> <a href="hidden.html#R1">=</a>  <a href="convfun1.html#K5">Sum</a> <font color="Maroon">y</font>
 <b>by </b><i>, , <a class="txt" href="mesfunc5.html#E16:99"><i><font color="Green">E94</font></i></a>, <a class="ref" href="mesfunc4.html#T4">MESFUNC4:4</a></i>;<br/>
<b>set </b><font color="Maroon">la</font> =  <a href="finseq_1.html#K3">len</a> <font color="Maroon">a</font>;<br/>
<b>set </b><font color="Maroon">lb</font> =  <a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font>;<br/>
<a NAME="E20:99"/><i><font color="Green">E95</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> &amp;  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</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">G</font></span>)</span> &amp;  <a href="relset_1.html#K4">dom</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">a</font> &amp;  <a href="relset_1.html#K4">dom</a> <font color="Maroon">G</font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">b</font> )
 
<b>by </b><i>, , <a class="ref" href="mesfunc3.html#D1">MESFUNC3:def 1</a></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="hidden.html#M1">set</a>  = <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K5">div</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></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> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>;<br/>
<b>consider </b><font color="Maroon">FG</font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E22:99"/><i><font color="Green">E98</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">FG</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">a</font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span>
 <b>and </b><br/><a NAME="E23:99"/><i><font color="Green">E99</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"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">a</font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> holds <br/><font color="Maroon">FG</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">k</font>)
 <b>from </b><i><a class="ref" href="finseq_1.html#S2">FINSEQ_1:sch 2</a>();<br/></i>
<a NAME="E24:99"/>
<font color="Maroon">FG</font> is    <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">S</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="99_1"><b>proof </b></a><div class="add">

<div><i><font color="Green">E100</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="99_1_1"><b>now </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="E1:99_1_1"/>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">FG</font>
 ;<br/><a NAME="E2:99_1_1"/><b>then </b><i><font color="Green">E101</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"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">a</font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E3:99_1_1"/><b>then </b><i><font color="Green">E102</font></i>: 
<font color="Maroon">FG</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> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K5">div</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></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> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E6:99_1_1"><i><font color="Green">E106</font></i></a></i>;<br/><b>set </b><font color="Maroon">i</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K5">div</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1;<br/><b>set </b><font color="Maroon">j</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1;<br/><a NAME="E4:99_1_1"/><i><font color="Green">E103</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K5">div</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 <a href="nat_1.html#K1">+</a> 1 &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E5:99_1_1"/><i><font color="Green">E104</font></i>: 
( 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> <font color="Maroon">a</font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span> )
 <b>by </b><i><a class="txt" href="mesfunc5.html#E10:99_1_1"><i><font color="Green">E108</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E6:99_1_1"/><i><font color="Green">E106</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i/>;<br/><a NAME="E7:99_1_1"/><b>then </b><i><font color="Green">E107</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><a NAME="E8:99_1_1"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E9:99_1_1"/><b>then </b>
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font> <a href="nat_1.html#R1">divides</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">a</font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">a</font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font> )
 <b>by </b><i>, <a class="ref" href="nat_1.html#D3">NAT_1:def 3</a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><a NAME="E10:99_1_1"/><b>then </b><i><font color="Green">E108</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">a</font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K5">div</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">a</font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K5">div</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> 1
 <b>by </b><i><a class="ref" href="nat_2.html#T17">NAT_2:17</a></i>;<br/><a NAME="E11:99_1_1"/>
<font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">a</font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1
 <b>by </b><i>, <a class="ref" href="jordan3.html#T5">JORDAN3:5</a></i>;<br/><a NAME="E12:99_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K5">div</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">a</font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K5">div</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span>
 <b>by </b><i><a class="ref" href="nat_2.html#T26">NAT_2:26</a></i>;<br/><a NAME="E13:99_1_1"/><b>then </b><i><font color="Green">E109</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K5">div</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">a</font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K5">div</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span>
 <b>by </b><i><a class="ref" href="mesfunc5.html#T27" target="_self">Th27</a>, <a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/><a NAME="E14:99_1_1"/><i><font color="Green">E121</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K5">div</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="mesfunc5.html#T26" target="_self">Th26</a>, <a class="txt" href="mesfunc5.html#E35"><i><font color="Green">Lemma115</font></i></a>, <a class="ref" href="nat_1.html#T68">NAT_1:68</a></i>;<br/><a NAME="E15:99_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font>
 <b>by </b><i>, <a class="ref" href="nat_1.html#T46">NAT_1:46</a></i>;<br/><a NAME="E16:99_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E17:99_1_1"/><b>then </b>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K5">div</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 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">a</font></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 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">b</font></span>)</span> )
 <b>by </b><i><a class="ref" href="mesfunc5.html#T28" target="_self">Th28</a>, <a class="ref" href="mesfunc5.html#T25" target="_self">Th25</a></i>;<br/><a NAME="E18:99_1_1"/><b>then </b>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K5">div</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">F</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">G</font> )
 <b>by </b><i><a class="txt" href="mesfunc5.html#E5:99_1_1"><i><font color="Green">E104</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E19:99_1_1"/><b>then </b>
( <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K5">div</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">F</font> &amp; <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <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="E20:99_1_1"/><b>then </b>
( <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K5">div</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="measure1.html#R2">is_measurable</a> <font color="Maroon">M</font> &amp; <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="measure1.html#R2">is_measurable</a> <font color="Maroon">M</font> )
 <b>by </b><i><a class="ref" href="measure1.html#D12">MEASURE1:def 12</a></i>;<br/><a NAME="E21:99_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K5">div</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></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> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="measure1.html#R2">is_measurable</a> <font color="Maroon">M</font>
 <b>by </b><i><a class="ref" href="measure1.html#T66">MEASURE1:66</a></i>;<br/><b>hence </b><a NAME="E22:99_1_1"/>
<font color="Maroon">FG</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>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E13:99_1_1"><i><font color="Green">E109</font></i></a>, <a class="ref" href="measure1.html#D12">MEASURE1:def 12</a></i>;<br/></div><b>end;</b></div>
<b>let </b><font color="Maroon">b</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>,<a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a><br/>

<b>assume </b><a NAME="E2:99_1"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">FG</font>
 ;<br/>

<a NAME="E3:99_1"/><b>then </b>
 ex <font color="Olive">a</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">a</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">FG</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">FG</font> <a href="funct_1.html#K1">.</a> <font color="Olive">a</font> )
 
<b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E4:99_1"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">FG</font> = <font color="Maroon">FG</font> as    <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">S</font> ;<br/>
<a NAME="E26:99"/>
for <font color="Olive">k</font>, <font color="Olive">l</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">FG</font> &amp; <font color="Olive">l</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">FG</font> &amp; <font color="Olive">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">l</font> holds <br/><font color="Maroon">FG</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">FG</font> <a href="funct_1.html#K1">.</a> <font color="Olive">l</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="99_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:99_2"/><i><font color="Green">E122</font></i>: 
( <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">FG</font> &amp; <font color="Maroon">l</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">FG</font> &amp; <font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">l</font> )
 ;<br/>

<b>set </b><font color="Maroon">i</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K5">div</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1;<br/>
<b>set </b><font color="Maroon">j</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1;<br/>
<b>set </b><font color="Maroon">n</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">l</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K5">div</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1;<br/>
<b>set </b><font color="Maroon">m</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">l</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1;<br/>
<a NAME="E2:99_2"/><i><font color="Green">E123</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"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">a</font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> &amp; <font color="Maroon">l</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">a</font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> )
 
<b>by </b><i>, , <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E3:99_2"/><b>then </b><i><font color="Green">E124</font></i>: 
( 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> <font color="Maroon">a</font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E4:99_2"/><b>then </b><i><font color="Green">E125</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
;<br/>
<a NAME="E5:99_2"/><b>then </b><i><font color="Green">E126</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
;<br/>
<a NAME="E6:99_2"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E7:99_2"/><b>then </b>
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font> <a href="nat_1.html#R1">divides</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">a</font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">a</font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font> )
 
<b>by </b><i>, <a class="ref" href="nat_1.html#D3">NAT_1:def 3</a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/>
<a NAME="E8:99_2"/><b>then </b><i><font color="Green">E127</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">a</font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K5">div</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">a</font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K5">div</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> 1
 
<b>by </b><i><a class="ref" href="nat_2.html#T17">NAT_2:17</a></i>;<br/>
<a NAME="E9:99_2"/>
<font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">a</font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1
 
<b>by </b><i>, <a class="ref" href="jordan3.html#T5">JORDAN3:5</a></i>;<br/>
<a NAME="E10:99_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K5">div</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">a</font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K5">div</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> 1
 
<b>by </b><i><a class="ref" href="mesfunc5.html#T26" target="_self">Th26</a>, <a class="ref" href="nat_2.html#T26">NAT_2:26</a></i>;<br/>
<a NAME="E11:99_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K5">div</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">a</font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K5">div</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/>
<a NAME="E12:99_2"/><b>then </b><i><font color="Green">E128</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K5">div</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">a</font>
 
<b>by </b><i><a class="ref" href="mesfunc5.html#T25" target="_self">Th25</a>, <a class="ref" href="nat_1.html#T68">NAT_1:68</a></i>;<br/>
<a NAME="E13:99_2"/><i><font color="Green">E129</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K5">div</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 <a href="nat_1.html#K1">+</a> 1 &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 <a href="nat_1.html#K1">+</a> 1 &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">l</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K5">div</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 <a href="nat_1.html#K1">+</a> 1 &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">l</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 <a href="nat_1.html#K1">+</a> 1 )
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E14:99_2"/>
( <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">l</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font> )
 
<b>by </b><i>, <a class="ref" href="nat_1.html#T46">NAT_1:46</a></i>;<br/>
<a NAME="E15:99_2"/><b>then </b>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">l</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font> )
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E16:99_2"/><b>then </b>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K5">div</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 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">a</font></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 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">b</font></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">l</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 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">b</font></span>)</span> )
 
<b>by </b><i>, <a class="ref" href="mesfunc5.html#T27" target="_self">Th27</a></i>;<br/>
<a NAME="E17:99_2"/><b>then </b><i><font color="Green">E130</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K5">div</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">F</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">G</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">l</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="relset_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="E18:99_2"/><i><font color="Green">E131</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">l</font> &amp; <font color="Maroon">l</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">a</font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span> )
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E19:99_2"/><b>then </b>
<font color="Maroon">l</font> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">a</font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1
 
<b>by </b><i><a class="ref" href="jordan3.html#T5">JORDAN3:5</a></i>;<br/>
<a NAME="E20:99_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">l</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K5">div</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">a</font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K5">div</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> 1
 
<b>by </b><i><a class="ref" href="mesfunc5.html#T26" target="_self">Th26</a>, <a class="ref" href="nat_2.html#T26">NAT_2:26</a></i>;<br/>
<a NAME="E21:99_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">l</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K5">div</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">a</font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K5">div</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/>
<a NAME="E22:99_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">l</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K5">div</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">a</font>
 
<b>by </b><i><a class="ref" href="mesfunc5.html#T25" target="_self">Th25</a>, <a class="ref" href="nat_1.html#T68">NAT_1:68</a></i>;<br/>
<a NAME="E23:99_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">l</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K5">div</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 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">a</font></span>)</span>
 
<b>by </b><i><a class="ref" href="mesfunc5.html#T27" target="_self">Th27</a></i>;<br/>
<a NAME="E24:99_2"/><b>then </b><i><font color="Green">E132</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">l</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K5">div</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="relset_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/>
<div><i><font color="Green">E133</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="99_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:99_2_1"/><i><font color="Green">E134</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K5">div</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">l</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K5">div</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">l</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 )
 ;<br/><a NAME="E2:99_2_1"/>
( <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K5">div</a> <span class="p0">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1 &amp; <span class="p1">(<span class="default"><font color="Maroon">l</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">l</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K5">div</a> <span class="p0">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">l</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1 )
 <b>by </b><i>, <a class="ref" href="nat_1.html#T47">NAT_1:47</a></i>;<br/><a NAME="E3:99_2_1"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K5">div</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">l</font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">l</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K5">div</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">l</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 <b>by </b><i>, <a class="ref" href="mesfunc5.html#T28" target="_self">Th28</a>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>;<br/><b>hence </b><a NAME="E4:99_2_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="mesfunc5.html#T30" target="_self">Th30</a></i>;<br/></div><b>end;</b></div>
<a NAME="E26:99_2"/>
<font color="Maroon">FG</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> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K5">div</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></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> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<b>then </b><i><font color="Green">E135</font></i>: <span class="p1">(<span class="default"><font color="Maroon">FG</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><font color="Maroon">FG</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">l</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K5">div</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">l</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K5">div</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">l</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K5">div</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">l</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K5">div</a> <span class="p0">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p3">(<span class="default"><font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">l</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p0">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T16">XBOOLE_1:16</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K5">div</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">l</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K5">div</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p0">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p3">(<span class="default"><font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">l</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p0">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
<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"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K5">div</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">l</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K5">div</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">l</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T16">XBOOLE_1:16</a></i>
;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="99_2_3"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:99_2_3"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K5">div</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">l</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K5">div</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 or <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">l</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="99_2_3_1"><b>suppose </b></a><a NAME="E1:99_2_3_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K5">div</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">l</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K5">div</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1
 ;<br/><div class="add"><a NAME="E2:99_2_3_1"/><b>then </b>
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K5">div</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">l</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K5">div</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E35"><i><font color="Green">Lemma115</font></i></a>, <a class="ref" href="mesfunc5.html#T29" target="_self">Th29</a>, <a class="ref" href="mesfunc3.html#T4">MESFUNC3:4</a></i>;<br/><a NAME="E3:99_2_3_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">FG</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><font color="Maroon">FG</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">l</font></span>)</span> <a href="hidden.html#R1">=</a> <a href="xboole_0.html#K1">{}</a>  <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">l</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="mesfunc5.html#T30" target="_self">Th30</a>, <a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/><b>hence </b><a NAME="E4:99_2_3_1"/>
<font color="Maroon">FG</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">FG</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">l</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="99_2_3_2"><b>suppose </b></a><a NAME="E1:99_2_3_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">l</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1
 ;<br/><div class="add"><a NAME="E2:99_2_3_2"/><b>then </b>
<font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">l</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E35"><i><font color="Green">Lemma115</font></i></a>, <a class="ref" href="mesfunc3.html#T4">MESFUNC3:4</a></i>;<br/><a NAME="E3:99_2_3_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">FG</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><font color="Maroon">FG</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">l</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K5">div</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">l</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K5">div</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="mesfunc5.html#T30" target="_self">Th30</a>, <a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/><b>hence </b><a NAME="E4:99_2_3_2"/>
<font color="Maroon">FG</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">FG</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">l</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></div><b>end;</b></div>

</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">FG</font> = <font color="Maroon">FG</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/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>,   <a href="hidden.html#M1">set</a> ] <b>means </b>( ( <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K5">div</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  implies <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> 0 ) &amp; ( <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K5">div</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  implies <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K5">div</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> ) );<br/>
<a NAME="E28:99"/><i><font color="Green">E136</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">FG</font></span>)</span> holds <br/> ex <font color="Olive">v</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">v</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="99_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="E1:99_3"/>
<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">FG</font></span>)</span>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="99_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:99_3_1"/>
( <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K5">div</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K5">div</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="99_3_1_1"><b>suppose </b></a><a NAME="E1:99_3_1_1"/><i><font color="Green">E137</font></i>: 
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K5">div</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><b>take </b><font color="Maroon">v</font> =  <a href="supinf_2.html#K1">0.</a> ;<br/><b>thus </b><a NAME="E2:99_3_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font>,<font color="Maroon">v</font>]
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="99_3_1_2"><b>suppose </b></a><a NAME="E1:99_3_1_2"/><i><font color="Green">E138</font></i>: 
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K5">div</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><b>take </b><font color="Maroon">v</font> = <font color="Maroon">a</font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K5">div</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>;<br/><b>thus </b><a NAME="E2:99_3_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font>,<font color="Maroon">v</font>]
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>consider </b><font color="Maroon">a1</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="E30:99"/><i><font color="Green">E139</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <font color="Maroon">a1</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">FG</font></span>)</span> &amp; ( 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">FG</font></span>)</span> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>,<font color="Maroon">a1</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">k</font>] ) )
 <b>from </b><i><a class="ref" href="finseq_1.html#S5">FINSEQ_1:sch 5</a>();<br/></i>
<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(  <a href="nat_1.html#NM1">Nat</a>) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K6">ExtREAL</a>  = <span class="p1">(<span class="default"><font color="Maroon">a1</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="finseqop.html#K5">*</a> <font color="Maroon">FG</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span>;<br/>
<b>consider </b><font color="Maroon">x1</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="E32:99"/><i><font color="Green">E140</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">FG</font> &amp; ( 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">FG</font></span>)</span> holds <br/><font color="Maroon">x1</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">k</font>) ) )
 <b>from </b><i><a class="ref" href="finseq_2.html#S1">FINSEQ_2:sch 1</a>();<br/></i>
<a NAME="E33:99"/><i><font color="Green">E141</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">FG</font>
 
<b>by </b><i>, <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/>
<a NAME="E34:99"/><i><font color="Green">E142</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="relset_1.html#K4">dom</a> <font color="Maroon">x1</font> holds <br/><font color="Maroon">x1</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a1</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">n</font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="finseqop.html#K5">*</a> <font color="Maroon">FG</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Olive">n</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="99_4"><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="E1:99_4"/>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">x1</font>
 ;<br/>

<a NAME="E2:99_4"/><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">x1</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="E3:99_4"/>
<font color="Maroon">x1</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a1</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">n</font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="finseqop.html#K5">*</a> <font color="Maroon">FG</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[  <a href="nat_1.html#NM1">Nat</a>,   <a href="hidden.html#M1">set</a> ] <b>means </b>( ( <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  implies <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> 0 ) &amp; ( <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  implies <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> ) );<br/>
<a NAME="E35:99"/><i><font color="Green">E143</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">FG</font></span>)</span> holds <br/> ex <font color="Olive">v</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K6">ExtREAL</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">k</font>,<font color="Olive">v</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="99_5"><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="E1:99_5"/>
<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">FG</font></span>)</span>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="99_5_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:99_5_1"/>
( <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="99_5_1_1"><b>suppose </b></a><a NAME="E1:99_5_1_1"/><i><font color="Green">E184</font></i>: 
<font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><b>take </b><font color="Maroon">v</font> =  <a href="measure6.html#K1">R_EAL</a> 0;<br/><b>thus </b><a NAME="E2:99_5_1_1"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">k</font>,<font color="Maroon">v</font>]
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="99_5_1_2"><b>suppose </b></a><a NAME="E1:99_5_1_2"/><i><font color="Green">E185</font></i>: 
<font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><b>take </b><font color="Maroon">v</font> = <font color="Maroon">b</font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>;<br/><b>thus </b><a NAME="E2:99_5_1_2"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">k</font>,<font color="Maroon">v</font>]
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>consider </b><font color="Maroon">b1</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="E37:99"/><i><font color="Green">E186</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <font color="Maroon">b1</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">FG</font></span>)</span> &amp; ( 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">FG</font></span>)</span> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">k</font>,<font color="Maroon">b1</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">k</font>] ) )
 <b>from </b><i><a class="ref" href="finseq_1.html#S5">FINSEQ_1:sch 5</a>(<a class="ref" href="mesfunc5.html#T26" target="_self">Th26</a>);<br/></i>
<b>deffunc </b><font color="Maroon">H<sub>3</sub></font>(  <a href="nat_1.html#NM1">Nat</a>) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K6">ExtREAL</a>  = <span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="finseqop.html#K5">*</a> <font color="Maroon">FG</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span>;<br/>
<b>consider </b><font color="Maroon">y1</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="E39:99"/><i><font color="Green">E187</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">y1</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">FG</font> &amp; ( 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">FG</font></span>)</span> holds <br/><font color="Maroon">y1</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Olive">k</font>) ) )
 <b>from </b><i><a class="ref" href="finseq_2.html#S1">FINSEQ_2:sch 1</a>();<br/></i>
<a NAME="E40:99"/><i><font color="Green">E188</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">y1</font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">FG</font>
 
<b>by </b><i><a class="ref" href="mesfunc5.html#T27" target="_self">Th27</a>, <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/>
<div><i><font color="Green">E189</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="99_6"><b>now </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="E1:99_6"/>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">y1</font>
 ;<br/><a NAME="E2:99_6"/><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">y1</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="E3:99_6"/>
<font color="Maroon">y1</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">n</font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="finseqop.html#K5">*</a> <font color="Maroon">FG</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i><a class="ref" href="mesfunc5.html#T27" target="_self">Th27</a></i>;<br/></div><b>end;</b></div>
<b>deffunc </b><font color="Maroon">H<sub>4</sub></font>(  <a href="nat_1.html#NM1">Nat</a>) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K6">ExtREAL</a>  = <span class="p1">(<span class="default"><font color="Maroon">a1</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span>;<br/>
<b>consider </b><font color="Maroon">c1</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="E43:99"/><i><font color="Green">E190</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c1</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">FG</font>
 <b>and </b><br/><a NAME="E44:99"/><i><font color="Green">E191</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">FG</font></span>)</span> holds <br/><font color="Maroon">c1</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>4</sub></font>(<font color="Olive">k</font>)
 <b>from </b><i><a class="ref" href="finseq_2.html#S1">FINSEQ_2:sch 1</a>();<br/></i>
<b>deffunc </b><font color="Maroon">H<sub>5</sub></font>(  <a href="nat_1.html#NM1">Nat</a>) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K6">ExtREAL</a>  = <span class="p1">(<span class="default"><font color="Maroon">c1</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="finseqop.html#K5">*</a> <font color="Maroon">FG</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span>;<br/>
<b>consider </b><font color="Maroon">z1</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="E46:99"/><i><font color="Green">E192</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">z1</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">FG</font> &amp; ( 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">FG</font></span>)</span> holds <br/><font color="Maroon">z1</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>5</sub></font>(<font color="Olive">k</font>) ) )
 <b>from </b><i><a class="ref" href="finseq_2.html#S1">FINSEQ_2:sch 1</a>();<br/></i>
<a NAME="E47:99"/><i><font color="Green">E193</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">z1</font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">FG</font>
 
<b>by </b><i><a class="ref" href="mesfunc5.html#T30" target="_self">Th30</a>, <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/>
<a NAME="E48:99"/><i><font color="Green">E194</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="relset_1.html#K4">dom</a> <font color="Maroon">z1</font> holds <br/><font color="Maroon">z1</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c1</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">n</font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="finseqop.html#K5">*</a> <font color="Maroon">FG</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Olive">n</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="99_7"><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="E1:99_7"/>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">z1</font>
 ;<br/>

<a NAME="E2:99_7"/><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">z1</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="E3:99_7"/>
<font color="Maroon">z1</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c1</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">n</font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="finseqop.html#K5">*</a> <font color="Maroon">FG</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i><a class="ref" href="mesfunc5.html#T30" target="_self">Th30</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E49:99"/><i><font color="Green">E195</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">FG</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="99_8"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:99_8"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</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">FG</font></span>)</span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="99_8_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:99_8_1"/><i><font color="Green">E196</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 ;<br/>

<b>then consider </b><font color="Maroon">Y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:99_8_1"/><i><font color="Green">E197</font></i>: 
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">Y</font> <a href="hidden.html#R2">in</a>  <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">i</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="E5:99_8_1"/><i><font color="Green">E198</font></i>: 
( <font color="Maroon">i</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">Y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> )
 <b>by </b><i>, <a class="ref" href="finseq_2.html#T11">FINSEQ_2:11</a></i>;<br/>
<b>consider </b><font color="Maroon">Z</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E7:99_8_1"/><i><font color="Green">E199</font></i>: 
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> &amp; <font color="Maroon">Z</font> <a href="hidden.html#R2">in</a>  <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">j</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="E9:99_8_1"/><i><font color="Green">E200</font></i>: 
( <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">G</font> &amp; <font color="Maroon">Z</font> <a href="hidden.html#R1">=</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="finseq_2.html#T11">FINSEQ_2:11</a></i>;<br/>
<a NAME="E10:99_8_1"/>
( <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">a</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">b</font></span>)</span> )
 
<b>by </b><i>, , , <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E11:99_8_1"/><b>then </b><i><font color="Green">E201</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">a</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font> &amp; <font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<b>then consider </b><font color="Maroon">i'</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E13:99_8_1"/><i><font color="Green">E202</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 1 <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">i'</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T28">NAT_1:28</a></i>;<br/>
<a NAME="E14:99_8_1"/>
<span class="p1">(<span class="default"><font color="Maroon">i'</font> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">j</font> is   <a href="nat_1.html#NM1">Nat</a>
 
;<br/>
<b>then reconsider </b><font color="Maroon">k</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">j</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/>
<b>consider </b><font color="Maroon">j'</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E17:99_8_1"/><i><font color="Green">E203</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R1">=</a> 1 <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">j'</font>
 <b>by </b><i>, <a class="ref" href="nat_1.html#T28">NAT_1:28</a></i>;<br/>
<a NAME="E18:99_8_1"/>
<font color="Maroon">i'</font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
;<br/>
<a NAME="E19:99_8_1"/><b>then </b>
<font color="Maroon">k</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 <a href="nat_1.html#K1">+</a> <font color="Maroon">j</font>
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E20:99_8_1"/><b>then </b><i><font color="Green">E204</font></i>: 
<font color="Maroon">k</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/>
<a NAME="E21:99_8_1"/><b>then </b>
<font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="real_1.html#K5">-</a> 1
 
<b>by </b><i><a class="ref" href="binarith.html#T50">BINARITH:50</a></i>;<br/>
<a NAME="E22:99_8_1"/><b>then </b><i><font color="Green">E205</font></i>: 
<font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">i'</font> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">j'</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E23:99_8_1"/><i><font color="Green">E206</font></i>: 
<font color="Maroon">j'</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font>
 
<b>by </b><i>, , <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E24:99_8_1"/><b>then </b><i><font color="Green">E207</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K5">div</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i>, , <a class="ref" href="nat_1.html#D1">NAT_1:def 1</a></i>;<br/>
<a NAME="E25:99_8_1"/><i><font color="Green">E208</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i>, , , <a class="ref" href="nat_1.html#D2">NAT_1:def 2</a></i>;<br/>
<a NAME="E26:99_8_1"/>
<font color="Maroon">i</font> <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">a</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/>
<a NAME="E27:99_8_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">a</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/>
<a NAME="E28:99_8_1"/><b>then </b><i><font color="Green">E209</font></i>: 
<font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">a</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">j</font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E29:99_8_1"/><i><font color="Green">E210</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">a</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">a</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E30:99_8_1"/>
<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> <font color="Maroon">a</font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E28:99"><i><font color="Green">E136</font></i></a>, , <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/>
<a NAME="E31:99_8_1"/><b>then </b><i><font color="Green">E211</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"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">a</font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E32:99_8_1"/><b>then </b>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">FG</font>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E33:99_8_1"/><b>then </b><i><font color="Green">E305</font></i>: 
<font color="Maroon">FG</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">FG</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E34:99_8_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</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="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>
 
<b>by </b><i>, , , , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E35:99_8_1"/><b>then </b>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FG</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>
 
<b>by </b><i>, , , </i>;<br/>
<b>hence </b><a NAME="E36:99_8_1"/>
<font color="Maroon">z</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">FG</font></span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E30:99"><i><font color="Green">E139</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>


</div><b>end;</b></div>

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

<b>assume </b><a NAME="E2:99_8"/>
<font color="Maroon">z</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">FG</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">Y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:99_8"/><i><font color="Green">E306</font></i>: 
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">Y</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">FG</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="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E6:99_8"/><i><font color="Green">E307</font></i>: 
( <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">FG</font> &amp; <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">FG</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> )
 <b>by </b><i>, <a class="ref" href="finseq_2.html#T11">FINSEQ_2:11</a></i>;<br/>
<b>set </b><font color="Maroon">i</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K5">div</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1;<br/>
<b>set </b><font color="Maroon">j</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1;<br/>
<a NAME="E7:99_8"/>
<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"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">a</font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E8:99_8"/><b>then </b>
<font color="Maroon">FG</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> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K5">div</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></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> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E9:99_8"/><b>then </b><i><font color="Green">E308</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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K5">div</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i>, , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E10:99_8"/>
<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">FG</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E11:99_8"/><b>then </b><i><font color="Green">E309</font></i>: 
( 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> <font color="Maroon">a</font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span> )
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E12:99_8"/><i><font color="Green">E310</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K5">div</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 <a href="nat_1.html#K1">+</a> 1 &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 <a href="nat_1.html#K1">+</a> 1 )
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E13:99_8"/><i><font color="Green">E311</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i/>;<br/>
<a NAME="E14:99_8"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
;<br/>
<a NAME="E15:99_8"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E16:99_8"/><b>then </b>
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font> <a href="nat_1.html#R1">divides</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">a</font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">a</font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font> )
 
<b>by </b><i>, <a class="ref" href="nat_1.html#D3">NAT_1:def 3</a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/>
<a NAME="E17:99_8"/><b>then </b><i><font color="Green">E312</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">a</font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K5">div</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">a</font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K5">div</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> 1
 
<b>by </b><i><a class="ref" href="nat_2.html#T17">NAT_2:17</a></i>;<br/>
<a NAME="E18:99_8"/>
<font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">a</font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1
 
<b>by </b><i>, <a class="ref" href="jordan3.html#T5">JORDAN3:5</a></i>;<br/>
<a NAME="E19:99_8"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K5">div</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">a</font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K5">div</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> 1
 
<b>by </b><i>, <a class="ref" href="nat_2.html#T26">NAT_2:26</a></i>;<br/>
<a NAME="E20:99_8"/><b>then </b><i><font color="Green">E313</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K5">div</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">a</font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K5">div</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/>
<a NAME="E21:99_8"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">a</font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K5">div</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">a</font>
 
<b>by </b><i>, <a class="ref" href="nat_1.html#T68">NAT_1:68</a></i>;<br/>
<a NAME="E22:99_8"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K5">div</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 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">a</font></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E23:99_8"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K5">div</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="relset_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="E24:99_8"/><b>then </b>
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K5">div</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <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#D5">FUNCT_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E25:99_8"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 <b>by </b><i>, , <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E50:99"/>
<font color="Maroon">g</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="E51:99"/><b>then </b><i><font color="Green">E314</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="mesfunc1.html#K6">-</a> <font color="Maroon">g</font></span>)</span> <a href="hidden.html#R1">=</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> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font></span>)</span>
 
<b>by </b><i><a class="ref" href="mesfunc2.html#T2">MESFUNC2:2</a></i>;<br/>
<a NAME="E52:99"/><i><font color="Green">E315</font></i>: 
<font color="Maroon">f</font> <a href="mesfunc1.html#K6">-</a> <font color="Maroon">g</font> <a href="mesfunc2.html#R1">is_finite</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="99_9"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="99_9_1"><b>now </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="E1:99_9_1"/><i><font color="Green">E316</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="mesfunc1.html#K6">-</a> <font color="Maroon">g</font></span>)</span>
 ;<br/><a NAME="E2:99_9_1"/><b>then </b><i><font color="Green">E317</font></i>: 
<span class="p1"><a href="extreal1.html#K4">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="mesfunc1.html#K6">-</a> <font color="Maroon">g</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font></span>)</span></span><a href="extreal1.html#K4">.|</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="extreal1.html#K4">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font></span>)</span></span>)</span></span><a href="extreal1.html#K4">.|</a></span>
 <b>by </b><i><a class="ref" href="mesfunc1.html#D4">MESFUNC1:def 4</a></i>;<br/><a NAME="E3:99_9_1"/>
( <font color="Maroon">f</font> <a href="mesfunc2.html#R1">is_finite</a>  &amp; <font color="Maroon">g</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="E4:99_9_1"/><b>then </b><i><font color="Green">E318</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">x</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>  &amp; <span class="p1"><a href="extreal1.html#K4">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</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="ref" href="mesfunc2.html#D1">MESFUNC2:def 1</a></i>;<br/><a NAME="E5:99_9_1"/><b>then </b>
( not <font color="Maroon">f</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  &amp; not <font color="Maroon">f</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  &amp; not <font color="Maroon">g</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  &amp; not <font color="Maroon">g</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  )
 <b>by </b><i><a class="ref" href="extreal2.html#T67">EXTREAL2:67</a></i>;<br/><a NAME="E6:99_9_1"/><b>then </b><i><font color="Green">E319</font></i>: 
<span class="p1"><a href="extreal1.html#K4">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="mesfunc1.html#K6">-</a> <font color="Maroon">g</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font></span>)</span></span><a href="extreal1.html#K4">.|</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <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">x</font></span>)</span></span><a href="extreal1.html#K4">.|</a></span> <a href="supinf_2.html#K2">+</a> <span class="p1"><a href="extreal1.html#K4">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font></span>)</span></span><a href="extreal1.html#K4">.|</a></span>
 <b>by </b><i>, <a class="ref" href="extreal2.html#T69">EXTREAL2:69</a></i>;<br/><a NAME="E7:99_9_1"/>
<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">x</font></span>)</span></span><a href="extreal1.html#K4">.|</a></span> <a href="supinf_2.html#K2">+</a> <span class="p1"><a href="extreal1.html#K4">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font></span>)</span></span><a href="extreal1.html#K4">.|</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K1">+infty</a> 
 <b>by </b><i>, <a class="ref" href="supinf_2.html#T8">SUPINF_2:8</a></i>;<br/><a NAME="E8:99_9_1"/><b>then </b>
<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">x</font></span>)</span></span><a href="extreal1.html#K4">.|</a></span> <a href="supinf_2.html#K2">+</a> <span class="p1"><a href="extreal1.html#K4">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</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="ref" href="supinf_1.html#T24">SUPINF_1:24</a></i>;<br/><b>hence </b><a NAME="E9:99_9_1"/>
<span class="p1"><a href="extreal1.html#K4">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="mesfunc1.html#K6">-</a> <font color="Maroon">g</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</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="ref" href="supinf_1.html#T29">SUPINF_1:29</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:99_9"/>
<font color="Maroon">f</font> <a href="mesfunc1.html#K6">-</a> <font color="Maroon">g</font> <a href="mesfunc2.html#R1">is_finite</a> 
 <b>by </b><i><a class="ref" href="mesfunc2.html#D1">MESFUNC2:def 1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E53:99"/>
for <font color="Olive">k</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">k</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">FG</font> &amp; <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FG</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FG</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="mesfunc1.html#K6">-</a> <font color="Maroon">g</font></span>)</span> <a href="mesfunc1.html#K4">.</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="mesfunc1.html#K6">-</a> <font color="Maroon">g</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Olive">y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="99_10"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</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="E1:99_10"/><i><font color="Green">E320</font></i>: 
( <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">FG</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FG</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FG</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> )
 ;<br/>

<a NAME="E2:99_10"/><b>then </b>
<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"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">a</font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E3:99_10"/><b>then </b><i><font color="Green">E321</font></i>: 
<font color="Maroon">FG</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> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K5">div</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></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> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i/>;<br/>
<b>set </b><font color="Maroon">i</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K5">div</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1;<br/>
<b>set </b><font color="Maroon">j</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1;<br/>
<a NAME="E4:99_10"/>
<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">FG</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E5:99_10"/><b>then </b><i><font color="Green">E322</font></i>: 
( 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> <font color="Maroon">a</font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span> )
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E6:99_10"/><i><font color="Green">E323</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K5">div</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 <a href="nat_1.html#K1">+</a> 1 &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 <a href="nat_1.html#K1">+</a> 1 )
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E7:99_10"/><i><font color="Green">E324</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i/>;<br/>
<a NAME="E8:99_10"/><b>then </b><i><font color="Green">E325</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
;<br/>
<a NAME="E9:99_10"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E10:99_10"/><b>then </b>
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font> <a href="nat_1.html#R1">divides</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">a</font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">a</font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font> )
 
<b>by </b><i>, <a class="ref" href="nat_1.html#D3">NAT_1:def 3</a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/>
<a NAME="E11:99_10"/><b>then </b><i><font color="Green">E326</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">a</font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K5">div</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">a</font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K5">div</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> 1
 
<b>by </b><i><a class="ref" href="nat_2.html#T17">NAT_2:17</a></i>;<br/>
<a NAME="E12:99_10"/>
<font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">a</font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1
 
<b>by </b><i>, <a class="ref" href="jordan3.html#T5">JORDAN3:5</a></i>;<br/>
<a NAME="E13:99_10"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K5">div</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">a</font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K5">div</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> 1
 
<b>by </b><i>, <a class="ref" href="nat_2.html#T26">NAT_2:26</a></i>;<br/>
<a NAME="E14:99_10"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K5">div</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">a</font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K5">div</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/>
<a NAME="E15:99_10"/><b>then </b><i><font color="Green">E327</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K5">div</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">a</font>
 
<b>by </b><i>, <a class="ref" href="nat_1.html#T68">NAT_1:68</a></i>;<br/>
<a NAME="E16:99_10"/>
<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font>
 
<b>by </b><i>, <a class="ref" href="nat_1.html#T46">NAT_1:46</a></i>;<br/>
<a NAME="E17:99_10"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E18:99_10"/><b>then </b>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K5">div</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 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">a</font></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 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">b</font></span>)</span> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E19:99_10"/><b>then </b><i><font color="Green">E328</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K5">div</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">F</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="relset_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="E20:99_10"/><i><font color="Green">E329</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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K5">div</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> &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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> &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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K5">div</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> &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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 
<b>by </b><i>, , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E21:99_10"/><b>then </b><i><font color="Green">E330</font></i>: 
( <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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K5">div</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <font color="Maroon">f</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K5">div</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 
<b>by </b><i>, , <a class="ref" href="mesfunc3.html#D1">MESFUNC3:def 1</a></i>;<br/>
<a NAME="E22:99_10"/>
<font color="Maroon">FG</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">FG</font>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E23:99_10"/><b>then </b><i><font color="Green">E331</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="mesfunc1.html#K6">-</a> <font color="Maroon">g</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="mesfunc1.html#K6">-</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/>
<b>then </b><span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="mesfunc1.html#K6">-</a> <font color="Maroon">g</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> = 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i><a class="ref" href="mesfunc1.html#D4">MESFUNC1:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="mesfunc1.html#K4">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K5">div</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="mesfunc1.html#K4">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i>, , , , <a class="ref" href="mesfunc3.html#D1">MESFUNC3:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">y</font></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">y</font></span>)</span>
<b>by </b><i>, , , , <a class="ref" href="mesfunc3.html#D1">MESFUNC3:def 1</a></i>
;<br/>
<b>hence </b><a NAME="E25:99_10"/>
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="mesfunc1.html#K6">-</a> <font color="Maroon">g</font></span>)</span> <a href="mesfunc1.html#K4">.</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="mesfunc1.html#K6">-</a> <font color="Maroon">g</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">y</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E28:99"><i><font color="Green">E136</font></i></a>, <a class="ref" href="mesfunc1.html#D4">MESFUNC1:def 4</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E54:99"/><i><font color="Green">E332</font></i>: 
<font color="Maroon">f</font> <a href="mesfunc1.html#K6">-</a> <font color="Maroon">g</font> <a href="mesfunc2.html#R2">is_simple_func_in</a> <font color="Maroon">S</font>
 <b>by </b><i>, , , , <a class="ref" href="mesfunc2.html#D5">MESFUNC2:def 5</a></i>;<br/>

<b>thus </b><a NAME="E55:99"/>
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="mesfunc1.html#K6">-</a> <font color="Maroon">g</font></span>)</span> <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="mesfunc5.html#E16:99"><i><font color="Green">E94</font></i></a></i>;<br/>

<a NAME="E56:99"/><i><font color="Green">E334</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> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="mesfunc1.html#K6">-</a> <font color="Maroon">g</font></span>)</span> holds <br/>0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="mesfunc1.html#K6">-</a> <font color="Maroon">g</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Olive">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="99_11"><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="E1:99_11"/><i><font color="Green">E336</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="mesfunc1.html#K6">-</a> <font color="Maroon">g</font></span>)</span>
 ;<br/>

<a NAME="E2:99_11"/>
<font color="Maroon">g</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="E3:99_11"/><b>then </b>
<span class="p1"><a href="extreal1.html#K4">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</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="ref" href="mesfunc2.html#D1">MESFUNC2:def 1</a></i>;<br/>
<a NAME="E4:99_11"/><b>then </b><i><font color="Green">E337</font></i>: 
( <font color="Maroon">g</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_1.html#K1">+infty</a>  &amp;  <a href="supinf_1.html#K2">-infty</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">f</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> )
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E20:99"><i><font color="Green">E95</font></i></a>, <a class="ref" href="extreal2.html#T58">EXTREAL2:58</a>, <a class="ref" href="supinf_2.html#T70">SUPINF_2:70</a></i>;<br/>
<a NAME="E5:99_11"/>
<font color="Maroon">g</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">f</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E23:99"><i><font color="Green">E99</font></i></a>, , , </i>;<br/>
<a NAME="E6:99_11"/><b>then </b>
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="extreal1.html#T40">EXTREAL1:40</a></i>;<br/>
<b>hence </b><a NAME="E7:99_11"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="mesfunc1.html#K6">-</a> <font color="Maroon">g</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font>
 <b>by </b><i>, <a class="ref" href="mesfunc1.html#D4">MESFUNC1:def 4</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E57:99"/>
<font color="Maroon">f</font> <a href="mesfunc1.html#K6">-</a> <font color="Maroon">g</font> is <a href="supinf_2.html#V6">nonnegative</a>
 <b>by </b><i><a class="ref" href="supinf_2.html#T71">SUPINF_2:71</a></i>;<br/>

<a NAME="E58:99"/><i><font color="Green">E338</font></i>: 
for <font color="Olive">k</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">FG</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">FG</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</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">a1</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">k</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="99_12"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:99_12"/><i><font color="Green">E339</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">FG</font>
 ;<br/>

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

<b>assume </b><a NAME="E2:99_12"/><i><font color="Green">E340</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FG</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>
 ;<br/>

<a NAME="E3:99_12"/>
<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"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">a</font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E4:99_12"/><b>then </b><i><font color="Green">E341</font></i>: 
<font color="Maroon">FG</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> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K5">div</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></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> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i/>;<br/>
<b>set </b><font color="Maroon">i</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K5">div</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1;<br/>
<b>set </b><font color="Maroon">j</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1;<br/>
<a NAME="E5:99_12"/>
<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">FG</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E6:99_12"/><b>then </b><i><font color="Green">E342</font></i>: 
( 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> <font color="Maroon">a</font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span> )
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E7:99_12"/><i><font color="Green">E343</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K5">div</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 <a href="nat_1.html#K1">+</a> 1 &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 <a href="nat_1.html#K1">+</a> 1 )
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E8:99_12"/><i><font color="Green">E344</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i/>;<br/>
<a NAME="E9:99_12"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
;<br/>
<a NAME="E10:99_12"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E11:99_12"/><b>then </b>
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font> <a href="nat_1.html#R1">divides</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">a</font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">a</font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font> )
 
<b>by </b><i>, <a class="ref" href="nat_1.html#D3">NAT_1:def 3</a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/>
<a NAME="E12:99_12"/><b>then </b><i><font color="Green">E345</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">a</font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K5">div</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">a</font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K5">div</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> 1
 
<b>by </b><i><a class="ref" href="nat_2.html#T17">NAT_2:17</a></i>;<br/>
<a NAME="E13:99_12"/>
<font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">a</font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1
 
<b>by </b><i>, <a class="ref" href="jordan3.html#T5">JORDAN3:5</a></i>;<br/>
<a NAME="E14:99_12"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K5">div</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">a</font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K5">div</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span>
 
<b>by </b><i><a class="ref" href="nat_2.html#T26">NAT_2:26</a></i>;<br/>
<a NAME="E15:99_12"/><b>then </b><i><font color="Green">E346</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K5">div</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">a</font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K5">div</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/>
<a NAME="E16:99_12"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">a</font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K5">div</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">a</font>
 
<b>by </b><i>, <a class="ref" href="nat_1.html#T68">NAT_1:68</a></i>;<br/>
<a NAME="E17:99_12"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K5">div</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 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">a</font></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E18:99_12"/><b>then </b><i><font color="Green">E347</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K5">div</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="relset_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="E19:99_12"/><i><font color="Green">E348</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">FG</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E20:99_12"/><i><font color="Green">E349</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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K5">div</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i>, , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E21:99_12"/><b>then </b>
<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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K5">div</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E28:99"><i><font color="Green">E136</font></i></a>, , <a class="ref" href="mesfunc3.html#D1">MESFUNC3:def 1</a></i>;<br/>
<b>hence </b><a NAME="E22:99_12"/>
<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">a1</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">k</font>
 <b>by </b><i>, , </i>;<br/>


</div><b>end;</b></div>
<a NAME="E59:99"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">FG</font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">a1</font>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E60:99"/><b>then </b>
<font color="Maroon">FG</font>,<font color="Maroon">a1</font> <a href="mesfunc3.html#R1">are_Re-presentation_of</a> <font color="Maroon">f</font>
 
<b>by </b><i>, , <a class="ref" href="mesfunc3.html#D1">MESFUNC3:def 1</a></i>;<br/>
<a NAME="E61:99"/><b>then </b><i><font color="Green">E350</font></i>: 
 <a href="mesfunc3.html#K1">integral</a> <font color="Maroon">X</font>,<font color="Maroon">S</font>,<font color="Maroon">M</font>,<font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="convfun1.html#K5">Sum</a> <font color="Maroon">x1</font>
 
<b>by </b><i>, <a class="txt" href="mesfunc5.html#E16:99"><i><font color="Green">E94</font></i></a>, , <a class="ref" href="mesfunc5.html#T25" target="_self">Th25</a>, , <a class="ref" href="mesfunc4.html#T3">MESFUNC4:3</a></i>;<br/>
<a NAME="E62:99"/><i><font color="Green">E351</font></i>: 
for <font color="Olive">k</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">FG</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">FG</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font> holds <br/><font color="Maroon">g</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b1</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">k</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="99_13"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:99_13"/><i><font color="Green">E352</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">FG</font>
 ;<br/>

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

<b>assume </b><a NAME="E2:99_13"/><i><font color="Green">E353</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FG</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>
 ;<br/>

<a NAME="E3:99_13"/>
<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"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">a</font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E4:99_13"/><b>then </b><i><font color="Green">E354</font></i>: 
<font color="Maroon">FG</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> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K5">div</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></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> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i/>;<br/>
<b>set </b><font color="Maroon">i</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K5">div</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1;<br/>
<b>set </b><font color="Maroon">j</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1;<br/>
<a NAME="E5:99_13"/>
<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">FG</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E6:99_13"/><b>then </b><i><font color="Green">E356</font></i>: 
( 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> <font color="Maroon">a</font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span> )
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E7:99_13"/><i><font color="Green">E357</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K5">div</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 <a href="nat_1.html#K1">+</a> 1 &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 <a href="nat_1.html#K1">+</a> 1 )
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E8:99_13"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i/>;<br/>
<a NAME="E9:99_13"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
;<br/>
<a NAME="E10:99_13"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T46">NAT_1:46</a></i>;<br/>
<a NAME="E11:99_13"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E12:99_13"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 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">b</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E13:99_13"/><b>then </b><i><font color="Green">E358</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="relset_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="E14:99_13"/><i><font color="Green">E359</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i>, , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E15:99_13"/><i><font color="Green">E360</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">FG</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<b>thus </b><font color="Maroon">g</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> = 
<font color="Maroon">b</font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="mesfunc5.html#E28:99"><i><font color="Green">E136</font></i></a>, , , <a class="ref" href="mesfunc3.html#D1">MESFUNC3:def 1</a></i>
<br/>.= 
<font color="Maroon">b1</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">k</font>
<b>by </b><i><a class="txt" href="mesfunc5.html#E28:99"><i><font color="Green">E136</font></i></a>, , </i>
;<br/>


</div><b>end;</b></div>
<a NAME="E63:99"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">FG</font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">b1</font>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E64:99"/><b>then </b>
<font color="Maroon">FG</font>,<font color="Maroon">b1</font> <a href="mesfunc3.html#R1">are_Re-presentation_of</a> <font color="Maroon">g</font>
 
<b>by </b><i>, , , <a class="ref" href="mesfunc3.html#D1">MESFUNC3:def 1</a></i>;<br/>
<a NAME="E65:99"/><b>then </b><i><font color="Green">E361</font></i>: 
 <a href="mesfunc3.html#K1">integral</a> <font color="Maroon">X</font>,<font color="Maroon">S</font>,<font color="Maroon">M</font>,<font color="Maroon">g</font> <a href="hidden.html#R1">=</a>  <a href="convfun1.html#K5">Sum</a> <font color="Maroon">y1</font>
 
<b>by </b><i>, , <a class="txt" href="mesfunc5.html#E16:99"><i><font color="Green">E94</font></i></a>, , <a class="txt" href="mesfunc5.html#E35"><i><font color="Green">Lemma115</font></i></a>, <a class="ref" href="mesfunc5.html#T28" target="_self">Th28</a>, <a class="ref" href="mesfunc4.html#T3">MESFUNC4:3</a></i>;<br/>
<a NAME="E66:99"/><i><font color="Green">E362</font></i>: 
for <font color="Olive">k</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">FG</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">FG</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="mesfunc1.html#K6">-</a> <font color="Maroon">g</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c1</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">k</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="99_14"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:99_14"/><i><font color="Green">E363</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">FG</font>
 ;<br/>

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

<b>assume </b><a NAME="E2:99_14"/><i><font color="Green">E364</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FG</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>
 ;<br/>

<a NAME="E3:99_14"/>
<font color="Maroon">FG</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">FG</font>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E4:99_14"/><b>then </b>
<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="mesfunc1.html#K6">-</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/>
<a NAME="E5:99_14"/><b>then </b><i><font color="Green">E365</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="mesfunc1.html#K6">-</a> <font color="Maroon">g</font></span>)</span> <a href="mesfunc1.html#K4">.</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="mesfunc1.html#K4">.</a> <font color="Maroon">x</font></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font></span>)</span>
 
<b>by </b><i><a class="ref" href="mesfunc1.html#D4">MESFUNC1:def 4</a></i>;<br/>
<a NAME="E6:99_14"/>
<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">FG</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E7:99_14"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">a1</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">k</font></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c1</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">k</font>
 
<b>by </b><i/>;<br/>
<a NAME="E8:99_14"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">a1</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">k</font></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c1</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">k</font>
 
<b>by </b><i>, , </i>;<br/>
<b>hence </b><a NAME="E9:99_14"/>
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="mesfunc1.html#K6">-</a> <font color="Maroon">g</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c1</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">k</font>
 <b>by </b><i>, , , </i>;<br/>


</div><b>end;</b></div>
<a NAME="E67:99"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">FG</font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c1</font>
 
<b>by </b><i><a class="ref" href="mesfunc5.html#T29" target="_self">Th29</a>, <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/>
<a NAME="E68:99"/><b>then </b>
<font color="Maroon">FG</font>,<font color="Maroon">c1</font> <a href="mesfunc3.html#R1">are_Re-presentation_of</a> <font color="Maroon">f</font> <a href="mesfunc1.html#K6">-</a> <font color="Maroon">g</font>
 
<b>by </b><i>, , , , <a class="ref" href="mesfunc3.html#D1">MESFUNC3:def 1</a></i>;<br/>
<a NAME="E69:99"/><b>then </b><i><font color="Green">E366</font></i>: 
 <a href="mesfunc3.html#K1">integral</a> <font color="Maroon">X</font>,<font color="Maroon">S</font>,<font color="Maroon">M</font>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="mesfunc1.html#K6">-</a> <font color="Maroon">g</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="convfun1.html#K5">Sum</a> <font color="Maroon">z1</font>
 
<b>by </b><i>, , , <a class="txt" href="mesfunc5.html#E16:99"><i><font color="Green">E94</font></i></a>, , <a class="ref" href="mesfunc5.html#T31" target="_self">Th31</a>, , <a class="ref" href="mesfunc4.html#T3">MESFUNC4:3</a></i>;<br/>
<a NAME="E70:99"/><i><font color="Green">E367</font></i>: 
for <font color="Olive">i</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="relset_1.html#K4">dom</a> <font color="Maroon">y1</font> holds <br/>0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">y1</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">i</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="99_15"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:99_15"/><i><font color="Green">E368</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">y1</font>
 ;<br/>

<a NAME="E2:99_15"/><i><font color="Green">E369</font></i>: 
<font color="Maroon">y1</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">i</font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="finseqop.html#K5">*</a> <font color="Maroon">FG</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">i</font></span>)</span>
 
<b>by </b><i><a class="ref" href="mesfunc5.html#T28" target="_self">Th28</a>, </i>;<br/>
<a NAME="E3:99_15"/><i><font color="Green">E370</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">FG</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="mesfunc5.html#T27" target="_self">Th27</a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<b>set </b><font color="Maroon">i'</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1;<br/>
<a NAME="E4:99_15"/><i><font color="Green">E371</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">a</font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E28:99"><i><font color="Green">E136</font></i></a>, , <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E5:99_15"/><i><font color="Green">E372</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E6:99_15"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i/>;<br/>
<a NAME="E7:99_15"/><b>then </b><i><font color="Green">E373</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
;<br/>
<a NAME="E8:99_15"/>
<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E30:99"><i><font color="Green">E139</font></i></a>, <a class="ref" href="nat_1.html#T46">NAT_1:46</a></i>;<br/>
<a NAME="E9:99_15"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E10:99_15"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 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">b</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E11:99_15"/><b>then </b><i><font color="Green">E375</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="relset_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/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="99_15_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:99_15_1"/>
( <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="99_15_1_1"><b>suppose </b></a><a NAME="E1:99_15_1_1"/><i><font color="Green">E376</font></i>: 
<font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">x'</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:99_15_1_1"/><i><font color="Green">E377</font></i>: 
<font color="Maroon">x'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/><font color="Maroon">g</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x'</font> = 
<font color="Maroon">b</font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="mesfunc5.html#E32:99"><i><font color="Green">E140</font></i></a>, <a class="txt" href="mesfunc5.html#E34:99"><i><font color="Green">E142</font></i></a>, , <a class="ref" href="mesfunc3.html#D1">MESFUNC3:def 1</a></i>
<br/>.= 
<font color="Maroon">b1</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">i</font>
<b>by </b><i><a class="txt" href="mesfunc5.html#E33:99"><i><font color="Green">E141</font></i></a>, <a class="txt" href="mesfunc5.html#E28:99"><i><font color="Green">E136</font></i></a>, </i>
;<br/><a NAME="E5:99_15_1_1"/><b>then </b><i><font color="Green">E378</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b1</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E22:99"><i><font color="Green">E98</font></i></a>, <a class="ref" href="supinf_2.html#T70">SUPINF_2:70</a></i>;<br/><a NAME="E6:99_15_1_1"/>
<font color="Maroon">FG</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">FG</font>
 <b>by </b><i>, <a class="txt" href="mesfunc5.html#E35"><i><font color="Green">Lemma115</font></i></a>, <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/><b>then reconsider </b><font color="Maroon">FGi</font> = <font color="Maroon">FG</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> as    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">S</font> ;<br/><b>reconsider </b><font color="Maroon">EMPTY</font> =  <a href="xboole_0.html#K1">{}</a>  as    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">S</font> <b>by </b><i><a class="ref" href="measure1.html#T21">MEASURE1:21</a></i>;<br/><a NAME="E9:99_15_1_1"/>
<font color="Maroon">EMPTY</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">FGi</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T2">XBOOLE_1:2</a></i>;<br/><a NAME="E10:99_15_1_1"/><b>then </b><i><font color="Green">E379</font></i>: 
<font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <a href="xboole_0.html#K1">{}</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">FGi</font>
 <b>by </b><i><a class="ref" href="measure1.html#T62">MEASURE1:62</a></i>;<br/><a NAME="E11:99_15_1_1"/>
<font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="measure1.html#D11">MEASURE1:def 11</a></i>;<br/><a NAME="E12:99_15_1_1"/><b>then </b>
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="finseqop.html#K5">*</a> <font color="Maroon">FG</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="ref" href="mesfunc5.html#T32" target="_self">Th32</a>, , <a class="txt" href="mesfunc5.html#E35"><i><font color="Green">Lemma115</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>;<br/><b>hence </b><a NAME="E13:99_15_1_1"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">y1</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, <a class="txt" href="mesfunc5.html#E35:99"><i><font color="Green">E143</font></i></a>, <a class="ref" href="extreal2.html#T45">EXTREAL2:45</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="99_15_1_2"><b>suppose </b></a><a NAME="E1:99_15_1_2"/><i><font color="Green">E380</font></i>: 
<font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:99_15_1_2"/>
<font color="Maroon">FG</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> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K5">div</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></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> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E28:99"><i><font color="Green">E136</font></i></a>, , </i>;<br/><a NAME="E3:99_15_1_2"/><b>then </b>
<font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><font color="Maroon">FG</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="txt" href="mesfunc5.html#E33:99"><i><font color="Green">E141</font></i></a>, <a class="ref" href="measure1.html#D11">MEASURE1:def 11</a></i>;<br/><a NAME="E4:99_15_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="finseqop.html#K5">*</a> <font color="Maroon">FG</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i>, <a class="txt" href="mesfunc5.html#E35"><i><font color="Green">Lemma115</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>;<br/><b>hence </b><a NAME="E5:99_15_1_2"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">y1</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, <a class="ref" href="extreal1.html#T22">EXTREAL1:22</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E71:99"/><i><font color="Green">E381</font></i>: 
for <font color="Olive">i</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="relset_1.html#K4">dom</a> <font color="Maroon">x1</font> holds <br/>0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">z1</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">i</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="99_16"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:99_16"/><i><font color="Green">E382</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">x1</font>
 ;<br/>

<a NAME="E2:99_16"/><b>then </b><i><font color="Green">E383</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="finseqop.html#K5">*</a> <font color="Maroon">FG</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><font color="Maroon">FG</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>
 
<b>by </b><i><a class="ref" href="mesfunc5.html#T25" target="_self">Th25</a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>;<br/>
<a NAME="E3:99_16"/>
( <font color="Maroon">FG</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">FG</font> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">FG</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">S</font> )
 
<b>by </b><i>, <a class="ref" href="mesfunc5.html#T25" target="_self">Th25</a>, <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">V</font> = <font color="Maroon">FG</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> as    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">S</font> ;<br/>
<b>reconsider </b><font color="Maroon">EMPTY</font> =  <a href="xboole_0.html#K1">{}</a>  as    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">S</font> <b>by </b><i><a class="ref" href="prob_1.html#T42">PROB_1:42</a></i>;<br/>
<a NAME="E6:99_16"/><i><font color="Green">E384</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">FG</font></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E7:99_16"/>
<font color="Maroon">EMPTY</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">V</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T2">XBOOLE_1:2</a></i>;<br/>
<a NAME="E8:99_16"/><b>then </b>
<font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">EMPTY</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">V</font>
 
<b>by </b><i><a class="ref" href="measure1.html#T62">MEASURE1:62</a></i>;<br/>
<a NAME="E9:99_16"/><b>then </b><i><font color="Green">E385</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="finseqop.html#K5">*</a> <font color="Maroon">FG</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">i</font>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E28:99"><i><font color="Green">E136</font></i></a>, <a class="ref" href="measure1.html#D11">MEASURE1:def 11</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="99_16_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:99_16_1"/>
( <font color="Maroon">FG</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">FG</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="99_16_1_1"><b>suppose </b></a><a NAME="E1:99_16_1_1"/>
<font color="Maroon">FG</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:99_16_1_1"/><i><font color="Green">E386</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FG</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/><a NAME="E4:99_16_1_1"/>
<font color="Maroon">FG</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">FG</font>
 <b>by </b><i>, <a class="ref" href="mesfunc5.html#T25" target="_self">Th25</a>, <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/><a NAME="E5:99_16_1_1"/><b>then </b><i><font color="Green">E415</font></i>: 
<font color="Maroon">x</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">FG</font></span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E30:99"><i><font color="Green">E139</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><a NAME="E6:99_16_1_1"/><b>then </b>
<font color="Maroon">g</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">f</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E23:99"><i><font color="Green">E99</font></i></a>, </i>;<br/><a NAME="E7:99_16_1_1"/><b>then </b>
<font color="Maroon">g</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a1</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, <a class="ref" href="mesfunc5.html#T25" target="_self">Th25</a>, , <a class="txt" href="mesfunc5.html#E30:99"><i><font color="Green">E139</font></i></a></i>;<br/><a NAME="E8:99_16_1_1"/><b>then </b><i><font color="Green">E416</font></i>: 
<font color="Maroon">b1</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a1</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E30:99"><i><font color="Green">E139</font></i></a>, , <a class="ref" href="mesfunc5.html#T25" target="_self">Th25</a>, </i>;<br/><a NAME="E9:99_16_1_1"/><i><font color="Green">E417</font></i>: 
<font color="Maroon">g</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/><b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font> <b>by </b><i><a class="txt" href="mesfunc5.html#E32:99"><i><font color="Green">E140</font></i></a>, </i>;<br/><a NAME="E11:99_16_1_1"/>
<span class="p1"><a href="extreal1.html#K4">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</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="mesfunc5.html#E32:99"><i><font color="Green">E140</font></i></a>, , , <a class="txt" href="mesfunc5.html#E34:99"><i><font color="Green">E142</font></i></a>, <a class="ref" href="mesfunc2.html#D1">MESFUNC2:def 1</a></i>;<br/><a NAME="E12:99_16_1_1"/><b>then </b>
<span class="p1"><a href="extreal1.html#K4">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">b1</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">i</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="mesfunc5.html#E30:99"><i><font color="Green">E139</font></i></a>, , <a class="ref" href="mesfunc5.html#T25" target="_self">Th25</a>, </i>;<br/><a NAME="E13:99_16_1_1"/><b>then </b>
(  <a href="supinf_2.html#K3">-</a> <a href="supinf_1.html#K1">+infty</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b1</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">b1</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">i</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="ref" href="extreal2.html#T58">EXTREAL2:58</a></i>;<br/><a NAME="E14:99_16_1_1"/><b>then </b>
( <font color="Maroon">b1</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K2">-infty</a>  &amp; <font color="Maroon">b1</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K1">+infty</a>  )
 <b>by </b><i><a class="ref" href="supinf_2.html#D3">SUPINF_2:def 3</a></i>;<br/><a NAME="E15:99_16_1_1"/><b>then </b>
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">a1</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">i</font></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">i</font></span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E33:99"><i><font color="Green">E141</font></i></a>, <a class="ref" href="measure5.html#T1">MEASURE5:1</a></i>;<br/><a NAME="E16:99_16_1_1"/><b>then </b>
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c1</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, </i>;<br/><a NAME="E17:99_16_1_1"/><b>then </b>
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c1</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">i</font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="finseqop.html#K5">*</a> <font color="Maroon">FG</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">i</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="extreal2.html#T45">EXTREAL2:45</a></i>;<br/><b>hence </b><a NAME="E18:99_16_1_1"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">z1</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, <a class="ref" href="mesfunc5.html#T25" target="_self">Th25</a>, <a class="ref" href="mesfunc5.html#T31" target="_self">Th31</a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="99_16_1_2"><b>suppose </b></a><a NAME="E1:99_16_1_2"/>
<font color="Maroon">FG</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:99_16_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="finseqop.html#K5">*</a> <font color="Maroon">FG</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="txt" href="mesfunc5.html#E28:99"><i><font color="Green">E136</font></i></a>, <a class="ref" href="measure1.html#D11">MEASURE1:def 11</a></i>;<br/><a NAME="E3:99_16_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c1</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">i</font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="finseqop.html#K5">*</a> <font color="Maroon">FG</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="extreal1.html#T22">EXTREAL1:22</a></i>;<br/><b>hence </b><a NAME="E4:99_16_1_2"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">z1</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, <a class="ref" href="mesfunc5.html#T25" target="_self">Th25</a>, <a class="ref" href="mesfunc5.html#T31" target="_self">Th31</a>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E72:99"/>
<font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z1</font> <a href="mesfunc1.html#K13">+</a> <font color="Maroon">y1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="99_17"><b>proof </b></a><div class="add">

<a NAME="E1:99_17"/>
not  <a href="supinf_1.html#K2">-infty</a>  <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">y1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="99_17_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:99_17_1"/>
 <a href="supinf_1.html#K2">-infty</a>  <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">y1</font>
 ;<br/>

<b>then consider </b><font color="Maroon">i</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:99_17_1"/><i><font color="Green">E418</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">y1</font>
 <b>and </b><br/><a NAME="E4:99_17_1"/><i><font color="Green">E419</font></i>: 
<font color="Maroon">y1</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a> 
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E5:99_17_1"/>
<font color="Maroon">i</font> is   <a href="nat_1.html#NM1">Nat</a>
 
<b>by </b><i>, <a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/>
<b>hence </b><a NAME="E6:99_17_1"/>
contradiction
 <b>by </b><i>, , <a class="txt" href="mesfunc5.html#E28:99"><i><font color="Green">E136</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then </b><i><font color="Green">E420</font></i>: <span class="p1">(<span class="default"><font color="Maroon">z1</font> <a href="relset_1.html#K11">"</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><a href="supinf_1.html#K1">+infty</a> </span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">y1</font> <a href="relset_1.html#K11">"</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><a href="supinf_1.html#K2">-infty</a> </span><a href="tarski.html#K1">}</a></span></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">z1</font> <a href="relset_1.html#K11">"</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><a href="supinf_1.html#K1">+infty</a> </span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="subset_1.html#K8">/\</a> <a href="xboole_0.html#K1">{}</a> 
<b>by </b><i><a class="ref" href="funct_1.html#T142">FUNCT_1:142</a></i>
<br/>.= 
 <a href="xboole_0.html#K1">{}</a> 

;<br/>
<a NAME="E3:99_17"/>
not  <a href="supinf_1.html#K2">-infty</a>  <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">z1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="99_17_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:99_17_3"/>
 <a href="supinf_1.html#K2">-infty</a>  <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">z1</font>
 ;<br/>

<b>then consider </b><font color="Maroon">i</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:99_17_3"/><i><font color="Green">E421</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">z1</font>
 <b>and </b><br/><a NAME="E4:99_17_3"/><i><font color="Green">E422</font></i>: 
<font color="Maroon">z1</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a> 
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E5:99_17_3"/>
<font color="Maroon">i</font> is   <a href="nat_1.html#NM1">Nat</a>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E28:99"><i><font color="Green">E136</font></i></a>, <a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/>
<b>hence </b><a NAME="E6:99_17_3"/>
contradiction
 <b>by </b><i>, <a class="ref" href="mesfunc5.html#T25" target="_self">Th25</a>, <a class="ref" href="mesfunc5.html#T31" target="_self">Th31</a>, <a class="txt" href="mesfunc5.html#E28:99"><i><font color="Green">E136</font></i></a>, </i>;<br/>


</div><b>end;</b></div>
<b>then </b><i><font color="Green">E423</font></i>: <span class="p1">(<span class="default"><font color="Maroon">z1</font> <a href="relset_1.html#K11">"</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><a href="supinf_1.html#K2">-infty</a> </span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">y1</font> <a href="relset_1.html#K11">"</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><a href="supinf_1.html#K1">+infty</a> </span><a href="tarski.html#K1">}</a></span></span>)</span> = 
<a href="xboole_0.html#K1">{}</a>  <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">y1</font> <a href="relset_1.html#K11">"</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><a href="supinf_1.html#K1">+infty</a> </span><a href="tarski.html#K1">}</a></span></span>)</span>
<b>by </b><i><a class="ref" href="funct_1.html#T142">FUNCT_1:142</a></i>
<br/>.= 
 <a href="xboole_0.html#K1">{}</a> 

;<br/>
<i><font color="Green">E424</font></i>:  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">z1</font> <a href="mesfunc1.html#K13">+</a> <font color="Maroon">y1</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">z1</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">y1</font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="xboole_0.html#K1">{}</a>  <a href="xboole_0.html#K2">\/</a> <a href="xboole_0.html#K1">{}</a> </span>)</span>
<b>by </b><i>, <a class="txt" href="mesfunc5.html#E28:99"><i><font color="Green">E136</font></i></a>, <a class="ref" href="mesfunc1.html#D3">MESFUNC1:def 3</a></i>
<br/>.= 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">x1</font>
<b>by </b><i>, <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a>, <a class="txt" href="mesfunc5.html#E35"><i><font color="Green">Lemma115</font></i></a>, <a class="ref" href="mesfunc5.html#T31" target="_self">Th31</a></i>
;<br/>
<a NAME="E6:99_17"/><i><font color="Green">E425</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="relset_1.html#K4">dom</a> <font color="Maroon">x1</font> holds <br/><font color="Maroon">x1</font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">z1</font> <a href="mesfunc1.html#K13">+</a> <font color="Maroon">y1</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Olive">k</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="99_17_6"><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="E1:99_17_6"/><i><font color="Green">E426</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">x1</font>
 ;<br/>

<a NAME="E2:99_17_6"/><b>then </b><i><font color="Green">E427</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">FG</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<b>set </b><font color="Maroon">p</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K5">div</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1;<br/>
<b>set </b><font color="Maroon">q</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1;<br/>
<a NAME="E3:99_17_6"/><i><font color="Green">E429</font></i>: 
( 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> <font color="Maroon">a</font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span> )
 
<b>by </b><i>, <a class="txt" href="mesfunc5.html#E32:99"><i><font color="Green">E140</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E4:99_17_6"/><i><font color="Green">E430</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K5">div</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 <a href="nat_1.html#K1">+</a> 1 &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 <a href="nat_1.html#K1">+</a> 1 )
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E5:99_17_6"/><i><font color="Green">E431</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E33:99"><i><font color="Green">E141</font></i></a></i>;<br/>
<a NAME="E6:99_17_6"/><b>then </b><i><font color="Green">E432</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
;<br/>
<a NAME="E7:99_17_6"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E8:99_17_6"/><b>then </b>
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font> <a href="nat_1.html#R1">divides</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">a</font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">a</font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font> )
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E33:99"><i><font color="Green">E141</font></i></a>, <a class="ref" href="nat_1.html#D3">NAT_1:def 3</a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/>
<a NAME="E9:99_17_6"/><b>then </b><i><font color="Green">E433</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">a</font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K5">div</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">a</font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K5">div</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> 1
 
<b>by </b><i><a class="ref" href="nat_2.html#T17">NAT_2:17</a></i>;<br/>
<a NAME="E10:99_17_6"/>
<font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">a</font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E33:99"><i><font color="Green">E141</font></i></a>, <a class="ref" href="jordan3.html#T5">JORDAN3:5</a></i>;<br/>
<a NAME="E11:99_17_6"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K5">div</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">a</font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K5">div</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> 1
 
<b>by </b><i>, <a class="ref" href="nat_2.html#T26">NAT_2:26</a></i>;<br/>
<a NAME="E12:99_17_6"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K5">div</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">a</font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K5">div</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/>
<a NAME="E13:99_17_6"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K5">div</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">a</font>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E35:99"><i><font color="Green">E143</font></i></a>, <a class="ref" href="nat_1.html#T68">NAT_1:68</a></i>;<br/>
<a NAME="E14:99_17_6"/><b>then </b><i><font color="Green">E434</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K5">div</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 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">a</font></span>)</span>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E34:99"><i><font color="Green">E142</font></i></a></i>;<br/>
<a NAME="E15:99_17_6"/>
<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font>
 
<b>by </b><i><a class="ref" href="mesfunc5.html#T32" target="_self">Th32</a>, <a class="ref" href="nat_1.html#T46">NAT_1:46</a></i>;<br/>
<a NAME="E16:99_17_6"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E17:99_17_6"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 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">b</font></span>)</span>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E34:99"><i><font color="Green">E142</font></i></a></i>;<br/>
<a NAME="E18:99_17_6"/><b>then </b><i><font color="Green">E435</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K5">div</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">F</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="relset_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>, <a class="ref" href="mesfunc5.html#T33" target="_self">Th33</a></i>;<br/>
<a NAME="E19:99_17_6"/><i><font color="Green">E436</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c1</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">k</font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">b1</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="finseqop.html#K5">*</a> <font color="Maroon">FG</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c1</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">k</font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">M</font> <a href="finseqop.html#K5">*</a> <font color="Maroon">FG</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b1</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">k</font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">M</font> <a href="finseqop.html#K5">*</a> <font color="Maroon">FG</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">k</font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="99_17_6_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="99_17_6_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:99_17_6_1_1"/>
( <font color="Maroon">FG</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>  or <font color="Maroon">FG</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="99_17_6_1_1_1"><b>suppose </b></a><a NAME="E1:99_17_6_1_1_1"/>
<font color="Maroon">FG</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> 
 ;<br/><div class="add"><a NAME="E2:99_17_6_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K5">div</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></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> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <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="mesfunc5.html#E32:99"><i><font color="Green">E140</font></i></a>, , </i>;<br/><b>then consider </b><font color="Maroon">v</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:99_17_6_1_1_1"/><i><font color="Green">E437</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K5">div</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></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> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/><a NAME="E5:99_17_6_1_1_1"/><i><font color="Green">E438</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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K5">div</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E6:99_17_6_1_1_1"/><b>then </b><i><font color="Green">E439</font></i>: 
( <font color="Maroon">a</font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K5">div</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">v</font> &amp; <font color="Maroon">b</font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">v</font> )
 <b>by </b><i>, , <a class="ref" href="mesfunc5.html#T34" target="_self">Th34</a>, <a class="ref" href="mesfunc3.html#D1">MESFUNC3:def 1</a></i>;<br/><a NAME="E7:99_17_6_1_1_1"/><i><font color="Green">E440</font></i>: 
( <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K5">div</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 <b>by </b><i/>;<br/><a NAME="E8:99_17_6_1_1_1"/>
( <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K5">div</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">F</font> &amp; <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <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="mesfunc5.html#T34" target="_self">Th34</a>, <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/><a NAME="E9:99_17_6_1_1_1"/><b>then </b><i><font color="Green">E441</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>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> )
 <b>by </b><i>, , <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><a NAME="E10:99_17_6_1_1_1"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b</font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i>, <a class="txt" href="mesfunc5.html#E22:99"><i><font color="Green">E98</font></i></a>, <a class="ref" href="supinf_2.html#T70">SUPINF_2:70</a></i>;<br/><a NAME="E11:99_17_6_1_1_1"/><b>then </b><i><font color="Green">E442</font></i>: 
( 0 <a href="hidden.html#R1">=</a> <font color="Maroon">b1</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">k</font> or 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b1</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">k</font> )
 <b>by </b><i>, , <a class="txt" href="mesfunc5.html#E32:99"><i><font color="Green">E140</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon">v</font> = <font color="Maroon">v</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font> <b>by </b><i/>;<br/><a NAME="E13:99_17_6_1_1_1"/>
<font color="Maroon">g</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="E14:99_17_6_1_1_1"/><b>then </b>
<span class="p1"><a href="extreal1.html#K4">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">v</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="ref" href="mesfunc2.html#D1">MESFUNC2:def 1</a></i>;<br/><a NAME="E15:99_17_6_1_1_1"/><b>then </b>
<span class="p1"><a href="extreal1.html#K4">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">b1</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">k</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="mesfunc5.html#E32:99"><i><font color="Green">E140</font></i></a></i>;<br/><a NAME="E16:99_17_6_1_1_1"/><b>then </b>
(  <a href="supinf_2.html#K3">-</a> <a href="supinf_1.html#K1">+infty</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b1</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">k</font> &amp; <font color="Maroon">b1</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">k</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="ref" href="extreal2.html#T58">EXTREAL2:58</a></i>;<br/><a NAME="E17:99_17_6_1_1_1"/><b>then </b><i><font color="Green">E443</font></i>: 
(  <a href="supinf_1.html#K2">-infty</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b1</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">k</font> &amp; <font color="Maroon">b1</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">k</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="ref" href="supinf_2.html#D3">SUPINF_2:def 3</a></i>;<br/><a NAME="E18:99_17_6_1_1_1"/>
<font color="Maroon">b</font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a</font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K5">div</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i>, , <a class="txt" href="mesfunc5.html#E23:99"><i><font color="Green">E99</font></i></a></i>;<br/><a NAME="E19:99_17_6_1_1_1"/><b>then </b>
<font color="Maroon">b1</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a</font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K5">div</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i>, , <a class="txt" href="mesfunc5.html#E32:99"><i><font color="Green">E140</font></i></a></i>;<br/><a NAME="E20:99_17_6_1_1_1"/><b>then </b>
<font color="Maroon">b1</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a1</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">k</font>
 <b>by </b><i>, , <a class="txt" href="mesfunc5.html#E32:99"><i><font color="Green">E140</font></i></a></i>;<br/><a NAME="E21:99_17_6_1_1_1"/><b>then </b>
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">a1</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">k</font></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">k</font></span>)</span>
 <b>by </b><i><a class="ref" href="mesfunc5.html#T37" target="_self">Th37</a>, <a class="ref" href="measure5.html#T1">MEASURE5:1</a></i>;<br/><a NAME="E22:99_17_6_1_1_1"/><b>then </b>
( 0 <a href="hidden.html#R1">=</a> <font color="Maroon">c1</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">k</font> or 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c1</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">k</font> )
 <b>by </b><i>, <a class="txt" href="mesfunc5.html#E32:99"><i><font color="Green">E140</font></i></a></i>;<br/><b>hence </b><a NAME="E23:99_17_6_1_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c1</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">k</font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">b1</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="finseqop.html#K5">*</a> <font color="Maroon">FG</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c1</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">k</font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">M</font> <a href="finseqop.html#K5">*</a> <font color="Maroon">FG</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b1</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">k</font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">M</font> <a href="finseqop.html#K5">*</a> <font color="Maroon">FG</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">k</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="mesfunc5.html#T36" target="_self">Th36</a>, <a class="ref" href="extreal1.html#T29">EXTREAL1:29</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="99_17_6_1_1_2"><b>suppose </b></a><a NAME="E1:99_17_6_1_1_2"/>
<font color="Maroon">FG</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:99_17_6_1_1_2"/><b>then </b>
<font color="Maroon">M</font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><font color="Maroon">FG</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="measure1.html#D11">MEASURE1:def 11</a></i>;<br/><a NAME="E3:99_17_6_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="finseqop.html#K5">*</a> <font color="Maroon">FG</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="txt" href="mesfunc5.html#E30:99"><i><font color="Green">E139</font></i></a>, <a class="ref" href="mesfunc5.html#T25" target="_self">Th25</a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>;<br/><b>hence </b><a NAME="E4:99_17_6_1_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c1</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">k</font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">b1</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="finseqop.html#K5">*</a> <font color="Maroon">FG</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c1</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">k</font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">M</font> <a href="finseqop.html#K5">*</a> <font color="Maroon">FG</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b1</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">k</font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">M</font> <a href="finseqop.html#K5">*</a> <font color="Maroon">FG</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">k</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="extreal1.html#T29">EXTREAL1:29</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E20:99_17_6"/><i><font color="Green">E444</font></i>: 
( <font color="Maroon">a1</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K1">+infty</a>  &amp; <font color="Maroon">a1</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K2">-infty</a>  &amp; <font color="Maroon">b1</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K1">+infty</a>  &amp; <font color="Maroon">b1</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K2">-infty</a>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="99_17_6_2"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="99_17_6_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="99_17_6_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:99_17_6_2_1_1"/>
( <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K5">div</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K5">div</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="99_17_6_2_1_1_1"><b>suppose </b></a><a NAME="E1:99_17_6_2_1_1_1"/><i><font color="Green">E445</font></i>: 
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K5">div</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">v</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:99_17_6_2_1_1_1"/><i><font color="Green">E446</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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K5">div</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/><a NAME="E4:99_17_6_2_1_1_1"/>
<font color="Maroon">a1</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K5">div</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i>, <a class="txt" href="mesfunc5.html#E32:99"><i><font color="Green">E140</font></i></a>, </i>;<br/><a NAME="E5:99_17_6_2_1_1_1"/><b>then </b><i><font color="Green">E447</font></i>: 
<font color="Maroon">a1</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">v</font>
 <b>by </b><i><a class="ref" href="mesfunc5.html#T34" target="_self">Th34</a>, , <a class="ref" href="mesfunc3.html#D1">MESFUNC3:def 1</a>, </i>;<br/><a NAME="E6:99_17_6_2_1_1_1"/>
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K5">div</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <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="mesfunc5.html#T34" target="_self">Th34</a>, <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/><a NAME="E7:99_17_6_2_1_1_1"/><b>then </b><i><font color="Green">E448</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>
 <b>by </b><i>, , <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><b>then reconsider </b><font color="Maroon">v</font> = <font color="Maroon">v</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font> ;<br/><a NAME="E9:99_17_6_2_1_1_1"/>
<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="E10:99_17_6_2_1_1_1"/><b>then </b>
<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">v</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="ref" href="mesfunc2.html#D1">MESFUNC2:def 1</a></i>;<br/><a NAME="E11:99_17_6_2_1_1_1"/><b>then </b>
(  <a href="supinf_2.html#K3">-</a> <a href="supinf_1.html#K1">+infty</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a1</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">k</font> &amp; <font color="Maroon">a1</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">k</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="ref" href="extreal2.html#T58">EXTREAL2:58</a></i>;<br/><b>hence </b><a NAME="E12:99_17_6_2_1_1_1"/>
( <font color="Maroon">a1</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K1">+infty</a>  &amp;  <a href="supinf_1.html#K2">-infty</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a1</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">k</font> )
 <b>by </b><i><a class="ref" href="supinf_2.html#D3">SUPINF_2:def 3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="99_17_6_2_1_1_2"><b>suppose </b></a><a NAME="E1:99_17_6_2_1_1_2"/>
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K5">div</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:99_17_6_2_1_1_2"/>
( <font color="Maroon">a1</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K1">+infty</a>  &amp;  <a href="supinf_1.html#K2">-infty</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a1</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">k</font> )
 <b>by </b><i><a class="txt" href="mesfunc5.html#E32:99"><i><font color="Green">E140</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="E2:99_17_6_2"/>
(  <a href="supinf_1.html#K1">+infty</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a1</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">k</font> &amp; <font color="Maroon">a1</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K2">-infty</a>  )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="99_17_6_2_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="99_17_6_2_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:99_17_6_2_2_1"/>
( <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="99_17_6_2_2_1_1"><b>suppose </b></a><a NAME="E1:99_17_6_2_2_1_1"/><i><font color="Green">E449</font></i>: 
<font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">v</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:99_17_6_2_2_1_1"/><i><font color="Green">E450</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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/><a NAME="E4:99_17_6_2_2_1_1"/>
<font color="Maroon">b1</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i>, <a class="txt" href="mesfunc5.html#E32:99"><i><font color="Green">E140</font></i></a>, </i>;<br/><a NAME="E5:99_17_6_2_2_1_1"/><b>then </b><i><font color="Green">E451</font></i>: 
<font color="Maroon">b1</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">v</font>
 <b>by </b><i>, <a class="ref" href="mesfunc5.html#T34" target="_self">Th34</a>, <a class="ref" href="mesfunc3.html#D1">MESFUNC3:def 1</a>, </i>;<br/><a NAME="E6:99_17_6_2_2_1_1"/>
<font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <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="mesfunc5.html#T34" target="_self">Th34</a>, <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/><a NAME="E7:99_17_6_2_2_1_1"/><b>then </b><i><font color="Green">E452</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">g</font>
 <b>by </b><i>, , <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><b>then reconsider </b><font color="Maroon">v</font> = <font color="Maroon">v</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font> ;<br/><a NAME="E9:99_17_6_2_2_1_1"/>
<font color="Maroon">g</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="E10:99_17_6_2_2_1_1"/><b>then </b>
<span class="p1"><a href="extreal1.html#K4">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">v</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="ref" href="mesfunc2.html#D1">MESFUNC2:def 1</a></i>;<br/><a NAME="E11:99_17_6_2_2_1_1"/><b>then </b>
(  <a href="supinf_2.html#K3">-</a> <a href="supinf_1.html#K1">+infty</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b1</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">k</font> &amp; <font color="Maroon">b1</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">k</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="ref" href="extreal2.html#T58">EXTREAL2:58</a></i>;<br/><b>hence </b><a NAME="E12:99_17_6_2_2_1_1"/>
( <font color="Maroon">b1</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K1">+infty</a>  &amp;  <a href="supinf_1.html#K2">-infty</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b1</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">k</font> )
 <b>by </b><i><a class="ref" href="supinf_2.html#D3">SUPINF_2:def 3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="99_17_6_2_2_1_2"><b>suppose </b></a><a NAME="E1:99_17_6_2_2_1_2"/>
<font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:99_17_6_2_2_1_2"/>
( <font color="Maroon">b1</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K1">+infty</a>  &amp;  <a href="supinf_1.html#K2">-infty</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b1</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">k</font> )
 <b>by </b><i><a class="txt" href="mesfunc5.html#E32:99"><i><font color="Green">E140</font></i></a>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E4:99_17_6_2"/>
(  <a href="supinf_1.html#K1">+infty</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b1</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">k</font> &amp; <font color="Maroon">b1</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K2">-infty</a>  )
 ;<br/>


</div><b>end;</b></div>
<a NAME="E21:99_17_6"/>
<font color="Maroon">c1</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a1</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">k</font></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">k</font></span>)</span>
 
<b>by </b><i><a class="txt" href="mesfunc5.html#E32:99"><i><font color="Green">E140</font></i></a>, </i>;<br/>
<b>then </b><span class="p1">(<span class="default"><font color="Maroon">c1</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">k</font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">k</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">a1</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">k</font></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b1</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">k</font></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p2">(<span class="default"><font color="Maroon">b1</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">k</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="extreal2.html#T41">EXTREAL2:41</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a1</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">k</font></span>)</span> <a href="supinf_2.html#K2">+</a> <a href="supinf_2.html#K1">0.</a> 
<b>by </b><i><a class="ref" href="extreal1.html#T9">EXTREAL1:9</a>, <a class="ref" href="extreal1.html#T24">EXTREAL1:24</a></i>
<br/>.= 
<font color="Maroon">a1</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">k</font>
<b>by </b><i><a class="ref" href="supinf_2.html#T18">SUPINF_2:18</a></i>
;<br/>
<b>hence </b><font color="Maroon">x1</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">k</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c1</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">k</font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">b1</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="finseqop.html#K5">*</a> <font color="Maroon">FG</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">k</font></span>)</span>
<b>by </b><i>, <a class="txt" href="mesfunc5.html#E30:99"><i><font color="Green">E139</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">z1</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">k</font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b1</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">k</font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">M</font> <a href="finseqop.html#K5">*</a> <font color="Maroon">FG</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">k</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="mesfunc5.html#T25" target="_self">Th25</a>, <a class="txt" href="mesfunc5.html#E30:99"><i><font color="Green">E139</font></i></a>, <a class="ref" href="mesfunc5.html#T31" target="_self">Th31</a>, , <a class="ref" href="mesfunc5.html#T35" target="_self">Th35</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">z1</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">k</font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">y1</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">k</font></span>)</span>
<b>by </b><i><a class="ref" href="mesfunc5.html#T25" target="_self">Th25</a>, <a class="txt" href="mesfunc5.html#E35"><i><font color="Green">Lemma115</font></i></a>, <a class="ref" href="mesfunc5.html#T28" target="_self">Th28</a>, <a class="txt" href="mesfunc5.html#E30:99"><i><font color="Green">E139</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">z1</font> <a href="mesfunc1.html#K13">+</a> <font color="Maroon">y1</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">k</font>
<b>by </b><i>, <a class="txt" href="mesfunc5.html#E30:99"><i><font color="Green">E139</font></i></a>, <a class="ref" href="mesfunc1.html#D3">MESFUNC1:def 3</a></i>
;<br/><br/>


</div><b>end;</b></div>
<a NAME="E7:99_17"/>
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">z1</font> <a href="mesfunc1.html#K13">+</a> <font color="Maroon">y1</font></span>)</span> <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">x1</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E8:99_17"/><b>then </b>
<font color="Maroon">z1</font> <a href="mesfunc1.html#K13">+</a> <font color="Maroon">y1</font> is   <a href="finseq_1.html#NM1">FinSequence</a>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D2">FINSEQ_1:def 2</a></i>;<br/>
<b>hence </b><a NAME="E9:99_17"/>
<font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z1</font> <a href="mesfunc1.html#K13">+</a> <font color="Maroon">y1</font>
 <b>by </b><i>, , <a class="ref" href="finseq_1.html#T17">FINSEQ_1:17</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E73:99"/>
 <a href="mesfunc3.html#K1">integral</a> <font color="Maroon">X</font>,<font color="Maroon">S</font>,<font color="Maroon">M</font>,<font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="mesfunc3.html#K1">integral</a> <font color="Maroon">X</font>,<font color="Maroon">S</font>,<font color="Maroon">M</font>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="mesfunc1.html#K6">-</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><a href="mesfunc3.html#K1">integral</a> <font color="Maroon">X</font>,<font color="Maroon">S</font>,<font color="Maroon">M</font>,<font color="Maroon">g</font></span>)</span>
 <b>by </b><i>, , , , <a class="ref" href="mesfunc5.html#T25" target="_self">Th25</a>, , <a class="txt" href="mesfunc5.html#E35"><i><font color="Green">Lemma115</font></i></a>, <a class="ref" href="mesfunc5.html#T31" target="_self">Th31</a>, <a class="ref" href="mesfunc4.html#T1">MESFUNC4:1</a></i>;<br/>


</div>
