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

<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="funct_2.html#NM2">Permutation</a> of  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K6">ExtREAL</a> ;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:32"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>3</sub></font>
 <b>and </b><br/><a NAME="E2:32"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>and </b><br/><a NAME="E3:32"/><i><font color="Green">E27</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1
 <b>and </b><br/><a NAME="E4:32"/><i><font color="Green">E28</font></i>: 
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">c<sub>5</sub></font>
 <b>and </b><br/><a NAME="E5:32"/><i><font color="Green">E29</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> holds <br/>( ( <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>2</sub></font> implies <font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> ) &amp; ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> implies <font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1 ) &amp; ( <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>2</sub></font> implies <font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="binop_2.html#K10">-</a> 1 ) )
 ;<br/>

<a NAME="E6:32"/><i><font color="Green">E30</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1 )
 
<b>by </b><i><a class="txt" href="convfun1.html#E2:32"><i><font color="Green">E26</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E7:32"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K10">-</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 <a href="binop_2.html#K10">-</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/>
<a NAME="E8:32"/><b>then </b><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="binarith.html#K5">-'</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K10">-</a> 1
 
<b>by </b><i><a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>;<br/>
<a NAME="E9:32"/><b>then </b><i><font color="Green">E32</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="binarith.html#K5">-'</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binop_2.html#K10">-</a> 1
 
<b>by </b><i><a class="txt" href="convfun1.html#E6:32"><i><font color="Green">E30</font></i></a>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/>
<a NAME="E10:32"/><b>then </b><i><font color="Green">E33</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="binarith.html#K5">-'</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font>
 
;<br/>
<a NAME="E11:32"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<a NAME="E12:32"/><b>then </b><i><font color="Green">E35</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="binarith.html#K5">-'</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="txt" href="convfun1.html#E9:32"><i><font color="Green">E32</font></i></a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/>
<a NAME="E13:32"/><i><font color="Green">E36</font></i>: 
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <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><a class="ref" href="finseq_1.html#T6">FINSEQ_1:6</a></i>;<br/>
<a NAME="E14:32"/><b>then </b><i><font color="Green">E37</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>3</sub></font> as    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <b>by </b><i><a class="txt" href="convfun1.html#E13:32"><i><font color="Green">E36</font></i></a>, <a class="ref" href="finseq_2.html#T28">FINSEQ_2:28</a></i>;<br/>
<a NAME="E16:32"/><i><font color="Green">E38</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="txt" href="convfun1.html#E14:32"><i><font color="Green">E37</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E17:32"/><i><font color="Green">E39</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="convfun1.html#E3:32"><i><font color="Green">E27</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E18:32"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> is   <a href="funct_2.html#NM1">Function</a> of  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>, <a href="numbers.html#K6">ExtREAL</a> 
 
<b>by </b><i><a class="ref" href="finseq_2.html#T30">FINSEQ_2:30</a></i>;<br/>
<a NAME="E19:32"/><b>then </b><i><font color="Green">E40</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="txt" href="convfun1.html#E1:32"><i><font color="Green">E25</font></i></a>, <a class="txt" href="convfun1.html#E16:32"><i><font color="Green">E38</font></i></a>, <a class="ref" href="finseq_2.html#T37">FINSEQ_2:37</a></i>;<br/>
<a NAME="E20:32"/><b>then </b><i><font color="Green">E41</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</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">c<sub>7</sub></font> = <font color="Maroon">c<sub>5</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>1</sub></font>;<br/>
<a NAME="E21:32"/>
0 <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E22:32"/><b>then </b><i><font color="Green">E42</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="convfun1.html#E3:32"><i><font color="Green">E27</font></i></a>, <a class="ref" href="finseq_1.html#T80">FINSEQ_1:80</a></i>;<br/>
<a NAME="E23:32"/><b>then </b><i><font color="Green">E43</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="binarith.html#K5">-'</a> 1 &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="binarith.html#K5">-'</a> 1 )
 
<b>by </b><i><a class="txt" href="convfun1.html#E10:32"><i><font color="Green">E33</font></i></a>, <a class="txt" href="convfun1.html#E12:32"><i><font color="Green">E35</font></i></a>, <a class="txt" href="convfun1.html#E19:32"><i><font color="Green">E40</font></i></a>, <a class="ref" href="finseq_1.html#T80">FINSEQ_1:80</a></i>;<br/>
<a NAME="E24:32"/><i><font color="Green">E44</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="binarith.html#K5">-'</a> 1 holds <br/><span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>8</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:32_1"/><i><font color="Green">E45</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="binarith.html#K5">-'</a> 1 )
 ;<br/>

<a NAME="E2:32_1"/><b>then </b><i><font color="Green">E46</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>8</sub></font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>8</sub></font> )
 
<b>by </b><i><a class="ref" href="finseq_3.html#T121">FINSEQ_3:121</a></i>;<br/>
<a NAME="E3:32_1"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="convfun1.html#E9:32"><i><font color="Green">E32</font></i></a>, <a class="txt" href="convfun1.html#E1:32_1"><i><font color="Green">E45</font></i></a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/>
<a NAME="E4:32_1"/><b>then </b><i><font color="Green">E48</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T121">FINSEQ_3:121</a></i>;<br/>
<a NAME="E5:32_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="txt" href="convfun1.html#E11:32"><i><font color="Green">E34</font></i></a>, <a class="txt" href="convfun1.html#E3:32_1"><i><font color="Green">E47</font></i></a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/>
<a NAME="E6:32_1"/><b>then </b><i><font color="Green">E49</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="convfun1.html#E1:32_1"><i><font color="Green">E45</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E7:32_1"/><b>then </b><i><font color="Green">E50</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="convfun1.html#E1:32"><i><font color="Green">E25</font></i></a>, <a class="txt" href="convfun1.html#E20:32"><i><font color="Green">E41</font></i></a>, <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>;<br/>
<a NAME="E8:32_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="txt" href="convfun1.html#E1:32_1"><i><font color="Green">E45</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<b>hence </b><a NAME="E9:32_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="convfun1.html#E5:32"><i><font color="Green">E29</font></i></a>, <a class="txt" href="convfun1.html#E8:32"><i><font color="Green">E31</font></i></a>, <a class="txt" href="convfun1.html#E2:32_1"><i><font color="Green">E46</font></i></a>, <a class="txt" href="convfun1.html#E4:32_1"><i><font color="Green">E48</font></i></a>, <a class="txt" href="convfun1.html#E6:32_1"><i><font color="Green">E49</font></i></a>, <a class="txt" href="convfun1.html#E7:32_1"><i><font color="Green">E50</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E25:32"/>
<font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="txt" href="convfun1.html#E2:32"><i><font color="Green">E26</font></i></a>, <a class="txt" href="convfun1.html#E5:32"><i><font color="Green">E29</font></i></a></i>;<br/>
<a NAME="E26:32"/><b>then </b><i><font color="Green">E45</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="convfun1.html#E1:32"><i><font color="Green">E25</font></i></a>, <a class="txt" href="convfun1.html#E2:32"><i><font color="Green">E26</font></i></a>, <a class="txt" href="convfun1.html#E20:32"><i><font color="Green">E41</font></i></a>, <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>;<br/>
<a NAME="E27:32"/><i><font color="Green">E46</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binop_2.html#K10">-</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="convfun1.html#E6:32"><i><font color="Green">E30</font></i></a>, <a class="txt" href="convfun1.html#E19:32"><i><font color="Green">E40</font></i></a>, <a class="ref" href="rfinseq.html#D2">RFINSEQ:def 2</a></i>;<br/>
<a NAME="E28:32"/><i><font color="Green">E47</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="rfinseq.html#K1">/^</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K10">-</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="convfun1.html#E8:32"><i><font color="Green">E31</font></i></a>, <a class="txt" href="convfun1.html#E9:32"><i><font color="Green">E32</font></i></a>, <a class="txt" href="convfun1.html#E22:32"><i><font color="Green">E42</font></i></a>, <a class="ref" href="rfinseq.html#D2">RFINSEQ:def 2</a></i>;<br/>
<a NAME="E29:32"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binop_2.html#K10">-</a> <font color="Maroon">c<sub>2</sub></font> holds <br/><span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="rfinseq.html#K1">/^</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>8</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:32_2"/><i><font color="Green">E48</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binop_2.html#K10">-</a> <font color="Maroon">c<sub>2</sub></font> )
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>9</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binop_2.html#K10">-</a> <font color="Maroon">c<sub>2</sub></font> as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="txt" href="convfun1.html#E6:32"><i><font color="Green">E30</font></i></a>, <a class="ref" href="int_1.html#T18">INT_1:18</a></i>;<br/>
<a NAME="E3:32_2"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="txt" href="convfun1.html#E1:32_2"><i><font color="Green">E48</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E4:32_2"/>
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="txt" href="convfun1.html#E27:32"><i><font color="Green">E46</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E5:32_2"/><b>then </b><i><font color="Green">E50</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="convfun1.html#E6:32"><i><font color="Green">E30</font></i></a>, <a class="txt" href="convfun1.html#E19:32"><i><font color="Green">E40</font></i></a>, <a class="txt" href="convfun1.html#E3:32_2"><i><font color="Green">E49</font></i></a>, <a class="ref" href="rfinseq.html#D2">RFINSEQ:def 2</a></i>;<br/>
<a NAME="E6:32_2"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="rfinseq.html#K1">/^</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="convfun1.html#E28:32"><i><font color="Green">E47</font></i></a>, <a class="txt" href="convfun1.html#E3:32_2"><i><font color="Green">E49</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E7:32_2"/><b>then </b><i><font color="Green">E51</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="rfinseq.html#K1">/^</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="convfun1.html#E9:32"><i><font color="Green">E32</font></i></a>, <a class="txt" href="convfun1.html#E22:32"><i><font color="Green">E42</font></i></a>, <a class="ref" href="rfinseq.html#D2">RFINSEQ:def 2</a></i>;<br/>
<a NAME="E8:32_2"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="binop_2.html#K10">-</a> 1
 
<b>by </b><i><a class="txt" href="convfun1.html#E8:32"><i><font color="Green">E31</font></i></a></i>;<br/>
<a NAME="E9:32_2"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="txt" href="convfun1.html#E1:32_2"><i><font color="Green">E48</font></i></a>, <a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/>
<a NAME="E10:32_2"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="convfun1.html#E8:32_2"><i><font color="Green">E52</font></i></a>, <a class="ref" href="xreal_1.html#T22">XREAL_1:22</a></i>;<br/>
<a NAME="E11:32_2"/><b>then </b><i><font color="Green">E54</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="rfinseq.html#K1">/^</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="binop_2.html#K10">-</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="convfun1.html#E8:32"><i><font color="Green">E31</font></i></a>, <a class="txt" href="convfun1.html#E7:32_2"><i><font color="Green">E51</font></i></a>, <a class="ref" href="finseq_3.html#T121">FINSEQ_3:121</a></i>;<br/>
<a NAME="E12:32_2"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="convfun1.html#E1:32_2"><i><font color="Green">E48</font></i></a>, <a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/>
<a NAME="E13:32_2"/><b>then </b><i><font color="Green">E55</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>4</sub></font> )
 
<b>by </b><i><a class="txt" href="convfun1.html#E20:32"><i><font color="Green">E41</font></i></a>, <a class="txt" href="convfun1.html#E9:32_2"><i><font color="Green">E53</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E14:32_2"/><b>then </b><i><font color="Green">E56</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="convfun1.html#E1:32"><i><font color="Green">E25</font></i></a>, <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>;<br/>
<a NAME="E15:32_2"/>
<font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="convfun1.html#E1:32_2"><i><font color="Green">E48</font></i></a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E16:32_2"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<b>hence </b><a NAME="E17:32_2"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="rfinseq.html#K1">/^</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="convfun1.html#E5:32"><i><font color="Green">E29</font></i></a>, <a class="txt" href="convfun1.html#E5:32_2"><i><font color="Green">E50</font></i></a>, <a class="txt" href="convfun1.html#E11:32_2"><i><font color="Green">E54</font></i></a>, <a class="txt" href="convfun1.html#E13:32_2"><i><font color="Green">E55</font></i></a>, <a class="txt" href="convfun1.html#E14:32_2"><i><font color="Green">E56</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E30:32"/><b>then </b><i><font color="Green">E48</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="rfinseq.html#K1">/^</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="convfun1.html#E27:32"><i><font color="Green">E46</font></i></a>, <a class="txt" href="convfun1.html#E28:32"><i><font color="Green">E47</font></i></a>, <a class="ref" href="finseq_1.html#T18">FINSEQ_1:18</a></i>;<br/>
<a NAME="E31:32"/><i><font color="Green">E49</font></i>: 
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">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="convfun1.html#E1:32"><i><font color="Green">E25</font></i></a>, <a class="txt" href="convfun1.html#E4:32"><i><font color="Green">E28</font></i></a>, <a class="ref" href="funct_1.html#T25">FUNCT_1:25</a></i>;<br/>
<a NAME="E32:32"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K17">|</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="binarith.html#K13">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="binarith.html#K13">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="convfun1.html#E6:32"><i><font color="Green">E30</font></i></a>, <a class="txt" href="convfun1.html#E19:32"><i><font color="Green">E40</font></i></a>, <a class="ref" href="polynom4.html#T3">POLYNOM4:3</a></i>;<br/>
<a NAME="E33:32"/><b>then </b>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K17">|</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="binarith.html#K13">&lt;*</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="binarith.html#K13">*&gt;</a></span></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T44">FINSEQ_1:44</a></i>;<br/>
<a NAME="E34:32"/><b>then </b><i><font color="Green">E51</font></i>: 
( not  <a href="supinf_1.html#K2">-infty</a>  <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K17">|</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="binarith.html#K13">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="binarith.html#K13">*&gt;</a></span></span>)</span> &amp; not  <a href="supinf_1.html#K2">-infty</a>  <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> )
 
<b>by </b><i><a class="txt" href="convfun1.html#E31:32"><i><font color="Green">E49</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E35:32"/><b>then </b>
not  <a href="supinf_1.html#K2">-infty</a>  <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K17">|</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2"><a href="binarith.html#K13">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="binarith.html#K13">*&gt;</a></span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T44">FINSEQ_1:44</a></i>;<br/>
<a NAME="E36:32"/><b>then </b><i><font color="Green">E52</font></i>: 
( not  <a href="supinf_1.html#K2">-infty</a>  <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span> &amp; not  <a href="supinf_1.html#K2">-infty</a>  <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1"><a href="binarith.html#K13">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="binarith.html#K13">*&gt;</a></span> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E37:32"/><b>then </b>
not  <a href="supinf_1.html#K2">-infty</a>  <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T56">FINSEQ_1:56</a></i>;<br/>
<a NAME="E38:32"/><b>then </b><i><font color="Green">E53</font></i>: 
 <a href="supinf_1.html#K2">-infty</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>4</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E39:32"/><i><font color="Green">E54</font></i>: 
(  <a href="convfun1.html#K5" target="_self">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K2">-infty</a>  &amp;  <a href="convfun1.html#K5" target="_self">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K2">-infty</a>  )
 
<b>by </b><i><a class="txt" href="convfun1.html#E34:32"><i><font color="Green">E51</font></i></a>, <a class="txt" href="convfun1.html#E36:32"><i><font color="Green">E52</font></i></a>, <a class="txt" href="convfun1.html#E19"><i><font color="Green">Lemma17</font></i></a></i>;<br/>
<a NAME="E40:32"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D15">FINSEQ_1:def 15</a></i>;<br/>
<a NAME="E41:32"/><b>then </b>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T99">RELAT_1:99</a></i>;<br/>
<a NAME="E42:32"/><b>then </b><i><font color="Green">E56</font></i>: 
not  <a href="supinf_1.html#K2">-infty</a>  <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="convfun1.html#E4:32"><i><font color="Green">E28</font></i></a></i>;<br/>
<a NAME="E43:32"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="rfinseq.html#K1">/^</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="rfinseq.html#T21">RFINSEQ:21</a></i>;<br/>
<a NAME="E44:32"/><b>then </b>
not  <a href="supinf_1.html#K2">-infty</a>  <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="rfinseq.html#K1">/^</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="convfun1.html#E42:32"><i><font color="Green">E56</font></i></a>, <a class="ref" href="finseq_1.html#T44">FINSEQ_1:44</a></i>;<br/>
<a NAME="E45:32"/><b>then </b><i><font color="Green">E58</font></i>: 
( not  <a href="supinf_1.html#K2">-infty</a>  <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span> &amp; not  <a href="supinf_1.html#K2">-infty</a>  <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="rfinseq.html#K1">/^</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E46:32"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="convfun1.html#E17:32"><i><font color="Green">E39</font></i></a>, <a class="ref" href="finseq_1.html#T6">FINSEQ_1:6</a></i>;<br/>
<a NAME="E47:32"/>
( <font color="Maroon">c<sub>5</sub></font> <a href="finseq_1.html#K17">|</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="finseq_1.html#K17">|</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> )
 
<b>by </b><i><a class="txt" href="convfun1.html#E3:32"><i><font color="Green">E27</font></i></a>, <a class="ref" href="finseq_1.html#T79">FINSEQ_1:79</a>, <a class="ref" href="finseq_1.html#D15">FINSEQ_1:def 15</a></i>;<br/>
<a NAME="E48:32"/><b>then </b><i><font color="Green">E60</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="binarith.html#K13">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="mesfunc1.html#K4">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="binarith.html#K13">*&gt;</a></span>
 
<b>by </b><i><a class="txt" href="convfun1.html#E40:32"><i><font color="Green">E55</font></i></a>, <a class="txt" href="convfun1.html#E46:32"><i><font color="Green">E59</font></i></a>, <a class="ref" href="finseq_5.html#T11">FINSEQ_5:11</a></i>;<br/>
<b>thus </b> <a href="convfun1.html#K5" target="_self">Sum</a> <font color="Maroon">c<sub>4</sub></font> = 
<span class="p1">(<span class="default"><a href="convfun1.html#K5" target="_self">Sum</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K17">|</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="binarith.html#K13">&lt;*</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="binarith.html#K13">*&gt;</a></span></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><a href="convfun1.html#K5" target="_self">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="convfun1.html#E32:32"><i><font color="Green">E50</font></i></a>, <a class="txt" href="convfun1.html#E34:32"><i><font color="Green">E51</font></i></a>, <a class="ref" href="convfun1.html#T7" target="_self">Th7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="convfun1.html#K5" target="_self">Sum</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K17">|</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><a href="convfun1.html#K5" target="_self">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="convfun1.html#E17"><i><font color="Green">Lemma15</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="convfun1.html#K5" target="_self">Sum</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K17">|</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p2">(<span class="default"><a href="convfun1.html#K5" target="_self">Sum</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="rfinseq.html#K1">/^</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="convfun1.html#E38:32"><i><font color="Green">E53</font></i></a>, <a class="txt" href="convfun1.html#E39:32"><i><font color="Green">E54</font></i></a>, <a class="ref" href="extreal1.html#T8">EXTREAL1:8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="convfun1.html#K5" target="_self">Sum</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p2">(<span class="default"><a href="convfun1.html#K5" target="_self">Sum</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="rfinseq.html#K1">/^</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="mesfunc1.html#K4">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="convfun1.html#E23:32"><i><font color="Green">E43</font></i></a>, <a class="txt" href="convfun1.html#E24:32"><i><font color="Green">E44</font></i></a>, <a class="txt" href="convfun1.html#E26:32"><i><font color="Green">E45</font></i></a>, <a class="txt" href="convfun1.html#E30:32"><i><font color="Green">E48</font></i></a>, <a class="ref" href="finseq_1.html#T18">FINSEQ_1:18</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="convfun1.html#K5" target="_self">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="mesfunc1.html#K4">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="convfun1.html#E43:32"><i><font color="Green">E57</font></i></a>, <a class="txt" href="convfun1.html#E45:32"><i><font color="Green">E58</font></i></a>, <a class="ref" href="convfun1.html#T7" target="_self">Th7</a></i>
<br/>.= 
 <a href="convfun1.html#K5" target="_self">Sum</a> <font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="txt" href="convfun1.html#E48:32"><i><font color="Green">E60</font></i></a>, <a class="txt" href="convfun1.html#E17"><i><font color="Green">Lemma15</font></i></a></i>
;<br/>


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