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

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>for <font color="Olive">a</font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K6">ExtREAL</a> <br/> for <font color="Olive">b</font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="supinf_1.html#K3">REAL</a>   st  <a href="finseq_1.html#K3">len</a> <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Olive">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> &amp; ( for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Olive">a</font> holds <br/><font color="Olive">a</font> <a href="supinf_2.html#K13">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Olive">b</font> <a href="seq_1.html#K2">.</a> <font color="Olive">i</font> ) holds <br/> <a href="convfun1.html#K5" target="_self">Sum</a> <font color="Olive">a</font> <a href="hidden.html#R1">=</a>  <a href="rvsum_1.html#K15">Sum</a> <font color="Olive">b</font>;<br/>
<a NAME="E1:27"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K6">ExtREAL</a> ;<br/><b>let </b><font color="Maroon">b</font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="supinf_1.html#K3">REAL</a> ;<br/>



<b>assume </b><a NAME="E1:27_1"/>
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> 0 &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> 0 &amp; ( for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">a</font> holds <br/><font color="Maroon">a</font> <a href="supinf_2.html#K13">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="seq_1.html#K2">.</a> <font color="Olive">i</font> ) )
 ;<br/>

<a NAME="E2:27_1"/><b>then </b>
(  <a href="convfun1.html#K5" target="_self">Sum</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a>  &amp;  <a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> 0 )
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T32">FINSEQ_1:32</a>, <a class="ref" href="rvsum_1.html#T102">RVSUM_1:102</a></i>;<br/>
<b>hence </b><a NAME="E3:27_1"/>
 <a href="convfun1.html#K5" target="_self">Sum</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="ref" href="supinf_2.html#D1">SUPINF_2:def 1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:27"/><i><font color="Green">E38</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:27_2"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font>]
 ;<br/>

<b>thus </b><a NAME="E2:27_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K6">ExtREAL</a> ;<br/><b>let </b><font color="Maroon">b</font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="supinf_1.html#K3">REAL</a> ;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:27_2_1"/><i><font color="Green">E45</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 )
 <b>and </b><br/><a NAME="E2:27_2_1"/><i><font color="Green">E46</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">a</font> holds <br/><font color="Maroon">a</font> <a href="supinf_2.html#K13">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="seq_1.html#K2">.</a> <font color="Olive">i</font>
 ;<br/>

<a NAME="E3:27_2_1"/><i><font color="Green">E47</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">n</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/>
<a NAME="E4:27_2_1"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">n</font> <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"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T6">FINSEQ_1:6</a></i>;<br/>
<a NAME="E5:27_2_1"/><i><font color="Green">E70</font></i>: 
0 <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E6:27_2_1"/><b>then </b><i><font color="Green">E71</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> )
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T80">FINSEQ_1:80</a></i>;<br/>
<a NAME="E7:27_2_1"/><i><font color="Green">E72</font></i>: 
 <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T7">FINSEQ_1:7</a></i>;<br/>
<a NAME="E8:27_2_1"/>
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n</font></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n</font></span>)</span> <a href="supinf_2.html#K13">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">i</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_2_1_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:27_2_1_1"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n</font></span>)</span>
 ;<br/>

<a NAME="E2:27_2_1_1"/><b>then </b><i><font color="Green">E73</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E3:27_2_1_1"/><b>then </b>
<font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E4:27_2_1_1"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n</font></span>)</span> <a href="supinf_2.html#K13">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="supinf_2.html#K13">.</a> <font color="Maroon">i</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font> )
 
<b>by </b><i><a class="ref" href="finseq_3.html#T121">FINSEQ_3:121</a></i>;<br/>
<b>hence </b><a NAME="E5:27_2_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n</font></span>)</span> <a href="supinf_2.html#K13">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, <a class="ref" href="convfun1.html#D1" target="_self">Def1</a>, , </i>;<br/>


</div><b>end;</b></div>
<a NAME="E9:27_2_1"/><b>then </b><i><font color="Green">E74</font></i>: 
 <a href="convfun1.html#K5" target="_self">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n</font></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E10:27_2_1"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">a</font> <a href="supinf_2.html#K13">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i>, <a class="ref" href="convfun1.html#D1" target="_self">Def1</a>, </i>;<br/>
<a NAME="E11:27_2_1"/>
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">a</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span> )
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D15">FINSEQ_1:def 15</a></i>;<br/>
<a NAME="E12:27_2_1"/><b>then </b>
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="supinf_2.html#K13">.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T61">FINSEQ_3:61</a></i>;<br/>
<b>then </b><i><font color="Green">E76</font></i>:  <a href="convfun1.html#K5" target="_self">Sum</a> <font color="Maroon">a</font> = 
<span class="p1">(<span class="default"><a href="convfun1.html#K5" target="_self">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="supinf_2.html#K13">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i>, , <a class="ref" href="supinf_2.html#T1">SUPINF_2:1</a></i>
;<br/>
<a NAME="E14:27_2_1"/>
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">b</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="rfunct_3.html#K23">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span> )
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D15">FINSEQ_1:def 15</a></i>;<br/>
<a NAME="E15:27_2_1"/><b>then </b>
<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T61">FINSEQ_3:61</a></i>;<br/>
<b>hence </b><a NAME="E16:27_2_1"/>
 <a href="convfun1.html#K5" target="_self">Sum</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">b</font>
 <b>by </b><i>, <a class="ref" href="rvsum_1.html#T104">RVSUM_1:104</a></i>;<br/>


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


</div><b>end;</b></div>
<b>let </b><font color="Maroon">a</font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K6">ExtREAL</a> ;<br/><b>let </b><font color="Maroon">b</font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="supinf_1.html#K3">REAL</a> ;<br/>



<a NAME="E3:27"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(, );<br/></i>
<b>hence </b><a NAME="E4:27"/>
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font> &amp; ( for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">a</font> holds <br/><font color="Maroon">a</font> <a href="supinf_2.html#K13">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="seq_1.html#K2">.</a> <font color="Olive">i</font> ) implies  <a href="convfun1.html#K5" target="_self">Sum</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">b</font> )
 ;<br/>


</div>
