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

<b>let </b><font color="Maroon">b</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="afinsq_1.html#NM2">XFinSequence</a> of  <a href="numbers.html#K1">REAL</a> ;<br/><b>let </b><font color="Maroon">a</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>let </b><font color="Maroon">m</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>





<b>assume </b><a NAME="E1:21"/><i><font color="Green">E25</font></i>: 
( <font color="Maroon">b</font> <a href="prgcor_2.html#K6" target="_self">.</a> 0 is   <a href="nat_1.html#NM1">Nat</a> &amp;  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b</font> <a href="prgcor_2.html#K6" target="_self">.</a> 0 &amp; <font color="Maroon">b</font> <a href="prgcor_2.html#K6" target="_self">.</a> 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m</font> )
 ;<br/>

<a NAME="E2:21"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">b</font> <a href="prgcor_2.html#K6" target="_self">.</a> 0 <a href="hidden.html#R2">in</a> <font color="Maroon">m</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E3:21"/><i><font color="Green">E33</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">b</font>
 
<b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">k</font> = <font color="Maroon">b</font> <a href="prgcor_2.html#K6" target="_self">.</a> 0 as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">c2</font> = <font color="Maroon">a</font> <a href="rvsum_1.html#K9">*</a> <span class="p1">(<span class="default"><a href="prgcor_2.html#K4" target="_self">XFS2FS*</a> <font color="Maroon">b</font></span>)</span> as    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a>  ;<br/>
<a NAME="E6:21"/><i><font color="Green">E35</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="prgcor_2.html#K4" target="_self">XFS2FS*</a> <font color="Maroon">b</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="prgcor_2.html#K6" target="_self">.</a> 0
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E7:21"/>
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="rvsum_1.html#K9">*</a> <span class="p2">(<span class="default"><a href="prgcor_2.html#K4" target="_self">XFS2FS*</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="prgcor_2.html#K4" target="_self">XFS2FS*</a> <font color="Maroon">b</font></span>)</span>
 
<b>by </b><i><a class="ref" href="rvsum_1.html#T65">RVSUM_1:65</a></i>;<br/>
<a NAME="E8:21"/><b>then </b>
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="rvsum_1.html#K9">*</a> <span class="p3">(<span class="default"><a href="prgcor_2.html#K4" target="_self">XFS2FS*</a> <font color="Maroon">b</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="prgcor_2.html#K4" target="_self">XFS2FS*</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="E9:21"/><b>then </b><i><font color="Green">E41</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<b>consider </b><font color="Maroon">p</font> being   <a href="afinsq_1.html#NM2">XFinSequence</a> of  <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E11:21"/><i><font color="Green">E43</font></i>: 
(  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> &amp; <font color="Maroon">p</font> <a href="prgcor_2.html#R1" target="_self">is_an_xrep_of</a> <font color="Maroon">c2</font> )
 <b>by </b><i>, , </i>;<br/>
<b>set </b><font color="Maroon">p2</font> =  <a href="afinsq_1.html#K11">Replace</a> <font color="Maroon">p</font>,0,<span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="prgcor_2.html#K6" target="_self">.</a> 0</span>)</span>;<br/>
<a NAME="E12:21"/><i><font color="Green">E44</font></i>: 
 <a href="afinsq_1.html#K1">len</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="afinsq_1.html#K1">len</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K11">Replace</a> <font color="Maroon">p</font>,0,<span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="prgcor_2.html#K6" target="_self">.</a> 0</span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="afinsq_1.html#T48">AFINSQ_1:48</a></i>;<br/>
<a NAME="E13:21"/>
 ex <font color="Olive">n</font> being  <a href="int_1.html#NM1">Integer</a> st <br/>( <span class="p1">(<span class="default"><a href="afinsq_1.html#K11">Replace</a> <font color="Maroon">p</font>,0,<span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="prgcor_2.html#K6" target="_self">.</a> 0</span>)</span></span>)</span> <a href="prgcor_2.html#K6" target="_self">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="prgcor_2.html#K6" target="_self">.</a> 0 &amp; <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="prgcor_2.html#K6" target="_self">.</a> 0 &amp; ( <font color="Olive">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 implies for <font color="Olive">i</font> being  <a href="nat_1.html#NM1">Nat</a>  st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">n</font> holds <br/><span class="p1">(<span class="default"><a href="afinsq_1.html#K11">Replace</a> <font color="Maroon">p</font>,0,<span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="prgcor_2.html#K6" target="_self">.</a> 0</span>)</span></span>)</span> <a href="prgcor_2.html#K6" target="_self">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="prgcor_2.html#K6" target="_self">.</a> <font color="Olive">i</font></span>)</span> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon">n</font> = <font color="Maroon">b</font> <a href="prgcor_2.html#K6" target="_self">.</a> 0 as   <a href="int_1.html#NM1">Integer</a> <b>by </b><i/>;<br/>
<a NAME="E2:21_1"/><i><font color="Green">E45</font></i>: 
( <font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 implies for <font color="Olive">i</font> being  <a href="nat_1.html#NM1">Nat</a>  st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> holds <br/><span class="p1">(<span class="default"><a href="afinsq_1.html#K11">Replace</a> <font color="Maroon">p</font>,0,<span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="prgcor_2.html#K6" target="_self">.</a> 0</span>)</span></span>)</span> <a href="prgcor_2.html#K6" target="_self">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="prgcor_2.html#K6" target="_self">.</a> <font color="Olive">i</font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:21_1_1_1"/>
<font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><b>thus </b><a NAME="E2:21_1_1_1"/>
for <font color="Olive">i</font> being  <a href="nat_1.html#NM1">Nat</a>  st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> holds <br/><span class="p1">(<span class="default"><a href="afinsq_1.html#K11">Replace</a> <font color="Maroon">p</font>,0,<span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="prgcor_2.html#K6" target="_self">.</a> 0</span>)</span></span>)</span> <a href="prgcor_2.html#K6" target="_self">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="prgcor_2.html#K6" target="_self">.</a> <font color="Olive">i</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_1_1_1"><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/>

<a NAME="E1:21_1_1_1_1"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/>
<b>assume </b><a NAME="E2:21_1_1_1_1"/><i><font color="Green">E49</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> <font color="Maroon">k</font> )
 ;<br/>

<a NAME="E3:21_1_1_1_1"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">p</font> <a href="prgcor_2.html#K6" target="_self">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c2</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font>
 
<b>by </b><i>, , , <a class="ref" href="prgcor_2.html#T1" target="_self">Th1</a></i>;<br/>
<a NAME="E4:21_1_1_1_1"/>
<span class="p1">(<span class="default"><a href="prgcor_2.html#K4" target="_self">XFS2FS*</a> <font color="Maroon">b</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="prgcor_2.html#K6" target="_self">.</a> <font color="Maroon">i</font>
 
<b>by </b><i>, , , <a class="ref" href="prgcor_2.html#T1" target="_self">Th1</a></i>;<br/>
<a NAME="E5:21_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="rvsum_1.html#K9">*</a> <span class="p2">(<span class="default"><a href="prgcor_2.html#K4" target="_self">XFS2FS*</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="prgcor_2.html#K6" target="_self">.</a> <font color="Maroon">i</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="rvsum_1.html#T66">RVSUM_1:66</a></i>;<br/>
<b>hence </b><a NAME="E6:21_1_1_1_1"/>
<span class="p1">(<span class="default"><a href="afinsq_1.html#K11">Replace</a> <font color="Maroon">p</font>,0,<span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="prgcor_2.html#K6" target="_self">.</a> 0</span>)</span></span>)</span> <a href="prgcor_2.html#K6" target="_self">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="prgcor_2.html#K6" target="_self">.</a> <font color="Maroon">i</font></span>)</span>
 <b>by </b><i><a class="ref" href="prgcor_2.html#T1" target="_self">Th1</a>, <a class="txt" href="prgcor_2.html#E1:21"><i><font color="Green">E25</font></i></a>, , <a class="ref" href="afinsq_1.html#T48">AFINSQ_1:48</a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E2:21_1_1"/>
( <font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 implies for <font color="Olive">i</font> being  <a href="nat_1.html#NM1">Nat</a>  st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> holds <br/><span class="p1">(<span class="default"><a href="afinsq_1.html#K11">Replace</a> <font color="Maroon">p</font>,0,<span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="prgcor_2.html#K6" target="_self">.</a> 0</span>)</span></span>)</span> <a href="prgcor_2.html#K6" target="_self">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="prgcor_2.html#K6" target="_self">.</a> <font color="Olive">i</font></span>)</span> )
 ;<br/>


</div><b>end;</b></div>
<a NAME="E3:21_1"/>
<span class="p1">(<span class="default"><a href="afinsq_1.html#K11">Replace</a> <font color="Maroon">p</font>,0,<span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="prgcor_2.html#K6" target="_self">.</a> 0</span>)</span></span>)</span> <a href="prgcor_2.html#K6" target="_self">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="prgcor_2.html#K6" target="_self">.</a> 0
 
<b>by </b><i>, , <a class="ref" href="afinsq_1.html#T48">AFINSQ_1:48</a></i>;<br/>
<b>hence </b><a NAME="E4:21_1"/>
 ex <font color="Olive">n</font> being  <a href="int_1.html#NM1">Integer</a> st <br/>( <span class="p1">(<span class="default"><a href="afinsq_1.html#K11">Replace</a> <font color="Maroon">p</font>,0,<span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="prgcor_2.html#K6" target="_self">.</a> 0</span>)</span></span>)</span> <a href="prgcor_2.html#K6" target="_self">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="prgcor_2.html#K6" target="_self">.</a> 0 &amp; <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="prgcor_2.html#K6" target="_self">.</a> 0 &amp; ( <font color="Olive">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 implies for <font color="Olive">i</font> being  <a href="nat_1.html#NM1">Nat</a>  st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">n</font> holds <br/><span class="p1">(<span class="default"><a href="afinsq_1.html#K11">Replace</a> <font color="Maroon">p</font>,0,<span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="prgcor_2.html#K6" target="_self">.</a> 0</span>)</span></span>)</span> <a href="prgcor_2.html#K6" target="_self">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="prgcor_2.html#K6" target="_self">.</a> <font color="Olive">i</font></span>)</span> ) )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E14:21"/><b>then </b>
<font color="Maroon">m</font> <a href="prgcor_2.html#R2" target="_self">scalar_prd_prg</a>  <a href="afinsq_1.html#K11">Replace</a> <font color="Maroon">p</font>,0,<span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="prgcor_2.html#K6" target="_self">.</a> 0</span>)</span>,<font color="Maroon">a</font>,<font color="Maroon">b</font>
 
<b>by </b><i>, , , </i>;<br/>
<b>hence </b><a NAME="E15:21"/>
 ex <font color="Olive">c</font> being  <a href="afinsq_1.html#NM2">XFinSequence</a> of  <a href="numbers.html#K1">REAL</a>  st <font color="Maroon">m</font> <a href="prgcor_2.html#R2" target="_self">scalar_prd_prg</a> <font color="Olive">c</font>,<font color="Maroon">a</font>,<font color="Maroon">b</font>
 ;<br/>

<b>thus </b><a NAME="E16:21"/>
for <font color="Olive">c</font> being non <a href="xboole_0.html#V1">empty</a> <a href="afinsq_1.html#NM2">XFinSequence</a> of  <a href="numbers.html#K1">REAL</a>   st <font color="Maroon">m</font> <a href="prgcor_2.html#R2" target="_self">scalar_prd_prg</a> <font color="Olive">c</font>,<font color="Maroon">a</font>,<font color="Maroon">b</font> holds <br/> <a href="prgcor_2.html#K4" target="_self">XFS2FS*</a> <font color="Olive">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="rvsum_1.html#K9">*</a> <span class="p1">(<span class="default"><a href="prgcor_2.html#K4" target="_self">XFS2FS*</a> <font color="Maroon">b</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="afinsq_1.html#NM2">XFinSequence</a> of  <a href="numbers.html#K1">REAL</a> ;<br/>

<b>assume </b><a NAME="E1:21_2"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">m</font> <a href="prgcor_2.html#R2" target="_self">scalar_prd_prg</a> <font color="Maroon">c</font>,<font color="Maroon">a</font>,<font color="Maroon">b</font>
 ;<br/>

<a NAME="E2:21_2"/><b>then </b><i><font color="Green">E61</font></i>: 
(  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> &amp;  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> &amp;  ex <font color="Olive">n</font> being  <a href="int_1.html#NM1">Integer</a> st <br/>( <font color="Maroon">c</font> <a href="prgcor_2.html#K6" target="_self">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="prgcor_2.html#K6" target="_self">.</a> 0 &amp; <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="prgcor_2.html#K6" target="_self">.</a> 0 &amp; ( <font color="Olive">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 implies for <font color="Olive">i</font> being  <a href="nat_1.html#NM1">Nat</a>  st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">n</font> holds <br/><font color="Maroon">c</font> <a href="prgcor_2.html#K6" target="_self">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="prgcor_2.html#K6" target="_self">.</a> <font color="Olive">i</font></span>)</span> ) ) )
 
<b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">n</font> being   <a href="int_1.html#NM1">Integer</a><b> such that </b><br/><a NAME="E4:21_2"/><i><font color="Green">E62</font></i>: 
( <font color="Maroon">c</font> <a href="prgcor_2.html#K6" target="_self">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="prgcor_2.html#K6" target="_self">.</a> 0 &amp; <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="prgcor_2.html#K6" target="_self">.</a> 0 &amp; ( <font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 implies for <font color="Olive">i</font> being  <a href="nat_1.html#NM1">Nat</a>  st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> holds <br/><font color="Maroon">c</font> <a href="prgcor_2.html#K6" target="_self">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="prgcor_2.html#K6" target="_self">.</a> <font color="Olive">i</font></span>)</span> ) )
 <b>by </b><i>, </i>;<br/>
<a NAME="E5:21_2"/><i><font color="Green">E63</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="prgcor_2.html#K4" target="_self">XFS2FS*</a> <font color="Maroon">c</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> <a href="prgcor_2.html#K6" target="_self">.</a> 0
 
<b>by </b><i>, , , </i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:21_2_1_1"/>
( <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_2_1_1_1"><b>case </b></a><a NAME="E1:21_2_1_1_1"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:21_2_1_1_1"/><b>then </b><i><font color="Green">E65</font></i>: 
 <a href="prgcor_2.html#K4" target="_self">XFS2FS*</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K6">&lt;*&gt;</a> <a href="numbers.html#K1">REAL</a> 
 <b>by </b><i>, <a class="ref" href="prgcor_2.html#T1" target="_self">Th1</a>, </i>;<br/><a NAME="E3:21_2_1_1_1"/>
 <a href="prgcor_2.html#K4" target="_self">XFS2FS*</a> <font color="Maroon">c</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K6">&lt;*&gt;</a> <a href="numbers.html#K1">REAL</a> 
 <b>by </b><i>, , <a class="ref" href="prgcor_2.html#T1" target="_self">Th1</a>, <a class="txt" href="prgcor_2.html#E2:21"><i><font color="Green">E26</font></i></a>, </i>;<br/><b>hence </b><a NAME="E4:21_2_1_1_1"/>
 <a href="prgcor_2.html#K4" target="_self">XFS2FS*</a> <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="rvsum_1.html#K9">*</a> <span class="p1">(<span class="default"><a href="prgcor_2.html#K4" target="_self">XFS2FS*</a> <font color="Maroon">b</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="rvsum_1.html#T68">RVSUM_1:68</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_2_1_1_2"><b>case </b></a><a NAME="E1:21_2_1_1_2"/>
<font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><div class="add"><b>set </b><font color="Maroon">p3</font> =  <a href="prgcor_2.html#K4" target="_self">XFS2FS*</a> <font color="Maroon">c</font>;<br/><a NAME="E2:21_2_1_1_2"/>
for <font color="Olive">k3</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k3</font> &amp; <font color="Olive">k3</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="prgcor_2.html#K4" target="_self">XFS2FS*</a> <font color="Maroon">c</font></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="prgcor_2.html#K4" target="_self">XFS2FS*</a> <font color="Maroon">c</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">k3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c2</font> <a href="seq_1.html#K2">.</a> <font color="Olive">k3</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_2_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k3</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:21_2_1_1_2_1"/><i><font color="Green">E66</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k3</font> &amp; <font color="Maroon">k3</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="prgcor_2.html#K4" target="_self">XFS2FS*</a> <font color="Maroon">c</font></span>)</span> )
 ;<br/>

<a NAME="E2:21_2_1_1_2_1"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">c</font> <a href="prgcor_2.html#K6" target="_self">.</a> 0 <a href="hidden.html#R2">in</a>  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">c</font>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E3:21_2_1_1_2_1"/><b>then </b><i><font color="Green">E68</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="prgcor_2.html#K4" target="_self">XFS2FS*</a> <font color="Maroon">c</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> &amp; ( for <font color="Olive">i</font> being  <a href="nat_1.html#NM1">Nat</a>  st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> holds <br/><span class="p1">(<span class="default"><a href="prgcor_2.html#K4" target="_self">XFS2FS*</a> <font color="Maroon">c</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> <a href="prgcor_2.html#K6" target="_self">.</a> <font color="Olive">i</font> ) )
 
<b>by </b><i>, <a class="ref" href="prgcor_2.html#T1" target="_self">Th1</a>, </i>;<br/>
<a NAME="E4:21_2_1_1_2_1"/><i><font color="Green">E69</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k3</font> &amp; <font color="Maroon">k3</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> )
 
<b>by </b><i><a class="txt" href="prgcor_2.html#E2:21"><i><font color="Green">E26</font></i></a>, , , <a class="ref" href="prgcor_2.html#T1" target="_self">Th1</a>, </i>;<br/>
<i><font color="Green">E70</font></i>: <span class="p1">(<span class="default"><a href="prgcor_2.html#K4" target="_self">XFS2FS*</a> <font color="Maroon">c</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">k3</font> = 
<font color="Maroon">c</font> <a href="prgcor_2.html#K6" target="_self">.</a> <font color="Maroon">k3</font>
<b>by </b><i><a class="txt" href="prgcor_2.html#E2:21"><i><font color="Green">E26</font></i></a>, </i>
<br/>.= 
<font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="prgcor_2.html#K6" target="_self">.</a> <font color="Maroon">k3</font></span>)</span>
<b>by </b><i><a class="ref" href="prgcor_2.html#T1" target="_self">Th1</a>, </i>
;<br/>
<a NAME="E6:21_2_1_1_2_1"/>
<font color="Maroon">b</font> <a href="prgcor_2.html#K6" target="_self">.</a> <font color="Maroon">k3</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="prgcor_2.html#K4" target="_self">XFS2FS*</a> <font color="Maroon">b</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">k3</font>
 
<b>by </b><i>, , , <a class="ref" href="prgcor_2.html#T1" target="_self">Th1</a>, </i>;<br/>
<b>hence </b><a NAME="E7:21_2_1_1_2_1"/>
<span class="p1">(<span class="default"><a href="prgcor_2.html#K4" target="_self">XFS2FS*</a> <font color="Maroon">c</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">k3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c2</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">k3</font>
 <b>by </b><i>, <a class="ref" href="rvsum_1.html#T66">RVSUM_1:66</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E3:21_2_1_1_2"/>
 <a href="prgcor_2.html#K4" target="_self">XFS2FS*</a> <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="rvsum_1.html#K9">*</a> <span class="p1">(<span class="default"><a href="prgcor_2.html#K4" target="_self">XFS2FS*</a> <font color="Maroon">b</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T18">FINSEQ_1:18</a>, , <a class="ref" href="prgcor_2.html#T1" target="_self">Th1</a>, <a class="txt" href="prgcor_2.html#E1:21"><i><font color="Green">E25</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="E7:21_2"/>
 <a href="prgcor_2.html#K4" target="_self">XFS2FS*</a> <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="rvsum_1.html#K9">*</a> <span class="p1">(<span class="default"><a href="prgcor_2.html#K4" target="_self">XFS2FS*</a> <font color="Maroon">b</font></span>)</span>
 ;<br/>


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


</div>
