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

<span class="kw">let </span><font color="Maroon" title="c1">b</font> be   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">m</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon" title="c1">b</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> is   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> &amp;  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c1">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">m</font> &amp; <font color="Maroon" title="c1">b</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">m</font> holds <br/>(  ex <font color="Olive" title="b2">c</font> being   <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of  st <font color="Olive" title="b1">m</font> <a href="prgcor_2.html#R3" title="PRGCOR_2:pred.3">vector_minus_prg</a> <font color="Olive" title="b2">c</font>,<font color="Maroon" title="c1">b</font> &amp; (  for <font color="Olive" title="b2">c</font> being   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of   st <font color="Olive" title="b1">m</font> <a href="prgcor_2.html#R3" title="PRGCOR_2:pred.3">vector_minus_prg</a> <font color="Olive" title="b2">c</font>,<font color="Maroon" title="c1">b</font> holds <br/> <a href="afinsq_1.html#K12" title="AFINSQ_1:func.12">XFS2FS*</a> <font color="Olive" title="b2">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K6" title="RVSUM_1:func.6">-</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K12" title="AFINSQ_1:func.12">XFS2FS*</a> <font color="Maroon" title="c1">b</font></span>)</span> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">m</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">b</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> is   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> &amp;  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c1">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">m</font> &amp; <font color="Maroon" title="c1">b</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">m</font> implies (  ex <font color="Olive" title="b1">c</font> being   <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of  st <font color="Maroon" title="c2">m</font> <a href="prgcor_2.html#R3" title="PRGCOR_2:pred.3">vector_minus_prg</a> <font color="Olive" title="b1">c</font>,<font color="Maroon" title="c1">b</font> &amp; (  for <font color="Olive" title="b1">c</font> being   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of   st <font color="Maroon" title="c2">m</font> <a href="prgcor_2.html#R3" title="PRGCOR_2:pred.3">vector_minus_prg</a> <font color="Olive" title="b1">c</font>,<font color="Maroon" title="c1">b</font> holds <br/> <a href="afinsq_1.html#K12" title="AFINSQ_1:func.12">XFS2FS*</a> <font color="Olive" title="b1">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K6" title="RVSUM_1:func.6">-</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K12" title="AFINSQ_1:func.12">XFS2FS*</a> <font color="Maroon" title="c1">b</font></span>)</span> ) ) )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:12"/><span class="lab"><font color="Green" title="E8">A1</font></span>: 
<font color="Maroon" title="c1">b</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> is   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>
 <span class="kw">and </span><br/><a NAME="E2:12"/><span class="lab"><font color="Green" title="E9">A2</font></span>: 
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c1">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">m</font>
 <span class="kw">and </span><br/><a NAME="E3:12"/><span class="lab"><font color="Green" title="E10">A3</font></span>: 
<font color="Maroon" title="c1">b</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">m</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  ex <font color="Olive" title="b1">c</font> being   <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of  st <font color="Maroon" title="c2">m</font> <a href="prgcor_2.html#R3" title="PRGCOR_2:pred.3">vector_minus_prg</a> <font color="Olive" title="b1">c</font>,<font color="Maroon" title="c1">b</font> &amp; (  for <font color="Olive" title="b1">c</font> being   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of   st <font color="Maroon" title="c2">m</font> <a href="prgcor_2.html#R3" title="PRGCOR_2:pred.3">vector_minus_prg</a> <font color="Olive" title="b1">c</font>,<font color="Maroon" title="c1">b</font> holds <br/> <a href="afinsq_1.html#K12" title="AFINSQ_1:func.12">XFS2FS*</a> <font color="Olive" title="b1">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K6" title="RVSUM_1:func.6">-</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K12" title="AFINSQ_1:func.12">XFS2FS*</a> <font color="Maroon" title="c1">b</font></span>)</span> ) )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c3">k</font> = <font color="Maroon" title="c1">b</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> as   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:12"><span class="lab"><font color="Green" title="E8">A1</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c4">c2</font> =  <a href="rvsum_1.html#K6" title="RVSUM_1:func.6">-</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K12" title="AFINSQ_1:func.12">XFS2FS*</a> <font color="Maroon" title="c1">b</font></span>)</span> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  ;<br/>
<a NAME="E6:12"/>
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K6" title="RVSUM_1:func.6">-</a> <span class="p2">(<span class="default"><a href="afinsq_1.html#K12" title="AFINSQ_1:func.12">XFS2FS*</a> <font color="Maroon" title="c1">b</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K12" title="AFINSQ_1:func.12">XFS2FS*</a> <font color="Maroon" title="c1">b</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#T8" title="VALUED_1:th.8">VALUED_1:8</a></span>;<br/>
<a NAME="E7:12"/><span class="kw">then </span><span class="lab"><font color="Green" title="E11">A4</font></span>: 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K6" title="RVSUM_1:func.6">-</a> <span class="p3">(<span class="default"><a href="afinsq_1.html#K12" title="AFINSQ_1:func.12">XFS2FS*</a> <font color="Maroon" title="c1">b</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K12" title="AFINSQ_1:func.12">XFS2FS*</a> <font color="Maroon" title="c1">b</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E8:12"/><span class="lab"><font color="Green" title="E12">A5</font></span>: 
<font color="Maroon" title="c1">b</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:12"><span class="lab"><font color="Green" title="E8">A1</font></span></a>, <a class="txt" href="#E3:12"><span class="lab"><font color="Green" title="E10">A3</font></span></a>, <a class="ref" href="nat_1.html#T44" title="NAT_1:th.44">NAT_1:44</a></span>;<br/>
<a NAME="E9:12"/><span class="kw">then </span>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K12" title="AFINSQ_1:func.12">XFS2FS*</a> <font color="Maroon" title="c1">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">b</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:12"><span class="lab"><font color="Green" title="E9">A2</font></span></a>, <a class="ref" href="afinsq_1.html#D11" title="AFINSQ_1:def.11">AFINSQ_1:def 11</a></span>;<br/>
<a NAME="E10:12"/><span class="kw">then </span><span class="lab"><font color="Green" title="E13">A6</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">c2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:12"><span class="lab"><font color="Green" title="E11">A4</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">p</font> being   <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of <span class="kw"> such that </span><br/><a NAME="E12:12"/><span class="lab"><font color="Green" title="E14">A7</font></span>: 
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c5">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">m</font>
 <span class="kw">and </span><br/><a NAME="E13:12"/><span class="lab"><font color="Green" title="E15">A8</font></span>: 
<font color="Maroon" title="c5">p</font> <a href="prgcor_2.html#R1" title="PRGCOR_2:pred.1">is_an_xrep_of</a> <font color="Maroon" title="c4">c2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:12"><span class="lab"><font color="Green" title="E10">A3</font></span></a>, <a class="ref" href="prgcor_2.html#T2" target="_self" title="PRGCOR_2:th.2">Th7</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c6">p2</font> =  <a href="afinsq_1.html#NK11" title="AFINSQ_1:NK.11">Replace</a> (<font color="Maroon" title="c5">p</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">b</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span>) as   <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of  ;<br/>
<a NAME="E15:12"/><span class="lab"><font color="Green" title="E16">A9</font></span>: 
( <font color="Maroon" title="c3">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies  for <font color="Olive" title="b1">i</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">k</font> holds <br/><font color="Maroon" title="c6">p2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">b</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:12_1"/>
<font color="Maroon" title="c3">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">i</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">k</font> holds <br/><font color="Maroon" title="c6">p2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">b</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span></span><br/>

<span class="kw">let </span><font color="Maroon" title="c7">i</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">i</font> &amp; <font color="Maroon" title="c7">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">k</font> implies <font color="Maroon" title="c6">p2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">b</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i</font></span>)</span> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E2:12_1"/><span class="lab"><font color="Green" title="E17">A10</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">i</font>
 <span class="kw">and </span><br/><a NAME="E3:12_1"/><span class="lab"><font color="Green" title="E18">A11</font></span>: 
<font color="Maroon" title="c7">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">k</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">p2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">b</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i</font></span>)</span></span><br/>

<a NAME="E4:12_1"/>
<span class="p1">(<span class="default"><a href="afinsq_1.html#K12" title="AFINSQ_1:func.12">XFS2FS*</a> <font color="Maroon" title="c1">b</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">b</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:12"><span class="lab"><font color="Green" title="E9">A2</font></span></a>, <a class="txt" href="#E8:12"><span class="lab"><font color="Green" title="E12">A5</font></span></a>, <a class="txt" href="#E2:12_1"><span class="lab"><font color="Green" title="E17">A10</font></span></a>, <a class="txt" href="#E3:12_1"><span class="lab"><font color="Green" title="E18">A11</font></span></a>, <a class="ref" href="afinsq_1.html#D11" title="AFINSQ_1:def.11">AFINSQ_1:def 11</a></span>;<br/>
<a NAME="E5:12_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E19">A12</font></span>: 
<span class="p1">(<span class="default"><a href="rvsum_1.html#K6" title="RVSUM_1:func.6">-</a> <span class="p2">(<span class="default"><a href="afinsq_1.html#K12" title="AFINSQ_1:func.12">XFS2FS*</a> <font color="Maroon" title="c1">b</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">b</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rvsum_1.html#T17" title="RVSUM_1:th.17">RVSUM_1:17</a></span>;<br/>
<a NAME="E6:12_1"/>
( <font color="Maroon" title="c7">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  &amp; <font color="Maroon" title="c5">p</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">c2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:12"><span class="lab"><font color="Green" title="E13">A6</font></span></a>, <a class="txt" href="#E13:12"><span class="lab"><font color="Green" title="E15">A8</font></span></a>, <a class="txt" href="#E2:12_1"><span class="lab"><font color="Green" title="E17">A10</font></span></a>, <a class="txt" href="#E3:12_1"><span class="lab"><font color="Green" title="E18">A11</font></span></a>, <a class="ref" href="prgcor_2.html#D1" target="_self" title="PRGCOR_2:def.1">Def5</a>, <a class="ref" href="ordinal1.html#D12" title="ORDINAL1:def.12">ORDINAL1:def 12</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:12_1"/>
<font color="Maroon" title="c6">p2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">b</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:12_1"><span class="lab"><font color="Green" title="E17">A10</font></span></a>, <a class="txt" href="#E5:12_1"><span class="lab"><font color="Green" title="E19">A12</font></span></a>, <a class="ref" href="afinsq_1.html#T44" title="AFINSQ_1:th.44">AFINSQ_1:44</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E16:12"/>
(  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c5">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c6">p2</font> &amp; <font color="Maroon" title="c6">p2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">b</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:12"><span class="lab"><font color="Green" title="E8">A1</font></span></a>, <a class="txt" href="#E3:12"><span class="lab"><font color="Green" title="E10">A3</font></span></a>, <a class="txt" href="#E12:12"><span class="lab"><font color="Green" title="E14">A7</font></span></a>, <a class="ref" href="afinsq_1.html#T44" title="AFINSQ_1:th.44">AFINSQ_1:44</a></span>;<br/>
<a NAME="E17:12"/><span class="kw">then </span>
<font color="Maroon" title="c2">m</font> <a href="prgcor_2.html#R3" title="PRGCOR_2:pred.3">vector_minus_prg</a> <font color="Maroon" title="c6">p2</font>,<font color="Maroon" title="c1">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:12"><span class="lab"><font color="Green" title="E9">A2</font></span></a>, <a class="txt" href="#E12:12"><span class="lab"><font color="Green" title="E14">A7</font></span></a>, <a class="txt" href="#E15:12"><span class="lab"><font color="Green" title="E16">A9</font></span></a>, <a class="ref" href="prgcor_2.html#D5" target="_self" title="PRGCOR_2:def.5">Def9</a></span>;<br/>
<span class="kw">hence </span><a NAME="E18:12"/>
 ex <font color="Olive" title="b1">c</font> being   <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of  st <font color="Maroon" title="c2">m</font> <a href="prgcor_2.html#R3" title="PRGCOR_2:pred.3">vector_minus_prg</a> <font color="Olive" title="b1">c</font>,<font color="Maroon" title="c1">b</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">c</font> being   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of   st <font color="Maroon" title="c2">m</font> <a href="prgcor_2.html#R3" title="PRGCOR_2:pred.3">vector_minus_prg</a> <font color="Olive" title="b1">c</font>,<font color="Maroon" title="c1">b</font> holds <br/> <a href="afinsq_1.html#K12" title="AFINSQ_1:func.12">XFS2FS*</a> <font color="Olive" title="b1">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K6" title="RVSUM_1:func.6">-</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K12" title="AFINSQ_1:func.12">XFS2FS*</a> <font color="Maroon" title="c1">b</font></span>)</span></span><br/>

<a NAME="E19:12"/><span class="lab"><font color="Green" title="E17">A13</font></span>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c1">b</font>
 
;<br/>
<span class="kw">thus </span><a NAME="E20:12"/>
 for <font color="Olive" title="b1">c</font> being   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of   st <font color="Maroon" title="c2">m</font> <a href="prgcor_2.html#R3" title="PRGCOR_2:pred.3">vector_minus_prg</a> <font color="Olive" title="b1">c</font>,<font color="Maroon" title="c1">b</font> holds <br/> <a href="afinsq_1.html#K12" title="AFINSQ_1:func.12">XFS2FS*</a> <font color="Olive" title="b1">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K6" title="RVSUM_1:func.6">-</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K12" title="AFINSQ_1:func.12">XFS2FS*</a> <font color="Maroon" title="c1">b</font></span>)</span>
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">c</font> be   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">m</font> <a href="prgcor_2.html#R3" title="PRGCOR_2:pred.3">vector_minus_prg</a> <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c1">b</font> implies  <a href="afinsq_1.html#K12" title="AFINSQ_1:func.12">XFS2FS*</a> <font color="Maroon" title="c7">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K6" title="RVSUM_1:func.6">-</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K12" title="AFINSQ_1:func.12">XFS2FS*</a> <font color="Maroon" title="c1">b</font></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:12_2"/><span class="lab"><font color="Green" title="E18">A14</font></span>: 
<font color="Maroon" title="c2">m</font> <a href="prgcor_2.html#R3" title="PRGCOR_2:pred.3">vector_minus_prg</a> <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c1">b</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="afinsq_1.html#K12" title="AFINSQ_1:func.12">XFS2FS*</a> <font color="Maroon" title="c7">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K6" title="RVSUM_1:func.6">-</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K12" title="AFINSQ_1:func.12">XFS2FS*</a> <font color="Maroon" title="c1">b</font></span>)</span></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">n</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a><span class="kw"> such that </span><br/><a NAME="E3:12_2"/><span class="lab"><font color="Green" title="E19">A15</font></span>: 
<font color="Maroon" title="c7">c</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">b</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>
 <span class="kw">and </span><br/><a NAME="E4:12_2"/><span class="lab"><font color="Green" title="E20">A16</font></span>: 
<font color="Maroon" title="c8">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">b</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>
 <span class="kw">and </span><br/><a NAME="E5:12_2"/><span class="lab"><font color="Green" title="E21">A17</font></span>: 
( <font color="Maroon" title="c8">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies  for <font color="Olive" title="b1">i</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">n</font> holds <br/><font color="Maroon" title="c7">c</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">b</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="prgcor_2.html#D5" target="_self" title="PRGCOR_2:def.5">Def9</a></span>;<br/>
<a NAME="E6:12_2"/><span class="lab"><font color="Green" title="E22">A18</font></span>: 
(  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c7">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">m</font> &amp;  ex <font color="Olive" title="b1">n</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> st <br/>( <font color="Maroon" title="c7">c</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">b</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> &amp; <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">b</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> &amp; ( <font color="Olive" title="b1">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies  for <font color="Olive" title="b2">i</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">i</font> &amp; <font color="Olive" title="b2">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> holds <br/><font color="Maroon" title="c7">c</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">b</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">i</font></span>)</span> ) ) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:12_2"><span class="lab"><font color="Green" title="E18">A14</font></span></a>, <a class="ref" href="prgcor_2.html#D5" target="_self" title="PRGCOR_2:def.5">Def9</a></span>;<br/>
<a NAME="E7:12_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E23">A19</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K12" title="AFINSQ_1:func.12">XFS2FS*</a> <font color="Maroon" title="c7">c</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">c</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:12"><span class="lab"><font color="Green" title="E12">A5</font></span></a>, <a class="ref" href="afinsq_1.html#D11" title="AFINSQ_1:def.11">AFINSQ_1:def 11</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_2_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_2_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:12_2_1_1"/>
( <font color="Maroon" title="c8">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c8">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_2_1_1_1"><span class="kw">case </span></a><a NAME="E1:12_2_1_1_1"/><span class="lab"><font color="Green" title="E24">A20</font></span>: 
<font color="Maroon" title="c8">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="afinsq_1.html#K12" title="AFINSQ_1:func.12">XFS2FS*</a> <font color="Maroon" title="c7">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K6" title="RVSUM_1:func.6">-</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K12" title="AFINSQ_1:func.12">XFS2FS*</a> <font color="Maroon" title="c1">b</font></span>)</span></span><br/><div class="add"><a NAME="E2:12_2_1_1_1"/><span class="kw">then </span>
 <a href="afinsq_1.html#K12" title="AFINSQ_1:func.12">XFS2FS*</a> <font color="Maroon" title="c1">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K6" title="FINSEQ_1:func.6">&lt;*&gt;</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:12"><span class="lab"><font color="Green" title="E17">A13</font></span></a>, <a class="txt" href="#E4:12_2"><span class="lab"><font color="Green" title="E20">A16</font></span></a>, <a class="ref" href="afinsq_1.html#T64" title="AFINSQ_1:th.64">AFINSQ_1:64</a></span>;<br/><span class="kw">hence </span><a NAME="E3:12_2_1_1_1"/>
 <a href="afinsq_1.html#K12" title="AFINSQ_1:func.12">XFS2FS*</a> <font color="Maroon" title="c7">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K6" title="RVSUM_1:func.6">-</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K12" title="AFINSQ_1:func.12">XFS2FS*</a> <font color="Maroon" title="c1">b</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:12_2"><span class="lab"><font color="Green" title="E22">A18</font></span></a>, <a class="txt" href="#E4:12_2"><span class="lab"><font color="Green" title="E20">A16</font></span></a>, <a class="txt" href="#E1:12_2_1_1_1"><span class="lab"><font color="Green" title="E24">A20</font></span></a>, <a class="ref" href="afinsq_1.html#T64" title="AFINSQ_1:th.64">AFINSQ_1:64</a>, <a class="ref" href="rvsum_1.html#T19" title="RVSUM_1:th.19">RVSUM_1:19</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_2_1_1_2"><span class="kw">case </span></a><a NAME="E1:12_2_1_1_2"/>
<font color="Maroon" title="c8">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="afinsq_1.html#K12" title="AFINSQ_1:func.12">XFS2FS*</a> <font color="Maroon" title="c7">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K6" title="RVSUM_1:func.6">-</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K12" title="AFINSQ_1:func.12">XFS2FS*</a> <font color="Maroon" title="c1">b</font></span>)</span></span><br/><div class="add"><span class="kw">set </span><font color="Maroon" title="c9">p3</font> =  <a href="afinsq_1.html#K12" title="AFINSQ_1:func.12">XFS2FS*</a> <font color="Maroon" title="c7">c</font>;<br/><a NAME="E2:12_2_1_1_2"/>
 for <font color="Olive" title="b1">k3</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k3</font> &amp; <font color="Olive" title="b1">k3</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K12" title="AFINSQ_1:func.12">XFS2FS*</a> <font color="Maroon" title="c7">c</font></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="afinsq_1.html#K12" title="AFINSQ_1:func.12">XFS2FS*</a> <font color="Maroon" title="c7">c</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">k3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">c2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">k3</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_2_1_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c10">k3</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">k3</font> &amp; <font color="Maroon" title="c10">k3</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K12" title="AFINSQ_1:func.12">XFS2FS*</a> <font color="Maroon" title="c7">c</font></span>)</span> implies <span class="p1">(<span class="default"><a href="afinsq_1.html#K12" title="AFINSQ_1:func.12">XFS2FS*</a> <font color="Maroon" title="c7">c</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">k3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">c2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">k3</font> )</span><br/>

<a NAME="E1:12_2_1_1_2_1"/><span class="lab"><font color="Green" title="E24">A21</font></span>: 
<font color="Maroon" title="c7">c</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c7">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:12"><span class="lab"><font color="Green" title="E8">A1</font></span></a>, <a class="txt" href="#E3:12"><span class="lab"><font color="Green" title="E10">A3</font></span></a>, <a class="txt" href="#E6:12_2"><span class="lab"><font color="Green" title="E22">A18</font></span></a>, <a class="ref" href="nat_1.html#T44" title="NAT_1:th.44">NAT_1:44</a></span>;<br/>
<a NAME="E2:12_2_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E25">A22</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K12" title="AFINSQ_1:func.12">XFS2FS*</a> <font color="Maroon" title="c7">c</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:12_2"><span class="lab"><font color="Green" title="E19">A15</font></span></a>, <a class="txt" href="#E4:12_2"><span class="lab"><font color="Green" title="E20">A16</font></span></a>, <a class="ref" href="afinsq_1.html#D11" title="AFINSQ_1:def.11">AFINSQ_1:def 11</a></span>;<br/>
<span class="kw">assume </span><a NAME="E3:12_2_1_1_2_1"/><span class="lab"><font color="Green" title="E26">A23</font></span>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">k3</font> &amp; <font color="Maroon" title="c10">k3</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K12" title="AFINSQ_1:func.12">XFS2FS*</a> <font color="Maroon" title="c7">c</font></span>)</span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="afinsq_1.html#K12" title="AFINSQ_1:func.12">XFS2FS*</a> <font color="Maroon" title="c7">c</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">k3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">c2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">k3</font></span><br/>

<a NAME="E4:12_2_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E27">A24</font></span>: 
<font color="Maroon" title="c1">b</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">k3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K12" title="AFINSQ_1:func.12">XFS2FS*</a> <font color="Maroon" title="c1">b</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">k3</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:12"><span class="lab"><font color="Green" title="E9">A2</font></span></a>, <a class="txt" href="#E8:12"><span class="lab"><font color="Green" title="E12">A5</font></span></a>, <a class="txt" href="#E4:12_2"><span class="lab"><font color="Green" title="E20">A16</font></span></a>, <a class="txt" href="#E2:12_2_1_1_2_1"><span class="lab"><font color="Green" title="E25">A22</font></span></a>, <a class="ref" href="afinsq_1.html#D11" title="AFINSQ_1:def.11">AFINSQ_1:def 11</a></span>;<br/>
<span class="p1">(<span class="default"><a href="afinsq_1.html#K12" title="AFINSQ_1:func.12">XFS2FS*</a> <font color="Maroon" title="c7">c</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">k3</font> = 
<font color="Maroon" title="c7">c</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">k3</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:12_2"><span class="lab"><font color="Green" title="E19">A15</font></span></a>, <a class="txt" href="#E4:12_2"><span class="lab"><font color="Green" title="E20">A16</font></span></a>, <a class="txt" href="#E3:12_2_1_1_2_1"><span class="lab"><font color="Green" title="E26">A23</font></span></a>, <a class="txt" href="#E1:12_2_1_1_2_1"><span class="lab"><font color="Green" title="E24">A21</font></span></a>, <a class="txt" href="#E2:12_2_1_1_2_1"><span class="lab"><font color="Green" title="E25">A22</font></span></a>, <a class="ref" href="afinsq_1.html#D11" title="AFINSQ_1:def.11">AFINSQ_1:def 11</a></span>
<br/>.= 
 <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">b</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">k3</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:12_2"><span class="lab"><font color="Green" title="E21">A17</font></span></a>, <a class="txt" href="#E3:12_2_1_1_2_1"><span class="lab"><font color="Green" title="E26">A23</font></span></a>, <a class="txt" href="#E2:12_2_1_1_2_1"><span class="lab"><font color="Green" title="E25">A22</font></span></a></span>
;<br/>
<span class="kw">hence </span><a NAME="E6:12_2_1_1_2_1"/>
<span class="p1">(<span class="default"><a href="afinsq_1.html#K12" title="AFINSQ_1:func.12">XFS2FS*</a> <font color="Maroon" title="c7">c</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">k3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">c2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">k3</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:12_2_1_1_2_1"><span class="lab"><font color="Green" title="E27">A24</font></span></a>, <a class="ref" href="rvsum_1.html#T17" title="RVSUM_1:th.17">RVSUM_1:17</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:12_2_1_1_2"/>
 <a href="afinsq_1.html#K12" title="AFINSQ_1:func.12">XFS2FS*</a> <font color="Maroon" title="c7">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K6" title="RVSUM_1:func.6">-</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K12" title="AFINSQ_1:func.12">XFS2FS*</a> <font color="Maroon" title="c1">b</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:12"><span class="lab"><font color="Green" title="E13">A6</font></span></a>, <a class="txt" href="#E3:12_2"><span class="lab"><font color="Green" title="E19">A15</font></span></a>, <a class="txt" href="#E7:12_2"><span class="lab"><font color="Green" title="E23">A19</font></span></a>, <a class="ref" href="finseq_1.html#T14" title="FINSEQ_1:th.14">FINSEQ_1:14</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E9:12_2"/>
 <a href="afinsq_1.html#K12" title="AFINSQ_1:func.12">XFS2FS*</a> <font color="Maroon" title="c7">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K6" title="RVSUM_1:func.6">-</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K12" title="AFINSQ_1:func.12">XFS2FS*</a> <font color="Maroon" title="c1">b</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>


</div>
