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

<span class="kw">let </span><font color="Maroon" title="c1">a</font>, <font color="Maroon" title="c2">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 href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="c2">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">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">m</font> &amp;  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c2">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">a</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="c2">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="Maroon" title="c2">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  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  st <font color="Olive" title="b1">m</font> <a href="prgcor_2.html#R4" title="PRGCOR_2:pred.4">vector_add_prg</a> <font color="Olive" title="b2">c</font>,<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">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  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>   st <font color="Olive" title="b1">m</font> <a href="prgcor_2.html#R4" title="PRGCOR_2:pred.4">vector_add_prg</a> <font color="Olive" title="b2">c</font>,<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">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> <span class="p1">(<span class="default"><a href="afinsq_1.html#K12" title="AFINSQ_1:func.12">XFS2FS*</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="rvsum_1.html#K4" title="RVSUM_1:func.4">+</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K12" title="AFINSQ_1:func.12">XFS2FS*</a> <font color="Maroon" title="c2">b</font></span>)</span> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">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="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">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">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">m</font> &amp;  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">m</font> &amp; <font color="Maroon" title="c1">a</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="c2">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="Maroon" title="c2">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="c3">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  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  st <font color="Maroon" title="c3">m</font> <a href="prgcor_2.html#R4" title="PRGCOR_2:pred.4">vector_add_prg</a> <font color="Olive" title="b1">c</font>,<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">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  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>   st <font color="Maroon" title="c3">m</font> <a href="prgcor_2.html#R4" title="PRGCOR_2:pred.4">vector_add_prg</a> <font color="Olive" title="b1">c</font>,<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">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> <span class="p1">(<span class="default"><a href="afinsq_1.html#K12" title="AFINSQ_1:func.12">XFS2FS*</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="rvsum_1.html#K4" title="RVSUM_1:func.4">+</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K12" title="AFINSQ_1:func.12">XFS2FS*</a> <font color="Maroon" title="c2">b</font></span>)</span> ) ) )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:19"/><span class="lab"><font color="Green" title="E9">A1</font></span>: 
<font color="Maroon" title="c2">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="E2:19"/><span class="lab"><font color="Green" title="E10">A2</font></span>: 
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">m</font>
 <span class="kw">and </span><br/><a NAME="E3:19"/><span class="lab"><font color="Green" title="E11">A3</font></span>: 
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">m</font>
 <span class="kw">and </span><br/><a NAME="E4:19"/><span class="lab"><font color="Green" title="E12">A4</font></span>: 
<font color="Maroon" title="c1">a</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="c2">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="E5:19"/><span class="lab"><font color="Green" title="E13">A5</font></span>: 
<font color="Maroon" title="c2">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="c3">m</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  st <font color="Maroon" title="c3">m</font> <a href="prgcor_2.html#R4" title="PRGCOR_2:pred.4">vector_add_prg</a> <font color="Olive" title="b1">c</font>,<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">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  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>   st <font color="Maroon" title="c3">m</font> <a href="prgcor_2.html#R4" title="PRGCOR_2:pred.4">vector_add_prg</a> <font color="Olive" title="b1">c</font>,<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">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> <span class="p1">(<span class="default"><a href="afinsq_1.html#K12" title="AFINSQ_1:func.12">XFS2FS*</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="rvsum_1.html#K4" title="RVSUM_1:func.4">+</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K12" title="AFINSQ_1:func.12">XFS2FS*</a> <font color="Maroon" title="c2">b</font></span>)</span> ) )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c4">k</font> = <font color="Maroon" title="c2">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="#E1:19"><span class="lab"><font color="Green" title="E9">A1</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c5">Fb</font> =  <a href="afinsq_1.html#K12" title="AFINSQ_1:func.12">XFS2FS*</a> <font color="Maroon" title="c2">b</font> 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/>
<span class="kw">reconsider </span><font color="Maroon" title="c6">Fa</font> =  <a href="afinsq_1.html#K12" title="AFINSQ_1:func.12">XFS2FS*</a> <font color="Maroon" title="c1">a</font> 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/>
<span class="kw">reconsider </span><font color="Maroon" title="c7">c2</font> = <font color="Maroon" title="c6">Fa</font> <a href="rvsum_1.html#K4" title="RVSUM_1:func.4">+</a> <font color="Maroon" title="c5">Fb</font> 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="E10:19"/><span class="lab"><font color="Green" title="E14">A6</font></span>: 
<font color="Maroon" title="c2">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="c3">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:19"><span class="lab"><font color="Green" title="E9">A1</font></span></a>, <a class="txt" href="#E5:19"><span class="lab"><font color="Green" title="E13">A5</font></span></a>, <a class="ref" href="nat_1.html#T45" title="NAT_1:th.45">NAT_1:45</a></span>;<br/>
<a NAME="E11:19"/><span class="kw">then </span><span class="lab"><font color="Green" title="E15">A7</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="c1">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">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="#E2:19"><span class="lab"><font color="Green" title="E10">A2</font></span></a>, <a class="txt" href="#E4:19"><span class="lab"><font color="Green" title="E12">A4</font></span></a>, <a class="ref" href="afinsq_1.html#D12" title="AFINSQ_1:def.12">AFINSQ_1:def 12</a></span>;<br/>
<a NAME="E12:19"/><span class="lab"><font color="Green" title="E16">A8</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="c2">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">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="#E3:19"><span class="lab"><font color="Green" title="E11">A3</font></span></a>, <a class="txt" href="#E10:19"><span class="lab"><font color="Green" title="E14">A6</font></span></a>, <a class="ref" href="afinsq_1.html#D12" title="AFINSQ_1:def.12">AFINSQ_1:def 12</a></span>;<br/>
<a NAME="E13:19"/><span class="kw">then </span><span class="lab"><font color="Green" title="E17">A9</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">c2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c1">a</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:19"><span class="lab"><font color="Green" title="E15">A7</font></span></a>, <a class="ref" href="rvsum_1.html#T145" title="RVSUM_1:th.145">RVSUM_1:145</a></span>;<br/>
<a NAME="E14:19"/><span class="kw">then </span>
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c7">c2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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="afinsq_1.html#K12" title="AFINSQ_1:func.12">XFS2FS*</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:19"><span class="lab"><font color="Green" title="E16">A8</font></span></a>, <a class="txt" href="#E11:19"><span class="lab"><font color="Green" title="E15">A7</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E15:19"/><span class="kw">then </span>
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_1.html#K12" title="AFINSQ_1:func.12">XFS2FS*</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="rvsum_1.html#K4" title="RVSUM_1:func.4">+</a> <span class="p2">(<span class="default"><a href="afinsq_1.html#K12" title="AFINSQ_1:func.12">XFS2FS*</a> <font color="Maroon" title="c2">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="c2">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="E16:19"/><span class="kw">then </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> <font color="Maroon" title="c7">c2</font></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="c2">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="E17:19"/><span class="kw">then </span><span class="lab"><font color="Green" title="E18">A10</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">c2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:19"><span class="lab"><font color="Green" title="E16">A8</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="c8">p</font> being   <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <span class="kw"> such that </span><br/><a NAME="E19:19"/><span class="lab"><font color="Green" title="E19">A11</font></span>: 
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c8">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">m</font>
 <span class="kw">and </span><br/><a NAME="E20:19"/><span class="lab"><font color="Green" title="E20">A12</font></span>: 
<font color="Maroon" title="c8">p</font> <a href="prgcor_2.html#R1" title="PRGCOR_2:pred.1">is_an_xrep_of</a> <font color="Maroon" title="c7">c2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:19"><span class="lab"><font color="Green" title="E13">A5</font></span></a>, <a class="ref" href="prgcor_2.html#T7" target="_self" title="PRGCOR_2:th.7">Th7</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c9">p2</font> =  <a href="afinsq_1.html#NK11" title="AFINSQ_1:NK.11">Replace</a> <font color="Maroon" title="c8">p</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p1">(<span class="default"><font color="Maroon" title="c2">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  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  ;<br/>
<div><span class="lab"><font color="Green" title="E21">A13</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:19_1"/>
<font color="Maroon" title="c4">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="Red">::  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="c4">k</font> holds <br/><font color="Maroon" title="c9">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> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">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">thus </span><a NAME="E2:19_1"/>
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="c4">k</font> holds <br/><font color="Maroon" title="c9">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> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span>
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c10">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="Red">::  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">i</font> &amp; <font color="Maroon" title="c10">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">k</font> implies <font color="Maroon" title="c9">p2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">i</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">i</font></span>)</span> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:19_1_1"/><span class="lab"><font color="Green" title="E22">A14</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">i</font>
 <span class="kw">and </span><br/><a NAME="E2:19_1_1"/><span class="lab"><font color="Green" title="E23">A15</font></span>: 
<font color="Maroon" title="c10">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">k</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c9">p2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">i</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">i</font></span>)</span></span><br/>

<a NAME="E3:19_1_1"/>
<font color="Maroon" title="c10">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <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> <font color="Maroon" title="c7">c2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:19"><span class="lab"><font color="Green" title="E15">A7</font></span></a>, <a class="txt" href="#E13:19"><span class="lab"><font color="Green" title="E17">A9</font></span></a>, <a class="txt" href="#E1:19_1_1"><span class="lab"><font color="Green" title="E22">A14</font></span></a>, <a class="txt" href="#E2:19_1_1"><span class="lab"><font color="Green" title="E23">A15</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/>
<a NAME="E4:19_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E24">A16</font></span>: 
<font color="Maroon" title="c10">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c7">c2</font>
 
<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="E5:19_1_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">a</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">a</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">i</font> &amp; <span class="p1">(<span class="default"><a href="afinsq_1.html#K12" title="AFINSQ_1:func.12">XFS2FS*</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">b</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">i</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:19"><span class="lab"><font color="Green" title="E10">A2</font></span></a>, <a class="txt" href="#E3:19"><span class="lab"><font color="Green" title="E11">A3</font></span></a>, <a class="txt" href="#E4:19"><span class="lab"><font color="Green" title="E12">A4</font></span></a>, <a class="txt" href="#E10:19"><span class="lab"><font color="Green" title="E14">A6</font></span></a>, <a class="txt" href="#E1:19_1_1"><span class="lab"><font color="Green" title="E22">A14</font></span></a>, <a class="txt" href="#E2:19_1_1"><span class="lab"><font color="Green" title="E23">A15</font></span></a>, <a class="ref" href="afinsq_1.html#D12" title="AFINSQ_1:def.12">AFINSQ_1:def 12</a></span>;<br/>
<a NAME="E6:19_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E25">A17</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_1.html#K12" title="AFINSQ_1:func.12">XFS2FS*</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="rvsum_1.html#K4" title="RVSUM_1:func.4">+</a> <span class="p2">(<span class="default"><a href="afinsq_1.html#K12" title="AFINSQ_1:func.12">XFS2FS*</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">i</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">i</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:19_1_1"><span class="lab"><font color="Green" title="E24">A16</font></span></a>, <a class="ref" href="valued_1.html#D1" title="VALUED_1:def.1">VALUED_1:def 1</a></span>;<br/>
<a NAME="E7:19_1_1"/>
( <font color="Maroon" title="c10">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="c8">p</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">c2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">i</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:19"><span class="lab"><font color="Green" title="E18">A10</font></span></a>, <a class="txt" href="#E20:19"><span class="lab"><font color="Green" title="E20">A12</font></span></a>, <a class="txt" href="#E1:19_1_1"><span class="lab"><font color="Green" title="E22">A14</font></span></a>, <a class="txt" href="#E2:19_1_1"><span class="lab"><font color="Green" title="E23">A15</font></span></a>, <a class="ref" href="prgcor_2.html#D5" target="_self" title="PRGCOR_2:def.5">Def5</a>, <a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:19_1_1"/>
<font color="Maroon" title="c9">p2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">i</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">i</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:19_1_1"><span class="lab"><font color="Green" title="E22">A14</font></span></a>, <a class="txt" href="#E6:19_1_1"><span class="lab"><font color="Green" title="E25">A17</font></span></a>, <a class="ref" href="afinsq_1.html#T48" title="AFINSQ_1:th.48">AFINSQ_1:48</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<a NAME="E23:19"/>
(  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c8">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="c9">p2</font> &amp; <font color="Maroon" title="c9">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="c2">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="#E1:19"><span class="lab"><font color="Green" title="E9">A1</font></span></a>, <a class="txt" href="#E5:19"><span class="lab"><font color="Green" title="E13">A5</font></span></a>, <a class="txt" href="#E19:19"><span class="lab"><font color="Green" title="E19">A11</font></span></a>, <a class="ref" href="afinsq_1.html#T48" title="AFINSQ_1:th.48">AFINSQ_1:48</a></span>;<br/>
<a NAME="E24:19"/><span class="kw">then </span>
<font color="Maroon" title="c3">m</font> <a href="prgcor_2.html#R4" title="PRGCOR_2:pred.4">vector_add_prg</a> <font color="Maroon" title="c9">p2</font>,<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:19"><span class="lab"><font color="Green" title="E10">A2</font></span></a>, <a class="txt" href="#E3:19"><span class="lab"><font color="Green" title="E11">A3</font></span></a>, <a class="txt" href="#E19:19"><span class="lab"><font color="Green" title="E19">A11</font></span></a>, <a class="txt" href="#E22:19"><span class="lab"><font color="Green" title="E21">A13</font></span></a>, <a class="ref" href="prgcor_2.html#D10" target="_self" title="PRGCOR_2:def.10">Def10</a></span>;<br/>
<span class="kw">hence </span><a NAME="E25:19"/>
 ex <font color="Olive" title="b1">c</font> being  <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  st <font color="Maroon" title="c3">m</font> <a href="prgcor_2.html#R4" title="PRGCOR_2:pred.4">vector_add_prg</a> <font color="Olive" title="b1">c</font>,<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>   st <font color="Maroon" title="c3">m</font> <a href="prgcor_2.html#R4" title="PRGCOR_2:pred.4">vector_add_prg</a> <font color="Olive" title="b1">c</font>,<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">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> <span class="p1">(<span class="default"><a href="afinsq_1.html#K12" title="AFINSQ_1:func.12">XFS2FS*</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="rvsum_1.html#K4" title="RVSUM_1:func.4">+</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K12" title="AFINSQ_1:func.12">XFS2FS*</a> <font color="Maroon" title="c2">b</font></span>)</span></span><br/>

<a NAME="E26:19"/><span class="lab"><font color="Green" title="E22">A18</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="c2">b</font>
 
;<br/>
<span class="kw">thus </span><a NAME="E27:19"/>
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  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>   st <font color="Maroon" title="c3">m</font> <a href="prgcor_2.html#R4" title="PRGCOR_2:pred.4">vector_add_prg</a> <font color="Olive" title="b1">c</font>,<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">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> <span class="p1">(<span class="default"><a href="afinsq_1.html#K12" title="AFINSQ_1:func.12">XFS2FS*</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="rvsum_1.html#K4" title="RVSUM_1:func.4">+</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K12" title="AFINSQ_1:func.12">XFS2FS*</a> <font color="Maroon" title="c2">b</font></span>)</span>
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c10">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 href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">m</font> <a href="prgcor_2.html#R4" title="PRGCOR_2:pred.4">vector_add_prg</a> <font color="Maroon" title="c10">c</font>,<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font> implies  <a href="afinsq_1.html#K12" title="AFINSQ_1:func.12">XFS2FS*</a> <font color="Maroon" title="c10">c</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">a</font></span>)</span> <a href="rvsum_1.html#K4" title="RVSUM_1:func.4">+</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K12" title="AFINSQ_1:func.12">XFS2FS*</a> <font color="Maroon" title="c2">b</font></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:19_2"/><span class="lab"><font color="Green" title="E23">A19</font></span>: 
<font color="Maroon" title="c3">m</font> <a href="prgcor_2.html#R4" title="PRGCOR_2:pred.4">vector_add_prg</a> <font color="Maroon" title="c10">c</font>,<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="afinsq_1.html#K12" title="AFINSQ_1:func.12">XFS2FS*</a> <font color="Maroon" title="c10">c</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">a</font></span>)</span> <a href="rvsum_1.html#K4" title="RVSUM_1:func.4">+</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K12" title="AFINSQ_1:func.12">XFS2FS*</a> <font color="Maroon" title="c2">b</font></span>)</span></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c11">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="E3:19_2"/><span class="lab"><font color="Green" title="E24">A21</font></span>: 
<font color="Maroon" title="c10">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="c2">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="E4:19_2"/><span class="lab"><font color="Green" title="E25">A22</font></span>: 
<font color="Maroon" title="c11">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">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="E5:19_2"/><span class="lab"><font color="Green" title="E26">A23</font></span>: 
( <font color="Maroon" title="c11">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="c11">n</font> holds <br/><font color="Maroon" title="c10">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> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">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="txt" href="#E1:19_2"><span class="lab"><font color="Green" title="E23">A19</font></span></a>, <a class="ref" href="prgcor_2.html#D10" target="_self" title="PRGCOR_2:def.10">Def10</a></span>;<br/>
<a NAME="E6:19_2"/><span class="lab"><font color="Green" title="E27">A24</font></span>: 
(  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c10">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">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="c10">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="c2">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="c2">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="c10">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> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">i</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">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="#E1:19_2"><span class="lab"><font color="Green" title="E23">A19</font></span></a>, <a class="ref" href="prgcor_2.html#D10" target="_self" title="PRGCOR_2:def.10">Def10</a></span>;<br/>
<a NAME="E7:19_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E28">A25</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="c10">c</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">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="#E10:19"><span class="lab"><font color="Green" title="E14">A6</font></span></a>, <a class="ref" href="afinsq_1.html#D12" title="AFINSQ_1:def.12">AFINSQ_1:def 12</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_2_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_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="E1:19_2_1_1"/>
( <font color="Maroon" title="c11">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="c11">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="19_2_1_1_1"><span class="kw">case </span></a><a NAME="E1:19_2_1_1_1"/>
<font color="Maroon" title="c11">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="Red">::  thesis: </font></span></a><span class="hide">  <a href="afinsq_1.html#K12" title="AFINSQ_1:func.12">XFS2FS*</a> <font color="Maroon" title="c10">c</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">a</font></span>)</span> <a href="rvsum_1.html#K4" title="RVSUM_1:func.4">+</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K12" title="AFINSQ_1:func.12">XFS2FS*</a> <font color="Maroon" title="c2">b</font></span>)</span></span><br/><div class="add"><a NAME="E2:19_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="c2">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>  &amp;  <a href="afinsq_1.html#K12" title="AFINSQ_1:func.12">XFS2FS*</a> <font color="Maroon" title="c10">c</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="#E26:19"><span class="lab"><font color="Green" title="E22">A18</font></span></a>, <a class="txt" href="#E6:19_2"><span class="lab"><font color="Green" title="E27">A24</font></span></a>, <a class="txt" href="#E4:19_2"><span class="lab"><font color="Green" title="E25">A22</font></span></a>, <a class="ref" href="afinsq_1.html#T68" title="AFINSQ_1:th.68">AFINSQ_1:68</a></span>;<br/><span class="kw">hence </span><a NAME="E3:19_2_1_1_1"/>
 <a href="afinsq_1.html#K12" title="AFINSQ_1:func.12">XFS2FS*</a> <font color="Maroon" title="c10">c</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">a</font></span>)</span> <a href="rvsum_1.html#K4" title="RVSUM_1:func.4">+</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K12" title="AFINSQ_1:func.12">XFS2FS*</a> <font color="Maroon" title="c2">b</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rvsum_1.html#T28" title="RVSUM_1:th.28">RVSUM_1:28</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="19_2_1_1_2"><span class="kw">case </span></a><a NAME="E1:19_2_1_1_2"/>
<font color="Maroon" title="c11">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="Red">::  thesis: </font></span></a><span class="hide">  <a href="afinsq_1.html#K12" title="AFINSQ_1:func.12">XFS2FS*</a> <font color="Maroon" title="c10">c</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">a</font></span>)</span> <a href="rvsum_1.html#K4" title="RVSUM_1:func.4">+</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K12" title="AFINSQ_1:func.12">XFS2FS*</a> <font color="Maroon" title="c2">b</font></span>)</span></span><br/><div class="add"><span class="kw">set </span><font color="Maroon" title="c12">p3</font> =  <a href="afinsq_1.html#K12" title="AFINSQ_1:func.12">XFS2FS*</a> <font color="Maroon" title="c10">c</font>;<br/><a NAME="E2:19_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="c10">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="c10">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="c7">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="19_2_1_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c13">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="Red">::  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="c13">k3</font> &amp; <font color="Maroon" title="c13">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="c10">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="c10">c</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c13">k3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">c2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c13">k3</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:19_2_1_1_2_1"/><span class="lab"><font color="Green" title="E29">A26</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">k3</font>
 <span class="kw">and </span><br/><a NAME="E2:19_2_1_1_2_1"/><span class="lab"><font color="Green" title="E30">A27</font></span>: 
<font color="Maroon" title="c13">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="c10">c</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="c10">c</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c13">k3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">c2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c13">k3</font></span><br/>

<a NAME="E3:19_2_1_1_2_1"/><span class="lab"><font color="Green" title="E31">A28</font></span>: 
<font color="Maroon" title="c10">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="c10">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:19"><span class="lab"><font color="Green" title="E9">A1</font></span></a>, <a class="txt" href="#E5:19"><span class="lab"><font color="Green" title="E13">A5</font></span></a>, <a class="txt" href="#E6:19_2"><span class="lab"><font color="Green" title="E27">A24</font></span></a>, <a class="ref" href="nat_1.html#T45" title="NAT_1:th.45">NAT_1:45</a></span>;<br/>
<a NAME="E4:19_2_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">A29</font></span>: 
<font color="Maroon" title="c13">k3</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:19_2"><span class="lab"><font color="Green" title="E24">A21</font></span></a>, <a class="txt" href="#E4:19_2"><span class="lab"><font color="Green" title="E25">A22</font></span></a>, <a class="txt" href="#E2:19_2_1_1_2_1"><span class="lab"><font color="Green" title="E30">A27</font></span></a>, <a class="ref" href="afinsq_1.html#D12" title="AFINSQ_1:def.12">AFINSQ_1:def 12</a></span>;<br/>
<a NAME="E5:19_2_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">A30</font></span>: 
( <font color="Maroon" title="c1">a</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c13">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">a</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c13">k3</font> &amp; <font color="Maroon" title="c2">b</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c13">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="c2">b</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c13">k3</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:19"><span class="lab"><font color="Green" title="E10">A2</font></span></a>, <a class="txt" href="#E3:19"><span class="lab"><font color="Green" title="E11">A3</font></span></a>, <a class="txt" href="#E4:19"><span class="lab"><font color="Green" title="E12">A4</font></span></a>, <a class="txt" href="#E10:19"><span class="lab"><font color="Green" title="E14">A6</font></span></a>, <a class="txt" href="#E4:19_2"><span class="lab"><font color="Green" title="E25">A22</font></span></a>, <a class="txt" href="#E1:19_2_1_1_2_1"><span class="lab"><font color="Green" title="E29">A26</font></span></a>, <a class="ref" href="afinsq_1.html#D12" title="AFINSQ_1:def.12">AFINSQ_1:def 12</a></span>;<br/>
<a NAME="E6:19_2_1_1_2_1"/>
<font color="Maroon" title="c13">k3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <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> <font color="Maroon" title="c7">c2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:19"><span class="lab"><font color="Green" title="E18">A10</font></span></a>, <a class="txt" href="#E4:19_2"><span class="lab"><font color="Green" title="E25">A22</font></span></a>, <a class="txt" href="#E1:19_2_1_1_2_1"><span class="lab"><font color="Green" title="E29">A26</font></span></a>, <a class="txt" href="#E4:19_2_1_1_2_1"><span class="lab"><font color="Green" title="E32">A29</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/>
<a NAME="E7:19_2_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E34">A31</font></span>: 
<font color="Maroon" title="c13">k3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c7">c2</font>
 
<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="E8:19_2_1_1_2_1"/>
 <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="c10">c</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:19_2"><span class="lab"><font color="Green" title="E24">A21</font></span></a>, <a class="txt" href="#E4:19_2"><span class="lab"><font color="Green" title="E25">A22</font></span></a>, <a class="txt" href="#E3:19_2_1_1_2_1"><span class="lab"><font color="Green" title="E31">A28</font></span></a>, <a class="ref" href="afinsq_1.html#D12" title="AFINSQ_1:def.12">AFINSQ_1:def 12</a></span>;<br/>
<span class="kw">then </span><span class="p1">(<span class="default"><a href="afinsq_1.html#K12" title="AFINSQ_1:func.12">XFS2FS*</a> <font color="Maroon" title="c10">c</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c13">k3</font> = 
<font color="Maroon" title="c10">c</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c13">k3</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:19_2"><span class="lab"><font color="Green" title="E24">A21</font></span></a>, <a class="txt" href="#E4:19_2"><span class="lab"><font color="Green" title="E25">A22</font></span></a>, <a class="txt" href="#E1:19_2_1_1_2_1"><span class="lab"><font color="Green" title="E29">A26</font></span></a>, <a class="txt" href="#E2:19_2_1_1_2_1"><span class="lab"><font color="Green" title="E30">A27</font></span></a>, <a class="txt" href="#E3:19_2_1_1_2_1"><span class="lab"><font color="Green" title="E31">A28</font></span></a>, <a class="ref" href="afinsq_1.html#D12" title="AFINSQ_1:def.12">AFINSQ_1:def 12</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c13">k3</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c13">k3</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:19_2"><span class="lab"><font color="Green" title="E26">A23</font></span></a>, <a class="txt" href="#E1:19_2_1_1_2_1"><span class="lab"><font color="Green" title="E29">A26</font></span></a>, <a class="txt" href="#E4:19_2_1_1_2_1"><span class="lab"><font color="Green" title="E32">A29</font></span></a></span>
;<br/>
<span class="kw">hence </span><a NAME="E10:19_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="c10">c</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c13">k3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">c2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c13">k3</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:19_2_1_1_2_1"><span class="lab"><font color="Green" title="E34">A31</font></span></a>, <a class="txt" href="#E5:19_2_1_1_2_1"><span class="lab"><font color="Green" title="E33">A30</font></span></a>, <a class="ref" href="valued_1.html#D1" title="VALUED_1:def.1">VALUED_1:def 1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:19_2_1_1_2"/>
 <a href="afinsq_1.html#K12" title="AFINSQ_1:func.12">XFS2FS*</a> <font color="Maroon" title="c10">c</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">a</font></span>)</span> <a href="rvsum_1.html#K4" title="RVSUM_1:func.4">+</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K12" title="AFINSQ_1:func.12">XFS2FS*</a> <font color="Maroon" title="c2">b</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:19"><span class="lab"><font color="Green" title="E18">A10</font></span></a>, <a class="txt" href="#E3:19_2"><span class="lab"><font color="Green" title="E24">A21</font></span></a>, <a class="txt" href="#E7:19_2"><span class="lab"><font color="Green" title="E28">A25</font></span></a>, <a class="ref" href="finseq_1.html#T18" title="FINSEQ_1:th.18">FINSEQ_1:18</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="E9:19_2"/>
 <a href="afinsq_1.html#K12" title="AFINSQ_1:func.12">XFS2FS*</a> <font color="Maroon" title="c10">c</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">a</font></span>)</span> <a href="rvsum_1.html#K4" title="RVSUM_1:func.4">+</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K12" title="AFINSQ_1:func.12">XFS2FS*</a> <font color="Maroon" title="c2">b</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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


</div>
