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<span class="kw">let </span><font color="Maroon" title="c1">n</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">  for <font color="Olive" title="b1">x0</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font> holds  <font color="Olive" title="b1">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_7.html#K5" title="EUCLID_7:func.5">Sum</a> <span class="p1">(<span class="default"><a href="euclid_7.html#K7" title="EUCLID_7:func.7">ProjFinSeq</a> <font color="Olive" title="b1">x0</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c2">x0</font> be    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</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="c2">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_7.html#K5" title="EUCLID_7:func.5">Sum</a> <span class="p1">(<span class="default"><a href="euclid_7.html#K7" title="EUCLID_7:func.7">ProjFinSeq</a> <font color="Maroon" title="c2">x0</font></span>)</span></span><br/>



<span class="kw">set </span><font color="Maroon" title="c3">f</font> =  <a href="euclid_7.html#K7" title="EUCLID_7:func.7">ProjFinSeq</a> <font color="Maroon" title="c2">x0</font>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c4">n2</font> = <font color="Maroon" title="c1">n</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" title="ORDINAL1:def.12">ORDINAL1:def 12</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="58_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="E1:58_1_1"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="euclid_7.html#K7" title="EUCLID_7:func.7">ProjFinSeq</a> <font color="Maroon" title="c2">x0</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="euclid_7.html#K7" title="EUCLID_7:func.7">ProjFinSeq</a> <font color="Maroon" title="c2">x0</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</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="58_1_1_1"><span class="kw">case </span></a><a NAME="E1:58_1_1_1"/><span class="lab"><font color="Green" title="E38">A1</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="euclid_7.html#K7" title="EUCLID_7:func.7">ProjFinSeq</a> <font color="Maroon" title="c2">x0</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&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"> <font color="Maroon" title="c2">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="euclid_7.html#K4" title="EUCLID_7:func.4">accum</a> <span class="p2">(<span class="default"><a href="euclid_7.html#K7" title="EUCLID_7:func.7">ProjFinSeq</a> <font color="Maroon" title="c2">x0</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</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="euclid_7.html#K7" title="EUCLID_7:func.7">ProjFinSeq</a> <font color="Maroon" title="c2">x0</font></span>)</span></span>)</span></span><br/><div class="add"><span class="kw">set </span><font color="Maroon" title="c5">g2</font> =  <a href="euclid_7.html#K4" title="EUCLID_7:func.4">accum</a> <span class="p1">(<span class="default"><a href="euclid_7.html#K7" title="EUCLID_7:func.7">ProjFinSeq</a> <font color="Maroon" title="c2">x0</font></span>)</span>;<br/><a NAME="E2:58_1_1_1"/><span class="lab"><font color="Green" title="E39">A2</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="euclid_7.html#K7" title="EUCLID_7:func.7">ProjFinSeq</a> <font color="Maroon" title="c2">x0</font></span>)</span> <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="euclid_7.html#K4" title="EUCLID_7:func.4">accum</a> <span class="p2">(<span class="default"><a href="euclid_7.html#K7" title="EUCLID_7:func.7">ProjFinSeq</a> <font color="Maroon" title="c2">x0</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="euclid_7.html#D10" target="_self" title="EUCLID_7:def.10">Def10</a></span>;<br/><a NAME="E3:58_1_1_1"/><span class="lab"><font color="Green" title="E40">A3</font></span>: 
<span class="p1">(<span class="default"><a href="euclid_7.html#K7" title="EUCLID_7:func.7">ProjFinSeq</a> <font color="Maroon" title="c2">x0</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="euclid_7.html#K4" title="EUCLID_7:func.4">accum</a> <span class="p2">(<span class="default"><a href="euclid_7.html#K7" title="EUCLID_7:func.7">ProjFinSeq</a> <font color="Maroon" title="c2">x0</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="euclid_7.html#D10" target="_self" title="EUCLID_7:def.10">Def10</a></span>;<br/><span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <span class="kw">means </span> 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>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="euclid_7.html#K7" title="EUCLID_7:func.7">ProjFinSeq</a> <font color="Maroon" title="c2">x0</font></span>)</span> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> $1 &amp; $1 <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="euclid_7.html#K7" title="EUCLID_7:func.7">ProjFinSeq</a> <font color="Maroon" title="c2">x0</font></span>)</span> holds <br/>( ( <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> $1 implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid_7.html#K4" title="EUCLID_7:func.4">accum</a> <span class="p3">(<span class="default"><a href="euclid_7.html#K7" title="EUCLID_7:func.7">ProjFinSeq</a> <font color="Maroon" title="c2">x0</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> $1</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x0</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> ) &amp; ( <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> $1 implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid_7.html#K4" title="EUCLID_7:func.4">accum</a> <span class="p3">(<span class="default"><a href="euclid_7.html#K7" title="EUCLID_7:func.7">ProjFinSeq</a> <font color="Maroon" title="c2">x0</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> $1</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) );<br/><a NAME="E4:58_1_1_1"/><span class="lab"><font color="Green" title="E41">A4</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="euclid_7.html#K7" title="EUCLID_7:func.7">ProjFinSeq</a> <font color="Maroon" title="c2">x0</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="euclid_7.html#D12" target="_self" title="EUCLID_7:def.12">Def12</a></span>;<br/><a NAME="E5:58_1_1_1"/><span class="lab"><font color="Green" title="E42">A5</font></span>: 
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <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="euclid_7.html#K7" title="EUCLID_7:func.7">ProjFinSeq</a> <font color="Maroon" title="c2">x0</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:58_1_1_1"><span class="lab"><font color="Green" title="E38">A1</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E6:58_1_1_1"/><span class="lab"><font color="Green" title="E43">A6</font></span>: 
 for <font color="Olive" title="b1">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">k</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">S<sub>1</sub></font>[<font color="Maroon" title="c6">k</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1] )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c7">k2</font> = <font color="Maroon" title="c6">k</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" title="ORDINAL1:def.12">ORDINAL1:def 12</a></span>;<br/>
<span class="kw">assume </span><a NAME="E2:58_1_1_1_1"/><span class="lab"><font color="Green" title="E44">A7</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">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">S<sub>1</sub></font>[<font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]</span><br/>

<a NAME="E3:58_1_1_1_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>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="euclid_7.html#K7" title="EUCLID_7:func.7">ProjFinSeq</a> <font color="Maroon" title="c2">x0</font></span>)</span> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <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="euclid_7.html#K7" title="EUCLID_7:func.7">ProjFinSeq</a> <font color="Maroon" title="c2">x0</font></span>)</span> holds <br/>( ( <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid_7.html#K4" title="EUCLID_7:func.4">accum</a> <span class="p3">(<span class="default"><a href="euclid_7.html#K7" title="EUCLID_7:func.7">ProjFinSeq</a> <font color="Maroon" title="c2">x0</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x0</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> ) &amp; ( <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid_7.html#K4" title="EUCLID_7:func.4">accum</a> <span class="p3">(<span class="default"><a href="euclid_7.html#K7" title="EUCLID_7:func.7">ProjFinSeq</a> <font color="Maroon" title="c2">x0</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1_1_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">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="c8">i</font> &amp; <font color="Maroon" title="c8">i</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="euclid_7.html#K7" title="EUCLID_7:func.7">ProjFinSeq</a> <font color="Maroon" title="c2">x0</font></span>)</span> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <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="euclid_7.html#K7" title="EUCLID_7:func.7">ProjFinSeq</a> <font color="Maroon" title="c2">x0</font></span>)</span> implies ( ( <font color="Maroon" title="c8">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid_7.html#K4" title="EUCLID_7:func.4">accum</a> <span class="p3">(<span class="default"><a href="euclid_7.html#K7" title="EUCLID_7:func.7">ProjFinSeq</a> <font color="Maroon" title="c2">x0</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x0</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">i</font> ) &amp; ( <font color="Maroon" title="c8">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid_7.html#K4" title="EUCLID_7:func.4">accum</a> <span class="p3">(<span class="default"><a href="euclid_7.html#K7" title="EUCLID_7:func.7">ProjFinSeq</a> <font color="Maroon" title="c2">x0</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) ) )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:58_1_1_1_1_1"/><span class="lab"><font color="Green" title="E45">A8</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">i</font>
 <span class="kw">and </span><br/><a NAME="E2:58_1_1_1_1_1"/><span class="lab"><font color="Green" title="E46">A9</font></span>: 
<font color="Maroon" title="c8">i</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="euclid_7.html#K7" title="EUCLID_7:func.7">ProjFinSeq</a> <font color="Maroon" title="c2">x0</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E3:58_1_1_1_1_1"/><span class="lab"><font color="Green" title="E47">A10</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">and </span><br/><a NAME="E4:58_1_1_1_1_1"/><span class="lab"><font color="Green" title="E48">A11</font></span>: 
<font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <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="euclid_7.html#K7" title="EUCLID_7:func.7">ProjFinSeq</a> <font color="Maroon" title="c2">x0</font></span>)</span>
 ; <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="c8">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid_7.html#K4" title="EUCLID_7:func.4">accum</a> <span class="p3">(<span class="default"><a href="euclid_7.html#K7" title="EUCLID_7:func.7">ProjFinSeq</a> <font color="Maroon" title="c2">x0</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x0</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">i</font> ) &amp; ( <font color="Maroon" title="c8">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid_7.html#K4" title="EUCLID_7:func.4">accum</a> <span class="p3">(<span class="default"><a href="euclid_7.html#K7" title="EUCLID_7:func.7">ProjFinSeq</a> <font color="Maroon" title="c2">x0</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) )</span><br/>

<a NAME="E5:58_1_1_1_1_1"/>
<span class="p1">(<span class="default"><a href="euclid_7.html#K4" title="EUCLID_7:func.4">accum</a> <span class="p2">(<span class="default"><a href="euclid_7.html#K7" title="EUCLID_7:func.7">ProjFinSeq</a> <font color="Maroon" title="c2">x0</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> is    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font>
 
;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c9">r</font> = <span class="p1">(<span class="default"><a href="euclid_7.html#K4" title="EUCLID_7:func.4">accum</a> <span class="p2">(<span class="default"><a href="euclid_7.html#K7" title="EUCLID_7:func.7">ProjFinSeq</a> <font color="Maroon" title="c2">x0</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:58_1_1_1"><span class="lab"><font color="Green" title="E39">A2</font></span></a>, <a class="txt" href="#E3:58_1_1_1_1_1"><span class="lab"><font color="Green" title="E47">A10</font></span></a>, <a class="txt" href="#E4:58_1_1_1_1_1"><span class="lab"><font color="Green" title="E48">A11</font></span></a>, <a class="ref" href="finseq_4.html#T15" title="FINSEQ_4:th.15">FINSEQ_4:15</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c10">i2</font> = <font color="Maroon" title="c8">i</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" title="ORDINAL1:def.12">ORDINAL1:def 12</a></span>;<br/>
<a NAME="E8:58_1_1_1_1_1"/><span class="lab"><font color="Green" title="E49">A12</font></span>: 
<span class="p1">(<span class="default"><a href="euclid_7.html#K4" title="EUCLID_7:func.4">accum</a> <span class="p2">(<span class="default"><a href="euclid_7.html#K7" title="EUCLID_7:func.7">ProjFinSeq</a> <font color="Maroon" title="c2">x0</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="euclid_7.html#K4" title="EUCLID_7:func.4">accum</a> <span class="p2">(<span class="default"><a href="euclid_7.html#K7" title="EUCLID_7:func.7">ProjFinSeq</a> <font color="Maroon" title="c2">x0</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:58_1_1_1"><span class="lab"><font color="Green" title="E39">A2</font></span></a>, <a class="txt" href="#E3:58_1_1_1_1_1"><span class="lab"><font color="Green" title="E47">A10</font></span></a>, <a class="txt" href="#E4:58_1_1_1_1_1"><span class="lab"><font color="Green" title="E48">A11</font></span></a>, <a class="ref" href="finseq_4.html#T15" title="FINSEQ_4:th.15">FINSEQ_4:15</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1_1_1_1_1_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="58_1_1_1_1_1_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="E1:58_1_1_1_1_1_1_1"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">k</font> or <font color="Maroon" title="c6">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="58_1_1_1_1_1_1_1_1"><span class="kw">case </span></a><a NAME="E1:58_1_1_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E50">A13</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">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="c8">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid_7.html#K4" title="EUCLID_7:func.4">accum</a> <span class="p3">(<span class="default"><a href="euclid_7.html#K7" title="EUCLID_7:func.7">ProjFinSeq</a> <font color="Maroon" title="c2">x0</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x0</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">i</font> ) &amp; ( <font color="Maroon" title="c8">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid_7.html#K4" title="EUCLID_7:func.4">accum</a> <span class="p3">(<span class="default"><a href="euclid_7.html#K7" title="EUCLID_7:func.7">ProjFinSeq</a> <font color="Maroon" title="c2">x0</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) )</span><br/><div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c11">r3</font> = <span class="p1">(<span class="default"><a href="euclid_7.html#K7" title="EUCLID_7:func.7">ProjFinSeq</a> <font color="Maroon" title="c2">x0</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font> ;<br/><span class="kw">reconsider </span><font color="Maroon" title="c12">r2</font> = <span class="p1">(<span class="default"><a href="euclid_7.html#K4" title="EUCLID_7:func.4">accum</a> <span class="p2">(<span class="default"><a href="euclid_7.html#K7" title="EUCLID_7:func.7">ProjFinSeq</a> <font color="Maroon" title="c2">x0</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">k</font> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font> ;<br/><span class="lab"><font color="Green" title="E51">A14</font></span>: <span class="p1">(<span class="default"><a href="euclid_7.html#K7" title="EUCLID_7:func.7">ProjFinSeq</a> <font color="Maroon" title="c2">x0</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><a href="euclid_7.html#K7" title="EUCLID_7:func.7">ProjFinSeq</a> <font color="Maroon" title="c2">x0</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:58_1_1_1_1_1"><span class="lab"><font color="Green" title="E47">A10</font></span></a>, <a class="txt" href="#E4:58_1_1_1_1_1"><span class="lab"><font color="Green" title="E48">A11</font></span></a>, <a class="ref" href="finseq_4.html#T15" title="FINSEQ_4:th.15">FINSEQ_4:15</a></span>
<br/>.= 
<span class="p1"><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">|(</a><span class="default"><font color="Maroon" title="c2">x0</font>,<span class="p2">(<span class="default"><a href="euclid_7.html#K6" title="EUCLID_7:func.6">Base_FinSeq</a> (<font color="Maroon" title="c4">n2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span></span><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">)|</a></span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p1">(<span class="default"><a href="euclid_7.html#K6" title="EUCLID_7:func.6">Base_FinSeq</a> (<font color="Maroon" title="c4">n2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:58_1_1_1"><span class="lab"><font color="Green" title="E41">A4</font></span></a>, <a class="txt" href="#E3:58_1_1_1_1_1"><span class="lab"><font color="Green" title="E47">A10</font></span></a>, <a class="txt" href="#E4:58_1_1_1_1_1"><span class="lab"><font color="Green" title="E48">A11</font></span></a>, <a class="ref" href="euclid_7.html#D12" target="_self" title="EUCLID_7:def.12">Def12</a></span>
;<br/><a NAME="E5:58_1_1_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E52">A15</font></span>: 
<font color="Maroon" title="c6">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T29" title="XREAL_1:th.29">XREAL_1:29</a></span>;<br/><a NAME="E6:58_1_1_1_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E53">A16</font></span>: 
<font color="Maroon" title="c6">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="euclid_7.html#K7" title="EUCLID_7:func.7">ProjFinSeq</a> <font color="Maroon" title="c2">x0</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:58_1_1_1_1_1"><span class="lab"><font color="Green" title="E48">A11</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E7:58_1_1_1_1_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c9">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid_7.html#K4" title="EUCLID_7:func.4">accum</a> <span class="p3">(<span class="default"><a href="euclid_7.html#K7" title="EUCLID_7:func.7">ProjFinSeq</a> <font color="Maroon" title="c2">x0</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid_7.html#K7" title="EUCLID_7:func.7">ProjFinSeq</a> <font color="Maroon" title="c2">x0</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="euclid_7.html#D10" target="_self" title="EUCLID_7:def.10">Def10</a>, <a class="txt" href="#E1:58_1_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E50">A13</font></span></a></span>;<br/><a NAME="E8:58_1_1_1_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E54">A17</font></span>: 
<font color="Maroon" title="c9">r</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">r2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">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="c11">r3</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">i</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rvsum_1.html#T11" title="RVSUM_1:th.11">RVSUM_1:11</a></span>;<br/><div><span class="lab"><font color="Green" title="E55">A18</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1_1_1_1_1_1_1_1_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:58_1_1_1_1_1_1_1_1_2"/><span class="lab"><font color="Green" title="E56">A19</font></span>: 
<font color="Maroon" title="c8">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ; <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"><span class="p2">(<span class="default"><a href="euclid_7.html#K4" title="EUCLID_7:func.4">accum</a> <span class="p3">(<span class="default"><a href="euclid_7.html#K7" title="EUCLID_7:func.7">ProjFinSeq</a> <font color="Maroon" title="c2">x0</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x0</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">i</font></span><br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="58_1_1_1_1_1_1_1_1_2_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="E1:58_1_1_1_1_1_1_1_1_2_1"/>
( <font color="Maroon" title="c8">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 or <font color="Maroon" title="c8">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:58_1_1_1_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E56">A19</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="58_1_1_1_1_1_1_1_1_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:58_1_1_1_1_1_1_1_1_2_1_1"/><span class="lab"><font color="Green" title="E57">A20</font></span>: 
<font color="Maroon" title="c8">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ; <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"><span class="p2">(<span class="default"><a href="euclid_7.html#K4" title="EUCLID_7:func.4">accum</a> <span class="p3">(<span class="default"><a href="euclid_7.html#K7" title="EUCLID_7:func.7">ProjFinSeq</a> <font color="Maroon" title="c2">x0</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x0</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">i</font></span><br/><div class="add"><a NAME="E2:58_1_1_1_1_1_1_1_1_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E58">A21</font></span>: 
<font color="Maroon" title="c8">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">k</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid_7.html#K7" title="EUCLID_7:func.7">ProjFinSeq</a> <font color="Maroon" title="c2">x0</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font> = 
<span class="p1"><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">|(</a><span class="default"><font color="Maroon" title="c2">x0</font>,<span class="p2">(<span class="default"><a href="euclid_7.html#K6" title="EUCLID_7:func.6">Base_FinSeq</a> (<font color="Maroon" title="c4">n2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span></span><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">)|</a></span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid_7.html#K6" title="EUCLID_7:func.6">Base_FinSeq</a> (<font color="Maroon" title="c4">n2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c7">k2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">i2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:58_1_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E51">A14</font></span></a>, <a class="ref" href="rvsum_1.html#T44" title="RVSUM_1:th.44">RVSUM_1:44</a></span>
<br/>.= 
<span class="p1"><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">|(</a><span class="default"><font color="Maroon" title="c2">x0</font>,<span class="p2">(<span class="default"><a href="euclid_7.html#K6" title="EUCLID_7:func.6">Base_FinSeq</a> (<font color="Maroon" title="c4">n2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span></span><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">)|</a></span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:58_1_1_1"><span class="lab"><font color="Green" title="E41">A4</font></span></a>, <a class="txt" href="#E1:58_1_1_1_1_1"><span class="lab"><font color="Green" title="E45">A8</font></span></a>, <a class="txt" href="#E2:58_1_1_1_1_1"><span class="lab"><font color="Green" title="E46">A9</font></span></a>, <a class="txt" href="#E1:58_1_1_1_1_1_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E57">A20</font></span></a>, <a class="ref" href="matrixr2.html#T76" title="MATRIXR2:th.76">MATRIXR2:76</a></span>
;<br/><span class="kw">hence </span><a NAME="E4:58_1_1_1_1_1_1_1_1_2_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid_7.html#K4" title="EUCLID_7:func.4">accum</a> <span class="p3">(<span class="default"><a href="euclid_7.html#K7" title="EUCLID_7:func.7">ProjFinSeq</a> <font color="Maroon" title="c2">x0</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x0</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:58_1_1_1_1"><span class="lab"><font color="Green" title="E44">A7</font></span></a>, <a class="txt" href="#E1:58_1_1_1_1_1"><span class="lab"><font color="Green" title="E45">A8</font></span></a>, <a class="txt" href="#E2:58_1_1_1_1_1"><span class="lab"><font color="Green" title="E46">A9</font></span></a>, <a class="txt" href="#E8:58_1_1_1_1_1"><span class="lab"><font color="Green" title="E49">A12</font></span></a>, <a class="txt" href="#E1:58_1_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E50">A13</font></span></a>, <a class="txt" href="#E6:58_1_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E53">A16</font></span></a>, <a class="txt" href="#E8:58_1_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E54">A17</font></span></a>, <a class="txt" href="#E2:58_1_1_1_1_1_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E58">A21</font></span></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="58_1_1_1_1_1_1_1_1_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:58_1_1_1_1_1_1_1_1_2_1_2"/><span class="lab"><font color="Green" title="E57">A22</font></span>: 
<font color="Maroon" title="c8">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ; <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"><span class="p2">(<span class="default"><a href="euclid_7.html#K4" title="EUCLID_7:func.4">accum</a> <span class="p3">(<span class="default"><a href="euclid_7.html#K7" title="EUCLID_7:func.7">ProjFinSeq</a> <font color="Maroon" title="c2">x0</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x0</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">i</font></span><br/><div class="add"><a NAME="E2:58_1_1_1_1_1_1_1_1_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E58">A23</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid_7.html#K4" title="EUCLID_7:func.4">accum</a> <span class="p3">(<span class="default"><a href="euclid_7.html#K7" title="EUCLID_7:func.7">ProjFinSeq</a> <font color="Maroon" title="c2">x0</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.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="#E2:58_1_1_1_1"><span class="lab"><font color="Green" title="E44">A7</font></span></a>, <a class="txt" href="#E1:58_1_1_1_1_1"><span class="lab"><font color="Green" title="E45">A8</font></span></a>, <a class="txt" href="#E2:58_1_1_1_1_1"><span class="lab"><font color="Green" title="E46">A9</font></span></a>, <a class="txt" href="#E1:58_1_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E50">A13</font></span></a>, <a class="txt" href="#E5:58_1_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E52">A15</font></span></a>, <a class="txt" href="#E6:58_1_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E53">A16</font></span></a></span>;<br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid_7.html#K7" title="EUCLID_7:func.7">ProjFinSeq</a> <font color="Maroon" title="c2">x0</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font> = 
<span class="p1"><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">|(</a><span class="default"><font color="Maroon" title="c2">x0</font>,<span class="p2">(<span class="default"><a href="euclid_7.html#K6" title="EUCLID_7:func.6">Base_FinSeq</a> (<font color="Maroon" title="c4">n2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span></span><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">)|</a></span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid_7.html#K6" title="EUCLID_7:func.6">Base_FinSeq</a> (<font color="Maroon" title="c4">n2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c7">k2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">i2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:58_1_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E51">A14</font></span></a>, <a class="ref" href="rvsum_1.html#T44" title="RVSUM_1:th.44">RVSUM_1:44</a></span>
<br/>.= 
<span class="p1"><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">|(</a><span class="default"><font color="Maroon" title="c2">x0</font>,<span class="p2">(<span class="default"><a href="euclid_7.html#K6" title="EUCLID_7:func.6">Base_FinSeq</a> (<font color="Maroon" title="c4">n2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span></span><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">)|</a></span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:58_1_1_1"><span class="lab"><font color="Green" title="E41">A4</font></span></a>, <a class="txt" href="#E1:58_1_1_1_1_1"><span class="lab"><font color="Green" title="E45">A8</font></span></a>, <a class="txt" href="#E2:58_1_1_1_1_1"><span class="lab"><font color="Green" title="E46">A9</font></span></a>, <a class="txt" href="#E1:58_1_1_1_1_1_1_1_1_2_1_2"><span class="lab"><font color="Green" title="E57">A22</font></span></a>, <a class="ref" href="matrixr2.html#T75" title="MATRIXR2:th.75">MATRIXR2:75</a></span>
<br/>.= 
<font color="Maroon" title="c2">x0</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:58_1_1_1"><span class="lab"><font color="Green" title="E41">A4</font></span></a>, <a class="txt" href="#E3:58_1_1_1_1_1"><span class="lab"><font color="Green" title="E47">A10</font></span></a>, <a class="txt" href="#E4:58_1_1_1_1_1"><span class="lab"><font color="Green" title="E48">A11</font></span></a>, <a class="ref" href="euclid_7.html#T30" target="_self" title="EUCLID_7:th.30">Th32</a></span>
;<br/><span class="kw">hence </span><a NAME="E4:58_1_1_1_1_1_1_1_1_2_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid_7.html#K4" title="EUCLID_7:func.4">accum</a> <span class="p3">(<span class="default"><a href="euclid_7.html#K7" title="EUCLID_7:func.7">ProjFinSeq</a> <font color="Maroon" title="c2">x0</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x0</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:58_1_1_1"><span class="lab"><font color="Green" title="E39">A2</font></span></a>, <a class="txt" href="#E1:58_1_1_1_1_1"><span class="lab"><font color="Green" title="E45">A8</font></span></a>, <a class="txt" href="#E2:58_1_1_1_1_1"><span class="lab"><font color="Green" title="E46">A9</font></span></a>, <a class="txt" href="#E8:58_1_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E54">A17</font></span></a>, <a class="txt" href="#E1:58_1_1_1_1_1_1_1_1_2_1_2"><span class="lab"><font color="Green" title="E57">A22</font></span></a>, <a class="txt" href="#E2:58_1_1_1_1_1_1_1_1_2_1_2"><span class="lab"><font color="Green" title="E58">A23</font></span></a>, <a class="ref" href="finseq_4.html#T15" title="FINSEQ_4:th.15">FINSEQ_4:15</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><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1_1_1_1_1_1_1_1_3"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:58_1_1_1_1_1_1_1_1_3"/><span class="lab"><font color="Green" title="E56">A24</font></span>: 
<font color="Maroon" title="c8">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ; <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"><span class="p2">(<span class="default"><a href="euclid_7.html#K4" title="EUCLID_7:func.4">accum</a> <span class="p3">(<span class="default"><a href="euclid_7.html#K7" title="EUCLID_7:func.7">ProjFinSeq</a> <font color="Maroon" title="c2">x0</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><br/><a NAME="E2:58_1_1_1_1_1_1_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E57">A25</font></span>: 
<font color="Maroon" title="c8">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c6">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:58_1_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E52">A15</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid_7.html#K7" title="EUCLID_7:func.7">ProjFinSeq</a> <font color="Maroon" title="c2">x0</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font> = 
<span class="p1"><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">|(</a><span class="default"><font color="Maroon" title="c2">x0</font>,<span class="p2">(<span class="default"><a href="euclid_7.html#K6" title="EUCLID_7:func.6">Base_FinSeq</a> (<font color="Maroon" title="c4">n2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span></span><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">)|</a></span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid_7.html#K6" title="EUCLID_7:func.6">Base_FinSeq</a> (<font color="Maroon" title="c4">n2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c7">k2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">i2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:58_1_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E51">A14</font></span></a>, <a class="ref" href="rvsum_1.html#T44" title="RVSUM_1:th.44">RVSUM_1:44</a></span>
<br/>.= 
<span class="p1"><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">|(</a><span class="default"><font color="Maroon" title="c2">x0</font>,<span class="p2">(<span class="default"><a href="euclid_7.html#K6" title="EUCLID_7:func.6">Base_FinSeq</a> (<font color="Maroon" title="c4">n2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span></span><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">)|</a></span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:58_1_1_1"><span class="lab"><font color="Green" title="E41">A4</font></span></a>, <a class="txt" href="#E1:58_1_1_1_1_1"><span class="lab"><font color="Green" title="E45">A8</font></span></a>, <a class="txt" href="#E2:58_1_1_1_1_1"><span class="lab"><font color="Green" title="E46">A9</font></span></a>, <a class="txt" href="#E1:58_1_1_1_1_1_1_1_1_3"><span class="lab"><font color="Green" title="E56">A24</font></span></a>, <a class="ref" href="matrixr2.html#T76" title="MATRIXR2:th.76">MATRIXR2:76</a></span>
<br/>.= 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 

;<br/><span class="kw">hence </span><a NAME="E4:58_1_1_1_1_1_1_1_1_3"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid_7.html#K4" title="EUCLID_7:func.4">accum</a> <span class="p3">(<span class="default"><a href="euclid_7.html#K7" title="EUCLID_7:func.7">ProjFinSeq</a> <font color="Maroon" title="c2">x0</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.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="#E2:58_1_1_1_1"><span class="lab"><font color="Green" title="E44">A7</font></span></a>, <a class="txt" href="#E1:58_1_1_1_1_1"><span class="lab"><font color="Green" title="E45">A8</font></span></a>, <a class="txt" href="#E2:58_1_1_1_1_1"><span class="lab"><font color="Green" title="E46">A9</font></span></a>, <a class="txt" href="#E8:58_1_1_1_1_1"><span class="lab"><font color="Green" title="E49">A12</font></span></a>, <a class="txt" href="#E1:58_1_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E50">A13</font></span></a>, <a class="txt" href="#E6:58_1_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E53">A16</font></span></a>, <a class="txt" href="#E8:58_1_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E54">A17</font></span></a>, <a class="txt" href="#E2:58_1_1_1_1_1_1_1_1_3"><span class="lab"><font color="Green" title="E57">A25</font></span></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="E11:58_1_1_1_1_1_1_1_1"/>
( ( <font color="Maroon" title="c8">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid_7.html#K4" title="EUCLID_7:func.4">accum</a> <span class="p3">(<span class="default"><a href="euclid_7.html#K7" title="EUCLID_7:func.7">ProjFinSeq</a> <font color="Maroon" title="c2">x0</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x0</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">i</font> ) &amp; ( <font color="Maroon" title="c8">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid_7.html#K4" title="EUCLID_7:func.4">accum</a> <span class="p3">(<span class="default"><a href="euclid_7.html#K7" title="EUCLID_7:func.7">ProjFinSeq</a> <font color="Maroon" title="c2">x0</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.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="#E9:58_1_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E55">A18</font></span></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="58_1_1_1_1_1_1_1_2"><span class="kw">case </span></a><a NAME="E1:58_1_1_1_1_1_1_1_2"/>
<font color="Maroon" title="c6">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1
 ; <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="c8">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid_7.html#K4" title="EUCLID_7:func.4">accum</a> <span class="p3">(<span class="default"><a href="euclid_7.html#K7" title="EUCLID_7:func.7">ProjFinSeq</a> <font color="Maroon" title="c2">x0</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x0</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">i</font> ) &amp; ( <font color="Maroon" title="c8">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid_7.html#K4" title="EUCLID_7:func.4">accum</a> <span class="p3">(<span class="default"><a href="euclid_7.html#K7" title="EUCLID_7:func.7">ProjFinSeq</a> <font color="Maroon" title="c2">x0</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) )</span><br/><div class="add"><a NAME="E2:58_1_1_1_1_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A26</font></span>: 
<font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E3:58_1_1_1_1_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A27</font></span>: 
<font color="Maroon" title="c6">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T6" title="XREAL_1:th.6">XREAL_1:6</a></span>;<br/><div><span class="lab"><font color="Green" title="E52">A28</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1_1_1_1_1_1_1_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:58_1_1_1_1_1_1_1_2_1"/><span class="lab"><font color="Green" title="E53">A29</font></span>: 
<font color="Maroon" title="c8">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ; <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"><span class="p2">(<span class="default"><a href="euclid_7.html#K4" title="EUCLID_7:func.4">accum</a> <span class="p3">(<span class="default"><a href="euclid_7.html#K7" title="EUCLID_7:func.7">ProjFinSeq</a> <font color="Maroon" title="c2">x0</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><br/><a NAME="E2:58_1_1_1_1_1_1_1_2_1"/>
<span class="p1">(<span class="default"><a href="euclid_7.html#K4" title="EUCLID_7:func.4">accum</a> <span class="p2">(<span class="default"><a href="euclid_7.html#K7" title="EUCLID_7:func.7">ProjFinSeq</a> <font color="Maroon" title="c2">x0</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="euclid_7.html#K7" title="EUCLID_7:func.7">ProjFinSeq</a> <font color="Maroon" title="c2">x0</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:58_1_1_1"><span class="lab"><font color="Green" title="E42">A5</font></span></a>, <a class="txt" href="#E2:58_1_1_1"><span class="lab"><font color="Green" title="E39">A2</font></span></a>, <a class="txt" href="#E3:58_1_1_1"><span class="lab"><font color="Green" title="E40">A3</font></span></a>, <a class="ref" href="finseq_4.html#T15" title="FINSEQ_4:th.15">FINSEQ_4:15</a></span>;<br/><span class="kw">then </span><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid_7.html#K4" title="EUCLID_7:func.4">accum</a> <span class="p3">(<span class="default"><a href="euclid_7.html#K7" title="EUCLID_7:func.7">ProjFinSeq</a> <font color="Maroon" title="c2">x0</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font> = 
<span class="p1">(<span class="default"><span class="p2"><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">|(</a><span class="default"><font color="Maroon" title="c2">x0</font>,<span class="p3">(<span class="default"><a href="euclid_7.html#K6" title="EUCLID_7:func.6">Base_FinSeq</a> (<font color="Maroon" title="c4">n2</font>,1)</span>)</span></span><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">)|</a></span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p2">(<span class="default"><a href="euclid_7.html#K6" title="EUCLID_7:func.6">Base_FinSeq</a> (<font color="Maroon" title="c4">n2</font>,1)</span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">i</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:58_1_1_1"><span class="lab"><font color="Green" title="E42">A5</font></span></a>, <a class="txt" href="#E4:58_1_1_1"><span class="lab"><font color="Green" title="E41">A4</font></span></a>, <a class="ref" href="euclid_7.html#D12" target="_self" title="EUCLID_7:def.12">Def12</a></span>
<br/>.= 
<span class="p1"><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">|(</a><span class="default"><font color="Maroon" title="c2">x0</font>,<span class="p2">(<span class="default"><a href="euclid_7.html#K6" title="EUCLID_7:func.6">Base_FinSeq</a> (<font color="Maroon" title="c4">n2</font>,1)</span>)</span></span><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">)|</a></span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid_7.html#K6" title="EUCLID_7:func.6">Base_FinSeq</a> (<font color="Maroon" title="c4">n2</font>,1)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">i2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rvsum_1.html#T44" title="RVSUM_1:th.44">RVSUM_1:44</a></span>
<br/>.= 
<span class="p1"><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">|(</a><span class="default"><font color="Maroon" title="c2">x0</font>,<span class="p2">(<span class="default"><a href="euclid_7.html#K6" title="EUCLID_7:func.6">Base_FinSeq</a> (<font color="Maroon" title="c4">n2</font>,1)</span>)</span></span><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">)|</a></span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:58_1_1_1"><span class="lab"><font color="Green" title="E41">A4</font></span></a>, <a class="txt" href="#E2:58_1_1_1_1_1"><span class="lab"><font color="Green" title="E46">A9</font></span></a>, <a class="txt" href="#E1:58_1_1_1_1_1_1_1_2_1"><span class="lab"><font color="Green" title="E53">A29</font></span></a>, <a class="ref" href="matrixr2.html#T76" title="MATRIXR2:th.76">MATRIXR2:76</a></span>
<br/>.= 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 

;<br/><span class="kw">hence </span><a NAME="E4:58_1_1_1_1_1_1_1_2_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid_7.html#K4" title="EUCLID_7:func.4">accum</a> <span class="p3">(<span class="default"><a href="euclid_7.html#K7" title="EUCLID_7:func.7">ProjFinSeq</a> <font color="Maroon" title="c2">x0</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.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="#E3:58_1_1_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E51">A27</font></span></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="lab"><font color="Green" title="E53">A30</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1_1_1_1_1_1_1_2_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:58_1_1_1_1_1_1_1_2_2"/>
<font color="Maroon" title="c8">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ; <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"><span class="p2">(<span class="default"><a href="euclid_7.html#K4" title="EUCLID_7:func.4">accum</a> <span class="p3">(<span class="default"><a href="euclid_7.html#K7" title="EUCLID_7:func.7">ProjFinSeq</a> <font color="Maroon" title="c2">x0</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x0</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">i</font></span><br/><a NAME="E2:58_1_1_1_1_1_1_1_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E54">A31</font></span>: 
<font color="Maroon" title="c8">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:58_1_1_1_1_1"><span class="lab"><font color="Green" title="E45">A8</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><a NAME="E3:58_1_1_1_1_1_1_1_2_2"/>
<span class="p1">(<span class="default"><a href="euclid_7.html#K4" title="EUCLID_7:func.4">accum</a> <span class="p2">(<span class="default"><a href="euclid_7.html#K7" title="EUCLID_7:func.7">ProjFinSeq</a> <font color="Maroon" title="c2">x0</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="euclid_7.html#K7" title="EUCLID_7:func.7">ProjFinSeq</a> <font color="Maroon" title="c2">x0</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:58_1_1_1"><span class="lab"><font color="Green" title="E42">A5</font></span></a>, <a class="txt" href="#E2:58_1_1_1"><span class="lab"><font color="Green" title="E39">A2</font></span></a>, <a class="txt" href="#E3:58_1_1_1"><span class="lab"><font color="Green" title="E40">A3</font></span></a>, <a class="ref" href="finseq_4.html#T15" title="FINSEQ_4:th.15">FINSEQ_4:15</a></span>;<br/><span class="kw">then </span><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid_7.html#K4" title="EUCLID_7:func.4">accum</a> <span class="p3">(<span class="default"><a href="euclid_7.html#K7" title="EUCLID_7:func.7">ProjFinSeq</a> <font color="Maroon" title="c2">x0</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 = 
<span class="p1">(<span class="default"><span class="p2"><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">|(</a><span class="default"><font color="Maroon" title="c2">x0</font>,<span class="p3">(<span class="default"><a href="euclid_7.html#K6" title="EUCLID_7:func.6">Base_FinSeq</a> (<font color="Maroon" title="c4">n2</font>,1)</span>)</span></span><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">)|</a></span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p2">(<span class="default"><a href="euclid_7.html#K6" title="EUCLID_7:func.6">Base_FinSeq</a> (<font color="Maroon" title="c4">n2</font>,1)</span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:58_1_1_1"><span class="lab"><font color="Green" title="E42">A5</font></span></a>, <a class="txt" href="#E4:58_1_1_1"><span class="lab"><font color="Green" title="E41">A4</font></span></a>, <a class="ref" href="euclid_7.html#D12" target="_self" title="EUCLID_7:def.12">Def12</a></span>
<br/>.= 
<span class="p1"><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">|(</a><span class="default"><font color="Maroon" title="c2">x0</font>,<span class="p2">(<span class="default"><a href="euclid_7.html#K6" title="EUCLID_7:func.6">Base_FinSeq</a> (<font color="Maroon" title="c4">n2</font>,1)</span>)</span></span><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">)|</a></span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid_7.html#K6" title="EUCLID_7:func.6">Base_FinSeq</a> (<font color="Maroon" title="c4">n2</font>,1)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rvsum_1.html#T44" title="RVSUM_1:th.44">RVSUM_1:44</a></span>
<br/>.= 
<span class="p1"><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">|(</a><span class="default"><font color="Maroon" title="c2">x0</font>,<span class="p2">(<span class="default"><a href="euclid_7.html#K6" title="EUCLID_7:func.6">Base_FinSeq</a> (<font color="Maroon" title="c4">n2</font>,1)</span>)</span></span><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">)|</a></span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:58_1_1_1"><span class="lab"><font color="Green" title="E42">A5</font></span></a>, <a class="txt" href="#E4:58_1_1_1"><span class="lab"><font color="Green" title="E41">A4</font></span></a>, <a class="ref" href="matrixr2.html#T75" title="MATRIXR2:th.75">MATRIXR2:75</a></span>
<br/>.= 
<font color="Maroon" title="c2">x0</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:58_1_1_1"><span class="lab"><font color="Green" title="E42">A5</font></span></a>, <a class="txt" href="#E4:58_1_1_1"><span class="lab"><font color="Green" title="E41">A4</font></span></a>, <a class="ref" href="euclid_7.html#T30" target="_self" title="EUCLID_7:th.30">Th32</a></span>
;<br/><span class="kw">hence </span><a NAME="E5:58_1_1_1_1_1_1_1_2_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid_7.html#K4" title="EUCLID_7:func.4">accum</a> <span class="p3">(<span class="default"><a href="euclid_7.html#K7" title="EUCLID_7:func.7">ProjFinSeq</a> <font color="Maroon" title="c2">x0</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x0</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:58_1_1_1_1_1_1_1_2_2"><span class="lab"><font color="Green" title="E54">A31</font></span></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="E6:58_1_1_1_1_1_1_1_2"/>
<font color="Maroon" title="c6">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:58_1_1_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E50">A26</font></span></a>, <a class="ref" href="xreal_1.html#T6" title="XREAL_1:th.6">XREAL_1:6</a></span>;<br/><span class="kw">hence </span><a NAME="E7:58_1_1_1_1_1_1_1_2"/>
( ( <font color="Maroon" title="c8">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid_7.html#K4" title="EUCLID_7:func.4">accum</a> <span class="p3">(<span class="default"><a href="euclid_7.html#K7" title="EUCLID_7:func.7">ProjFinSeq</a> <font color="Maroon" title="c2">x0</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x0</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">i</font> ) &amp; ( <font color="Maroon" title="c8">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid_7.html#K4" title="EUCLID_7:func.4">accum</a> <span class="p3">(<span class="default"><a href="euclid_7.html#K7" title="EUCLID_7:func.7">ProjFinSeq</a> <font color="Maroon" title="c2">x0</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.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="#E5:58_1_1_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E53">A30</font></span></a>, <a class="txt" href="#E4:58_1_1_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E52">A28</font></span></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="E10:58_1_1_1_1_1"/>
( ( <font color="Maroon" title="c8">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid_7.html#K4" title="EUCLID_7:func.4">accum</a> <span class="p3">(<span class="default"><a href="euclid_7.html#K7" title="EUCLID_7:func.7">ProjFinSeq</a> <font color="Maroon" title="c2">x0</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x0</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">i</font> ) &amp; ( <font color="Maroon" title="c8">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid_7.html#K4" title="EUCLID_7:func.4">accum</a> <span class="p3">(<span class="default"><a href="euclid_7.html#K7" title="EUCLID_7:func.7">ProjFinSeq</a> <font color="Maroon" title="c2">x0</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</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"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E4:58_1_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 ; <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">reconsider </span><font color="Maroon" title="c6">r4</font> = <span class="p1">(<span class="default"><a href="euclid_7.html#K4" title="EUCLID_7:func.4">accum</a> <span class="p2">(<span class="default"><a href="euclid_7.html#K7" title="EUCLID_7:func.7">ProjFinSeq</a> <font color="Maroon" title="c2">x0</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</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="euclid_7.html#K7" title="EUCLID_7:func.7">ProjFinSeq</a> <font color="Maroon" title="c2">x0</font></span>)</span></span>)</span> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font> ;<br/><a NAME="E8:58_1_1_1"/><span class="lab"><font color="Green" title="E44">A32</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#D7" title="CARD_1:def.7">CARD_1:def 7</a></span>;<br/><a NAME="E9:58_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E45">A33</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">r4</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#D7" title="CARD_1:def.7">CARD_1:def 7</a></span>;<br/><a NAME="E10:58_1_1_1"/><span class="lab"><font color="Green" title="E46">A34</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 ;<br/><a NAME="E11:58_1_1_1"/><span class="lab"><font color="Green" title="E47">A35</font></span>: 
 for <font color="Olive" title="b1">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S2" title="NAT_1:sch.2">NAT_1:sch 2</a>(<a class="txt" href="#E10:58_1_1_1"><span class="lab"><font color="Green" title="E46">A34</font></span></a>, <a class="txt" href="#E6:58_1_1_1"><span class="lab"><font color="Green" title="E43">A6</font></span></a>);<br/></span><a NAME="E12:58_1_1_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>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">r4</font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid_7.html#K4" title="EUCLID_7:func.4">accum</a> <span class="p3">(<span class="default"><a href="euclid_7.html#K7" title="EUCLID_7:func.7">ProjFinSeq</a> <font color="Maroon" title="c2">x0</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="euclid_7.html#K7" title="EUCLID_7:func.7">ProjFinSeq</a> <font color="Maroon" title="c2">x0</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x0</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1_1_1_2"><span class="kw">proof </span></a><div class="add">

<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>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">r4</font> implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid_7.html#K4" title="EUCLID_7:func.4">accum</a> <span class="p3">(<span class="default"><a href="euclid_7.html#K7" title="EUCLID_7:func.7">ProjFinSeq</a> <font color="Maroon" title="c2">x0</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="euclid_7.html#K7" title="EUCLID_7:func.7">ProjFinSeq</a> <font color="Maroon" title="c2">x0</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_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="c2">x0</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:58_1_1_1_2"/><span class="lab"><font color="Green" title="E48">A36</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="E2:58_1_1_1_2"/><span class="lab"><font color="Green" title="E49">A37</font></span>: 
<font color="Maroon" title="c7">i</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> <font color="Maroon" title="c6">r4</font>
 ; <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"><span class="p2">(<span class="default"><a href="euclid_7.html#K4" title="EUCLID_7:func.4">accum</a> <span class="p3">(<span class="default"><a href="euclid_7.html#K7" title="EUCLID_7:func.7">ProjFinSeq</a> <font color="Maroon" title="c2">x0</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="euclid_7.html#K7" title="EUCLID_7:func.7">ProjFinSeq</a> <font color="Maroon" title="c2">x0</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_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="c2">x0</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">i</font></span><br/>

<a NAME="E3:58_1_1_1_2"/><span class="lab"><font color="Green" title="E50">A38</font></span>: 
<font color="Maroon" title="c7">i</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="euclid_7.html#K7" title="EUCLID_7:func.7">ProjFinSeq</a> <font color="Maroon" title="c2">x0</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:58_1_1_1"><span class="lab"><font color="Green" title="E41">A4</font></span></a>, <a class="txt" href="#E2:58_1_1_1_2"><span class="lab"><font color="Green" title="E49">A37</font></span></a>, <a class="ref" href="card_1.html#D7" title="CARD_1:def.7">CARD_1:def 7</a></span>;<br/>
<a NAME="E4:58_1_1_1_2"/>
1 <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="euclid_7.html#K7" title="EUCLID_7:func.7">ProjFinSeq</a> <font color="Maroon" title="c2">x0</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:58_1_1_1"><span class="lab"><font color="Green" title="E41">A4</font></span></a>, <a class="txt" href="#E8:58_1_1_1"><span class="lab"><font color="Green" title="E44">A32</font></span></a>, <a class="txt" href="#E9:58_1_1_1"><span class="lab"><font color="Green" title="E45">A33</font></span></a>, <a class="txt" href="#E1:58_1_1_1_2"><span class="lab"><font color="Green" title="E48">A36</font></span></a>, <a class="txt" href="#E2:58_1_1_1_2"><span class="lab"><font color="Green" title="E49">A37</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:58_1_1_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid_7.html#K4" title="EUCLID_7:func.4">accum</a> <span class="p3">(<span class="default"><a href="euclid_7.html#K7" title="EUCLID_7:func.7">ProjFinSeq</a> <font color="Maroon" title="c2">x0</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="euclid_7.html#K7" title="EUCLID_7:func.7">ProjFinSeq</a> <font color="Maroon" title="c2">x0</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_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="c2">x0</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="#E11:58_1_1_1"><span class="lab"><font color="Green" title="E47">A35</font></span></a>, <a class="txt" href="#E1:58_1_1_1_2"><span class="lab"><font color="Green" title="E48">A36</font></span></a>, <a class="txt" href="#E3:58_1_1_1_2"><span class="lab"><font color="Green" title="E50">A38</font></span></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="E13:58_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="euclid_7.html#K4" title="EUCLID_7:func.4">accum</a> <span class="p2">(<span class="default"><a href="euclid_7.html#K7" title="EUCLID_7:func.7">ProjFinSeq</a> <font color="Maroon" title="c2">x0</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</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="euclid_7.html#K7" title="EUCLID_7:func.7">ProjFinSeq</a> <font color="Maroon" title="c2">x0</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:58_1_1_1"><span class="lab"><font color="Green" title="E45">A33</font></span></a>, <a class="ref" href="finseq_1.html#T14" title="FINSEQ_1:th.14">FINSEQ_1:14</a></span>;<br/><span class="kw">hence </span><a NAME="E14:58_1_1_1"/>
<font color="Maroon" title="c2">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="euclid_7.html#K4" title="EUCLID_7:func.4">accum</a> <span class="p2">(<span class="default"><a href="euclid_7.html#K7" title="EUCLID_7:func.7">ProjFinSeq</a> <font color="Maroon" title="c2">x0</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</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="euclid_7.html#K7" title="EUCLID_7:func.7">ProjFinSeq</a> <font color="Maroon" title="c2">x0</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:58_1_1_1"><span class="lab"><font color="Green" title="E42">A5</font></span></a>, <a class="txt" href="#E2:58_1_1_1"><span class="lab"><font color="Green" title="E39">A2</font></span></a>, <a class="ref" href="finseq_4.html#T15" title="FINSEQ_4:th.15">FINSEQ_4:15</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="58_1_1_2"><span class="kw">case </span></a><a NAME="E1:58_1_1_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="euclid_7.html#K7" title="EUCLID_7:func.7">ProjFinSeq</a> <font color="Maroon" title="c2">x0</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</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"> <font color="Maroon" title="c2">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">n</font></span><br/><div class="add"><a NAME="E2:58_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E38">A39</font></span>: 
<font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="euclid_7.html#D12" target="_self" title="EUCLID_7:def.12">Def12</a></span>;<br/><a NAME="E3:58_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c2">x0</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>
 ;<br/><span class="kw">hence </span><a NAME="E4:58_1_1_2"/>
<font color="Maroon" title="c2">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:58_1_1_2"><span class="lab"><font color="Green" title="E38">A39</font></span></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="E3:58"/>
<font color="Maroon" title="c2">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_7.html#K5" title="EUCLID_7:func.5">Sum</a> <span class="p1">(<span class="default"><a href="euclid_7.html#K7" title="EUCLID_7:func.7">ProjFinSeq</a> <font color="Maroon" title="c2">x0</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="euclid_7.html#D11" target="_self" title="EUCLID_7:def.11">Def11</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>
