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<b>let </b><font color="Maroon" title="c1">a</font>, <font color="Maroon" title="c2">b</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">a</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="c2">b</font> &amp; <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">a</font> holds <br/><span class="p1"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="prgcor_2.html#K6" title="PRGCOR_2:func.6">inner_prd_prg</a> <span class="p1">(<span class="default"><a href="prgcor_2.html#K3" title="PRGCOR_2:func.3">FS2XFS*</a> <font color="Maroon" title="c1">a</font>,<font color="Olive" title="b1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="prgcor_2.html#K3" title="PRGCOR_2:func.3">FS2XFS*</a> <font color="Maroon" title="c2">b</font>,<font color="Olive" title="b1">n</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c3">n2</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">a</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="c2">b</font> &amp; <font color="Maroon" title="c3">n2</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">a</font> implies <span class="p1"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="prgcor_2.html#K6" title="PRGCOR_2:func.6">inner_prd_prg</a> <span class="p1">(<span class="default"><a href="prgcor_2.html#K3" title="PRGCOR_2:func.3">FS2XFS*</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c3">n2</font></span>)</span>,<span class="p1">(<span class="default"><a href="prgcor_2.html#K3" title="PRGCOR_2:func.3">FS2XFS*</a> <font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">n2</font></span>)</span> )</span><br/>



<b>assume </b><a NAME="E1:17"/><i><font color="Green" title="E13">A1</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">a</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="c2">b</font> &amp; <font color="Maroon" title="c3">n2</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">a</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="prgcor_2.html#K6" title="PRGCOR_2:func.6">inner_prd_prg</a> <span class="p1">(<span class="default"><a href="prgcor_2.html#K3" title="PRGCOR_2:func.3">FS2XFS*</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c3">n2</font></span>)</span>,<span class="p1">(<span class="default"><a href="prgcor_2.html#K3" title="PRGCOR_2:func.3">FS2XFS*</a> <font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">n2</font></span>)</span></span><br/>

<b>set </b><font color="Maroon" title="c4">pa</font> =  <a href="prgcor_2.html#K3" title="PRGCOR_2:func.3">FS2XFS*</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c3">n2</font>;<br/>
<b>set </b><font color="Maroon" title="c5">pb</font> =  <a href="prgcor_2.html#K3" title="PRGCOR_2:func.3">FS2XFS*</a> <font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">n2</font>;<br/>
<b>set </b><font color="Maroon" title="c6">ri</font> =  <a href="prgcor_2.html#K6" title="PRGCOR_2:func.6">inner_prd_prg</a> <span class="p1">(<span class="default"><a href="prgcor_2.html#K3" title="PRGCOR_2:func.3">FS2XFS*</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c3">n2</font></span>)</span>,<span class="p1">(<span class="default"><a href="prgcor_2.html#K3" title="PRGCOR_2:func.3">FS2XFS*</a> <font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">n2</font></span>)</span>;<br/>
<a NAME="E2:17"/><i><font color="Green" title="E14">A2</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="prgcor_2.html#K3" title="PRGCOR_2:func.3">FS2XFS*</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c3">n2</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp;  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><a href="prgcor_2.html#K3" title="PRGCOR_2:func.3">FS2XFS*</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c3">n2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">n2</font> &amp; ( 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="c1">a</font> holds <br/><span class="p1">(<span class="default"><a href="prgcor_2.html#K3" title="PRGCOR_2:func.3">FS2XFS*</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c3">n2</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">a</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> ) &amp; ( for <font color="Olive" title="b1">j</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">j</font> &amp; <font color="Olive" title="b1">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">n2</font> holds <br/><span class="p1">(<span class="default"><a href="prgcor_2.html#K3" title="PRGCOR_2:func.3">FS2XFS*</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c3">n2</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) )
 
<b>by </b><i><a class="txt" href="prgcor_2.html#E1:17"><i><font color="Green" title="E13">A1</font></i></a>, <a class="ref" href="prgcor_2.html#D3" target="_self" title="PRGCOR_2:def.3">Def3</a></i>;<br/>
<a NAME="E3:17"/><i><font color="Green" title="E15">A3</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="prgcor_2.html#K3" title="PRGCOR_2:func.3">FS2XFS*</a> <font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">n2</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp;  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><a href="prgcor_2.html#K3" title="PRGCOR_2:func.3">FS2XFS*</a> <font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">n2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">n2</font> &amp; ( 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="c2">b</font> holds <br/><span class="p1">(<span class="default"><a href="prgcor_2.html#K3" title="PRGCOR_2:func.3">FS2XFS*</a> <font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">n2</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">b</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> ) &amp; ( for <font color="Olive" title="b1">j</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">b</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">j</font> &amp; <font color="Olive" title="b1">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">n2</font> holds <br/><span class="p1">(<span class="default"><a href="prgcor_2.html#K3" title="PRGCOR_2:func.3">FS2XFS*</a> <font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">n2</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) )
 
<b>by </b><i><a class="txt" href="prgcor_2.html#E1:17"><i><font color="Green" title="E13">A1</font></i></a>, <a class="ref" href="prgcor_2.html#D3" target="_self" title="PRGCOR_2:def.3">Def3</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c7">s</font> being   <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> , <font color="Maroon" title="c8">n</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a><b> such that </b><br/><a NAME="E5:17"/><i><font color="Green" title="E16">A4</font></i>: 
(  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c7">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><a href="prgcor_2.html#K3" title="PRGCOR_2:func.3">FS2XFS*</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c3">n2</font></span>)</span> &amp; <font color="Maroon" title="c7">s</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c8">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="prgcor_2.html#K3" title="PRGCOR_2:func.3">FS2XFS*</a> <font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">n2</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; ( <font color="Maroon" title="c8">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">n</font> holds <br/><font color="Maroon" title="c7">s</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">i</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"><font color="Maroon" title="c7">s</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="prgcor_2.html#K3" title="PRGCOR_2:func.3">FS2XFS*</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c3">n2</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="prgcor_2.html#K3" title="PRGCOR_2:func.3">FS2XFS*</a> <font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">n2</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span> ) &amp;  <a href="prgcor_2.html#K6" title="PRGCOR_2:func.6">inner_prd_prg</a> <span class="p1">(<span class="default"><a href="prgcor_2.html#K3" title="PRGCOR_2:func.3">FS2XFS*</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c3">n2</font></span>)</span>,<span class="p1">(<span class="default"><a href="prgcor_2.html#K3" title="PRGCOR_2:func.3">FS2XFS*</a> <font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">n2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">s</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">n</font> )
 <b>by </b><i><a class="txt" href="prgcor_2.html#E1:17"><i><font color="Green" title="E13">A1</font></i></a>, <a class="txt" href="prgcor_2.html#E2:17"><i><font color="Green" title="E14">A2</font></i></a>, <a class="ref" href="prgcor_2.html#D7" target="_self" title="PRGCOR_2:def.7">Def7</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c9">ra</font> = <font color="Maroon" title="c1">a</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  <b>by </b><i><a class="ref" href="finseq_2.html#T110" title="FINSEQ_2:th.110">FINSEQ_2:110</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c10">rb</font> = <font color="Maroon" title="c2">b</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  <b>by </b><i><a class="txt" href="prgcor_2.html#E1:17"><i><font color="Green" title="E13">A1</font></i></a>, <a class="ref" href="finseq_2.html#T110" title="FINSEQ_2:th.110">FINSEQ_2:110</a></i>;<br/>
<a NAME="E8:17"/><i><font color="Green" title="E17">A5</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15" title="RVSUM_1:func.15">mlt</a> <font color="Maroon" title="c9">ra</font>,<font color="Maroon" title="c10">rb</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> <font color="Maroon" title="c1">a</font>
 
<b>by </b><i><a class="ref" href="finseq_2.html#T109" title="FINSEQ_2:th.109">FINSEQ_2:109</a></i>;<br/>
<a NAME="E9:17"/><b>then </b><i><font color="Green" title="E18">A6</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15" title="RVSUM_1:func.15">mlt</a> <font color="Maroon" title="c9">ra</font>,<font color="Maroon" title="c10">rb</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">a</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E10:17"/>
for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">i</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="rvsum_1.html#K14" title="RVSUM_1:func.14">mlt</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span>)</span> holds <br/><font color="Maroon" title="c7">s</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">i</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"><font color="Maroon" title="c7">s</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rvsum_1.html#K14" title="RVSUM_1:func.14">mlt</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">i</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c11">i</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="rvsum_1.html#K14" title="RVSUM_1:func.14">mlt</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span>)</span> implies <font color="Maroon" title="c7">s</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">i</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"><font color="Maroon" title="c7">s</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">i</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rvsum_1.html#K14" title="RVSUM_1:func.14">mlt</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> )</span><br/>

<b>assume </b><a NAME="E1:17_1"/><i><font color="Green" title="E19">A7</font></i>: 
<font color="Maroon" title="c11">i</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="rvsum_1.html#K14" title="RVSUM_1:func.14">mlt</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c7">s</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">i</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"><font color="Maroon" title="c7">s</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">i</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rvsum_1.html#K14" title="RVSUM_1:func.14">mlt</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><br/>

<a NAME="E2:17_1"/><i><font color="Green" title="E20">A8</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/>
<a NAME="E3:17_1"/><i><font color="Green" title="E21">A9</font></i>: 
<font color="Maroon" title="c11">i</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> <font color="Maroon" title="c1">a</font>
 
<b>by </b><i><a class="txt" href="prgcor_2.html#E8:17"><i><font color="Green" title="E17">A5</font></i></a>, <a class="txt" href="prgcor_2.html#E1:17_1"><i><font color="Green" title="E19">A7</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E4:17_1"/><b>then </b><i><font color="Green" title="E22">A10</font></i>: 
<span class="p1">(<span class="default"><a href="prgcor_2.html#K3" title="PRGCOR_2:func.3">FS2XFS*</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c3">n2</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">i</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> <font color="Maroon" title="c1">a</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="prgcor_2.html#E1:17"><i><font color="Green" title="E13">A1</font></i></a>, <a class="txt" href="prgcor_2.html#E2:17_1"><i><font color="Green" title="E20">A8</font></i></a>, <a class="ref" href="prgcor_2.html#D3" target="_self" title="PRGCOR_2:def.3">Def3</a></i>;<br/>
<a NAME="E5:17_1"/><i><font color="Green" title="E23">A11</font></i>: 
<span class="p1">(<span class="default"><a href="prgcor_2.html#K3" title="PRGCOR_2:func.3">FS2XFS*</a> <font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">n2</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">i</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> <font color="Maroon" title="c2">b</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="prgcor_2.html#E1:17"><i><font color="Green" title="E13">A1</font></i></a>, <a class="txt" href="prgcor_2.html#E2:17_1"><i><font color="Green" title="E20">A8</font></i></a>, <a class="txt" href="prgcor_2.html#E3:17_1"><i><font color="Green" title="E21">A9</font></i></a>, <a class="ref" href="prgcor_2.html#D3" target="_self" title="PRGCOR_2:def.3">Def3</a></i>;<br/>
<a NAME="E6:17_1"/><i><font color="Green" title="E24">A12</font></i>: 
<font color="Maroon" title="c11">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K14" title="RVSUM_1:func.14">mlt</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span>)</span>
 
<b>by </b><i><a class="txt" href="prgcor_2.html#E9:17"><i><font color="Green" title="E18">A6</font></i></a>, <a class="txt" href="prgcor_2.html#E2:17_1"><i><font color="Green" title="E20">A8</font></i></a>, <a class="txt" href="prgcor_2.html#E3:17_1"><i><font color="Green" title="E21">A9</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E7:17_1"/><i><font color="Green" title="E25">A13</font></i>: 
<font color="Maroon" title="c11">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">n</font>
 
<b>by </b><i><a class="txt" href="prgcor_2.html#E1:17"><i><font color="Green" title="E13">A1</font></i></a>, <a class="txt" href="prgcor_2.html#E5:17"><i><font color="Green" title="E16">A4</font></i></a>, <a class="txt" href="prgcor_2.html#E8:17"><i><font color="Green" title="E17">A5</font></i></a>, <a class="txt" href="prgcor_2.html#E1:17_1"><i><font color="Green" title="E19">A7</font></i></a>, <a class="ref" href="prgcor_2.html#D3" target="_self" title="PRGCOR_2:def.3">Def3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:17_1_1_1"/>
( <font color="Maroon" title="c8">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c8">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_1_1"><b>case </b></a><a NAME="E1:17_1_1_1_1"/>
<font color="Maroon" title="c8">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c7">s</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">i</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"><font color="Maroon" title="c7">s</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">i</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rvsum_1.html#K14" title="RVSUM_1:func.14">mlt</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><br/><div class="add"><font color="Maroon" title="c7">s</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">s</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">i</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="prgcor_2.html#K3" title="PRGCOR_2:func.3">FS2XFS*</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c3">n2</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="prgcor_2.html#K3" title="PRGCOR_2:func.3">FS2XFS*</a> <font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">n2</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="prgcor_2.html#E5:17"><i><font color="Green" title="E16">A4</font></i></a>, <a class="txt" href="prgcor_2.html#E7:17_1"><i><font color="Green" title="E25">A13</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">s</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">i</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rvsum_1.html#K14" title="RVSUM_1:func.14">mlt</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="prgcor_2.html#E4:17_1"><i><font color="Green" title="E22">A10</font></i></a>, <a class="txt" href="prgcor_2.html#E5:17_1"><i><font color="Green" title="E23">A11</font></i></a>, <a class="txt" href="prgcor_2.html#E6:17_1"><i><font color="Green" title="E24">A12</font></i></a>, <a class="ref" href="rvsum_1.html#T86" title="RVSUM_1:th.86">RVSUM_1:86</a></i>
;<br/><b>hence </b><a NAME="E3:17_1_1_1_1"/>
<font color="Maroon" title="c7">s</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">i</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"><font color="Maroon" title="c7">s</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">i</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rvsum_1.html#K14" title="RVSUM_1:func.14">mlt</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_1_2"><b>case </b></a><a NAME="E1:17_1_1_1_2"/>
<font color="Maroon" title="c8">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><div class="add"><b>hence </b><a NAME="E2:17_1_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="prgcor_2.html#E1:17"><i><font color="Green" title="E13">A1</font></i></a>, <a class="txt" href="prgcor_2.html#E5:17"><i><font color="Green" title="E16">A4</font></i></a>, <a class="txt" href="prgcor_2.html#E8:17"><i><font color="Green" title="E17">A5</font></i></a>, <a class="txt" href="prgcor_2.html#E1:17_1"><i><font color="Green" title="E19">A7</font></i></a>, <a class="ref" href="prgcor_2.html#D3" target="_self" title="PRGCOR_2:def.3">Def3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E9:17_1"/>
<font color="Maroon" title="c7">s</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">i</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"><font color="Maroon" title="c7">s</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">i</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rvsum_1.html#K14" title="RVSUM_1:func.14">mlt</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E11:17"/><b>then </b>
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K14" title="RVSUM_1:func.14">mlt</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">s</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">n</font>
 
<b>by </b><i><a class="txt" href="prgcor_2.html#E1:17"><i><font color="Green" title="E13">A1</font></i></a>, <a class="txt" href="prgcor_2.html#E2:17"><i><font color="Green" title="E14">A2</font></i></a>, <a class="txt" href="prgcor_2.html#E3:17"><i><font color="Green" title="E15">A3</font></i></a>, <a class="txt" href="prgcor_2.html#E5:17"><i><font color="Green" title="E16">A4</font></i></a>, <a class="txt" href="prgcor_2.html#E8:17"><i><font color="Green" title="E17">A5</font></i></a>, <a class="ref" href="prgcor_2.html#T8" target="_self" title="PRGCOR_2:th.8">Th8</a></i>;<br/>
<b>hence </b><a NAME="E12:17"/>
<span class="p1"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="prgcor_2.html#K6" title="PRGCOR_2:func.6">inner_prd_prg</a> <span class="p1">(<span class="default"><a href="prgcor_2.html#K3" title="PRGCOR_2:func.3">FS2XFS*</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c3">n2</font></span>)</span>,<span class="p1">(<span class="default"><a href="prgcor_2.html#K3" title="PRGCOR_2:func.3">FS2XFS*</a> <font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">n2</font></span>)</span>
 <b>by </b><i><a class="txt" href="prgcor_2.html#E5:17"><i><font color="Green" title="E16">A4</font></i></a>, <a class="ref" href="euclid_2.html#D1" title="EUCLID_2:def.1">EUCLID_2:def 1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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