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<span class="kw">let </span><font color="Maroon" title="c1">i</font>, <font color="Maroon" title="c2">j</font> be    <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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">M1</font>, <font color="Olive" title="b2">M2</font> being  <a href="matrix_1.html#NM1" title="MATRIX_1:NM.1">Matrix</a> of <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a>   st  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">M2</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>  holds <br/> ex <font color="Olive" title="b3">s</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a>  st <br/>(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b3">s</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="Olive" title="b2">M2</font> &amp; <font color="Olive" title="b3">s</font> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">M1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c1">i</font>,1</span>)</span> <a href="binop_2.html#K5" title="BINOP_2:func.5">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">M2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c2">j</font></span>)</span> &amp; ( for <font color="Olive" title="b4">k</font> being   <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>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b4">k</font> &amp; <font color="Olive" title="b4">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> <font color="Olive" title="b2">M2</font> holds <br/><font color="Olive" title="b3">s</font> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b4">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">s</font> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Olive" title="b4">k</font></span>)</span> <a href="binop_2.html#K3" title="BINOP_2:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">M1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c1">i</font>,<span class="p3">(<span class="default"><font color="Olive" title="b4">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K5" title="BINOP_2:func.5">*</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">M2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p3">(<span class="default"><font color="Olive" title="b4">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>,<font color="Maroon" title="c2">j</font></span>)</span></span>)</span> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">M1</font>, <font color="Maroon" title="c4">M2</font> be   <a href="matrix_1.html#NM1" title="MATRIX_1:NM.1">Matrix</a> of <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">M2</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>  implies  ex <font color="Olive" title="b1">s</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a>  st <br/>(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">s</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="c4">M2</font> &amp; <font color="Olive" title="b1">s</font> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">M1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c1">i</font>,1</span>)</span> <a href="binop_2.html#K5" title="BINOP_2:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">M2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c2">j</font></span>)</span> &amp; ( for <font color="Olive" title="b2">k</font> being   <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>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">k</font> &amp; <font color="Olive" title="b2">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> <font color="Maroon" title="c4">M2</font> holds <br/><font color="Olive" title="b1">s</font> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">s</font> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Olive" title="b2">k</font></span>)</span> <a href="binop_2.html#K3" title="BINOP_2:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">M1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c1">i</font>,<span class="p3">(<span class="default"><font color="Olive" title="b2">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K5" title="BINOP_2:func.5">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">M2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p3">(<span class="default"><font color="Olive" title="b2">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>,<font color="Maroon" title="c2">j</font></span>)</span></span>)</span> ) ) )</span><br/>



<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <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">means </span> ex <font color="Olive" title="b1">q</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a>  st <br/>( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> $1 <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 &amp; $1 <a href="binop_2.html#K23" title="BINOP_2:func.23">+</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="c4">M2</font> implies (  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 &amp; <font color="Olive" title="b1">q</font> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">M1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c1">i</font>,1</span>)</span> <a href="binop_2.html#K5" title="BINOP_2:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">M2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c2">j</font></span>)</span> &amp; ( for <font color="Olive" title="b2">k</font> being   <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>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">k</font> &amp; <font color="Olive" title="b2">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> $1 <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 holds <br/><font color="Olive" title="b1">q</font> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">q</font> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Olive" title="b2">k</font></span>)</span> <a href="binop_2.html#K3" title="BINOP_2:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">M1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c1">i</font>,<span class="p3">(<span class="default"><font color="Olive" title="b2">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K5" title="BINOP_2:func.5">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">M2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p3">(<span class="default"><font color="Olive" title="b2">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>,<font color="Maroon" title="c2">j</font></span>)</span></span>)</span> ) ) );<br/>
<span class="kw">set </span><font color="Maroon" title="c5">q0</font> = <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">M1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c1">i</font>,1</span>)</span> <a href="binop_2.html#K5" title="BINOP_2:func.5">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">M2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c2">j</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>;<br/>
<a NAME="E1:41"/><span class="lab"><font color="Green" title="E7">A1</font></span>: 
for <font color="Olive" title="b1">k</font> being   <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>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 holds <br/><span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">M1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c1">i</font>,1</span>)</span> <a href="binop_2.html#K5" title="BINOP_2:func.5">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">M2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c2">j</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">M1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c1">i</font>,1</span>)</span> <a href="binop_2.html#K5" title="BINOP_2:func.5">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">M2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c2">j</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="binop_2.html#K3" title="BINOP_2:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">M1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c1">i</font>,<span class="p3">(<span class="default"><font color="Olive" title="b1">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K5" title="BINOP_2:func.5">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">M2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>,<font color="Maroon" title="c2">j</font></span>)</span></span>)</span>
 
;<br/>
<a NAME="E2:41"/><span class="lab"><font color="Green" title="E8">A2</font></span>: 
for <font color="Olive" title="b1">i2</font> being   <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>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">i2</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">i2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">i2</font> be    <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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">i2</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">i2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1] )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c7">k0</font> = <font color="Maroon" title="c6">i2</font>;<br/>
<span class="kw">assume </span><a NAME="E1:41_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">i2</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">i2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1]</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">q2</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> <span class="kw"> such that </span><br/><a NAME="E3:41_1"/><span class="lab"><font color="Green" title="E9">A3</font></span>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">i2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 &amp; <font color="Maroon" title="c6">i2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</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="c4">M2</font> implies (  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">i2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 &amp; <font color="Maroon" title="c8">q2</font> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">M1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c1">i</font>,1</span>)</span> <a href="binop_2.html#K5" title="BINOP_2:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">M2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c2">j</font></span>)</span> &amp; ( for <font color="Olive" title="b1">k2</font> being   <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>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k2</font> &amp; <font color="Olive" title="b1">k2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">i2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 holds <br/><font color="Maroon" title="c8">q2</font> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">k2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">q2</font> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Olive" title="b1">k2</font></span>)</span> <a href="binop_2.html#K3" title="BINOP_2:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">M1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c1">i</font>,<span class="p3">(<span class="default"><font color="Olive" title="b1">k2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K5" title="BINOP_2:func.5">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">M2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">k2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>,<font color="Maroon" title="c2">j</font></span>)</span></span>)</span> ) ) )
 ;<br/>
<span class="kw">set </span><font color="Maroon" title="c9">q3</font> = <font color="Maroon" title="c8">q2</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c8">q2</font> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">i2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K3" title="BINOP_2:func.3">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">M1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c1">i</font>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c6">i2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K5" title="BINOP_2:func.5">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">M2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c6">i2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>,<font color="Maroon" title="c2">j</font></span>)</span></span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>;<br/>
<a NAME="E4:41_1"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">i2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c6">i2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</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="c4">M2</font> implies (  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">q2</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c8">q2</font> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c6">i2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K3" title="BINOP_2:func.3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">M1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c1">i</font>,<span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c6">i2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K5" title="BINOP_2:func.5">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">M2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c6">i2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>,<font color="Maroon" title="c2">j</font></span>)</span></span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">i2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c8">q2</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c8">q2</font> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c6">i2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K3" title="BINOP_2:func.3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">M1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c1">i</font>,<span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c6">i2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K5" title="BINOP_2:func.5">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">M2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c6">i2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>,<font color="Maroon" title="c2">j</font></span>)</span></span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">M1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c1">i</font>,1</span>)</span> <a href="binop_2.html#K5" title="BINOP_2:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">M2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c2">j</font></span>)</span> &amp; ( for <font color="Olive" title="b1">k</font> being   <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>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">i2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c8">q2</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c8">q2</font> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c6">i2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K3" title="BINOP_2:func.3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">M1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c1">i</font>,<span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c6">i2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K5" title="BINOP_2:func.5">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">M2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c6">i2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>,<font color="Maroon" title="c2">j</font></span>)</span></span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">q2</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p3"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c8">q2</font> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <span class="p0">(<span class="default"><font color="Maroon" title="c6">i2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K3" title="BINOP_2:func.3">+</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">M1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c1">i</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">i2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K5" title="BINOP_2:func.5">*</a> <span class="p0">(<span class="default"><font color="Maroon" title="c4">M2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">i2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>,<font color="Maroon" title="c2">j</font></span>)</span></span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="binop_2.html#K3" title="BINOP_2:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">M1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c1">i</font>,<span class="p3">(<span class="default"><font color="Olive" title="b1">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K5" title="BINOP_2:func.5">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">M2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>,<font color="Maroon" title="c2">j</font></span>)</span></span>)</span> ) ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:41_1_1"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">i2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1
 <span class="kw">and </span><br/><a NAME="E2:41_1_1"/><span class="lab"><font color="Green" title="E10">A4</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">i2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</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="c4">M2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">q2</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c8">q2</font> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c6">i2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K3" title="BINOP_2:func.3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">M1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c1">i</font>,<span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c6">i2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K5" title="BINOP_2:func.5">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">M2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c6">i2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>,<font color="Maroon" title="c2">j</font></span>)</span></span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">i2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c8">q2</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c8">q2</font> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c6">i2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K3" title="BINOP_2:func.3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">M1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c1">i</font>,<span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c6">i2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K5" title="BINOP_2:func.5">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">M2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c6">i2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>,<font color="Maroon" title="c2">j</font></span>)</span></span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">M1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c1">i</font>,1</span>)</span> <a href="binop_2.html#K5" title="BINOP_2:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">M2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c2">j</font></span>)</span> &amp; ( for <font color="Olive" title="b1">k</font> being   <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>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">i2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c8">q2</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c8">q2</font> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c6">i2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K3" title="BINOP_2:func.3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">M1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c1">i</font>,<span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c6">i2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K5" title="BINOP_2:func.5">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">M2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c6">i2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>,<font color="Maroon" title="c2">j</font></span>)</span></span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">q2</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p3"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c8">q2</font> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <span class="p0">(<span class="default"><font color="Maroon" title="c6">i2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K3" title="BINOP_2:func.3">+</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">M1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c1">i</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">i2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K5" title="BINOP_2:func.5">*</a> <span class="p0">(<span class="default"><font color="Maroon" title="c4">M2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">i2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>,<font color="Maroon" title="c2">j</font></span>)</span></span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="binop_2.html#K3" title="BINOP_2:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">M1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c1">i</font>,<span class="p3">(<span class="default"><font color="Olive" title="b1">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K5" title="BINOP_2:func.5">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">M2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>,<font color="Maroon" title="c2">j</font></span>)</span></span>)</span> ) )</span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_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="E1:41_1_1_1"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">i2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1
 </span><span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:41_1_1_1_1"/><span class="lab"><font color="Green" title="E11">A5</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">i2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">q2</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c8">q2</font> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c6">i2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K3" title="BINOP_2:func.3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">M1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c1">i</font>,<span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c6">i2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K5" title="BINOP_2:func.5">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">M2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c6">i2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>,<font color="Maroon" title="c2">j</font></span>)</span></span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">i2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c8">q2</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c8">q2</font> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c6">i2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K3" title="BINOP_2:func.3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">M1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c1">i</font>,<span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c6">i2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K5" title="BINOP_2:func.5">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">M2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c6">i2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>,<font color="Maroon" title="c2">j</font></span>)</span></span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">M1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c1">i</font>,1</span>)</span> <a href="binop_2.html#K5" title="BINOP_2:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">M2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c2">j</font></span>)</span> &amp; ( for <font color="Olive" title="b1">k</font> being   <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>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">i2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c8">q2</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c8">q2</font> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c6">i2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K3" title="BINOP_2:func.3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">M1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c1">i</font>,<span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c6">i2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K5" title="BINOP_2:func.5">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">M2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c6">i2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>,<font color="Maroon" title="c2">j</font></span>)</span></span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">q2</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p3"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c8">q2</font> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <span class="p0">(<span class="default"><font color="Maroon" title="c6">i2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K3" title="BINOP_2:func.3">+</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">M1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c1">i</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">i2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K5" title="BINOP_2:func.5">*</a> <span class="p0">(<span class="default"><font color="Maroon" title="c4">M2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">i2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>,<font color="Maroon" title="c2">j</font></span>)</span></span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="binop_2.html#K3" title="BINOP_2:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">M1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c1">i</font>,<span class="p3">(<span class="default"><font color="Olive" title="b1">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K5" title="BINOP_2:func.5">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">M2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>,<font color="Maroon" title="c2">j</font></span>)</span></span>)</span> ) )</span><br/><div class="add"><span class="lab"><font color="Green" title="E12">A6</font></span>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">q2</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c8">q2</font> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c6">i2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K3" title="BINOP_2:func.3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">M1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c1">i</font>,<span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c6">i2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K5" title="BINOP_2:func.5">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">M2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c6">i2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>,<font color="Maroon" title="c2">j</font></span>)</span></span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">q2</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c8">q2</font> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c6">i2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K3" title="BINOP_2:func.3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">M1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c1">i</font>,<span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c6">i2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K5" title="BINOP_2:func.5">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">M2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c6">i2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>,<font color="Maroon" title="c2">j</font></span>)</span></span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">i2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:41_1"><span class="lab"><font color="Green" title="E9">A3</font></span></a>, <a class="txt" href="#E2:41_1_1"><span class="lab"><font color="Green" title="E10">A4</font></span></a>, <a class="txt" href="#E1:41_1_1_1_1"><span class="lab"><font color="Green" title="E11">A5</font></span></a>, <a class="ref" href="finseq_1.html#T56" title="FINSEQ_1:th.56">FINSEQ_1:56</a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>
;<br/><a NAME="E3:41_1_1_1_1"/><span class="lab"><font color="Green" title="E13">A7</font></span>: 
for <font color="Olive" title="b1">k2</font> being   <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>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k2</font> &amp; <font color="Olive" title="b1">k2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">i2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c8">q2</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c8">q2</font> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c6">i2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K3" title="BINOP_2:func.3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">M1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c1">i</font>,<span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c6">i2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K5" title="BINOP_2:func.5">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">M2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c6">i2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>,<font color="Maroon" title="c2">j</font></span>)</span></span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">k2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">q2</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p3"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c8">q2</font> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <span class="p0">(<span class="default"><font color="Maroon" title="c6">i2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K3" title="BINOP_2:func.3">+</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">M1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c1">i</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">i2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K5" title="BINOP_2:func.5">*</a> <span class="p0">(<span class="default"><font color="Maroon" title="c4">M2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">i2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>,<font color="Maroon" title="c2">j</font></span>)</span></span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Olive" title="b1">k2</font></span>)</span> <a href="binop_2.html#K3" title="BINOP_2:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">M1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c1">i</font>,<span class="p3">(<span class="default"><font color="Olive" title="b1">k2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K5" title="BINOP_2:func.5">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">M2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">k2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>,<font color="Maroon" title="c2">j</font></span>)</span></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_1_1_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c10">k2</font> be    <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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">k2</font> &amp; <font color="Maroon" title="c10">k2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">i2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 implies <span class="p1">(<span class="default"><font color="Maroon" title="c8">q2</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c8">q2</font> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c6">i2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K3" title="BINOP_2:func.3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">M1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c1">i</font>,<span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c6">i2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K5" title="BINOP_2:func.5">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">M2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c6">i2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>,<font color="Maroon" title="c2">j</font></span>)</span></span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">q2</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p3"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c8">q2</font> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <span class="p0">(<span class="default"><font color="Maroon" title="c6">i2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K3" title="BINOP_2:func.3">+</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">M1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c1">i</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">i2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K5" title="BINOP_2:func.5">*</a> <span class="p0">(<span class="default"><font color="Maroon" title="c4">M2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">i2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>,<font color="Maroon" title="c2">j</font></span>)</span></span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Maroon" title="c10">k2</font></span>)</span> <a href="binop_2.html#K3" title="BINOP_2:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">M1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c1">i</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c10">k2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K5" title="BINOP_2:func.5">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">M2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">k2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>,<font color="Maroon" title="c2">j</font></span>)</span></span>)</span> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:41_1_1_1_1_2"/><span class="lab"><font color="Green" title="E14">A8</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">k2</font>
 <span class="kw">and </span><br/><a NAME="E2:41_1_1_1_1_2"/><span class="lab"><font color="Green" title="E15">A9</font></span>: 
<font color="Maroon" title="c10">k2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">i2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c8">q2</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c8">q2</font> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c6">i2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K3" title="BINOP_2:func.3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">M1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c1">i</font>,<span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c6">i2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K5" title="BINOP_2:func.5">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">M2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c6">i2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>,<font color="Maroon" title="c2">j</font></span>)</span></span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">q2</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p3"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c8">q2</font> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <span class="p0">(<span class="default"><font color="Maroon" title="c6">i2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K3" title="BINOP_2:func.3">+</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">M1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c1">i</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">i2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K5" title="BINOP_2:func.5">*</a> <span class="p0">(<span class="default"><font color="Maroon" title="c4">M2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">i2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>,<font color="Maroon" title="c2">j</font></span>)</span></span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Maroon" title="c10">k2</font></span>)</span> <a href="binop_2.html#K3" title="BINOP_2:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">M1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c1">i</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c10">k2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K5" title="BINOP_2:func.5">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">M2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">k2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>,<font color="Maroon" title="c2">j</font></span>)</span></span>)</span></span><br/>

<a NAME="E3:41_1_1_1_1_2"/><span class="lab"><font color="Green" title="E16">A10</font></span>: 
<font color="Maroon" title="c10">k2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">i2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:41_1_1_1_1_2"><span class="lab"><font color="Green" title="E15">A9</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_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="E1:41_1_1_1_1_2_1"/>
( <font color="Maroon" title="c10">k2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">i2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 or <font color="Maroon" title="c10">k2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">i2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:41_1_1_1_1_2"><span class="lab"><font color="Green" title="E16">A10</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="41_1_1_1_1_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:41_1_1_1_1_2_1_1"/><span class="lab"><font color="Green" title="E17">A11</font></span>: 
<font color="Maroon" title="c10">k2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">i2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c8">q2</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c8">q2</font> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c6">i2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K3" title="BINOP_2:func.3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">M1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c1">i</font>,<span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c6">i2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K5" title="BINOP_2:func.5">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">M2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c6">i2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>,<font color="Maroon" title="c2">j</font></span>)</span></span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">q2</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p3"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c8">q2</font> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <span class="p0">(<span class="default"><font color="Maroon" title="c6">i2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K3" title="BINOP_2:func.3">+</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">M1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c1">i</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">i2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K5" title="BINOP_2:func.5">*</a> <span class="p0">(<span class="default"><font color="Maroon" title="c4">M2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">i2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>,<font color="Maroon" title="c2">j</font></span>)</span></span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Maroon" title="c10">k2</font></span>)</span> <a href="binop_2.html#K3" title="BINOP_2:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">M1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c1">i</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c10">k2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K5" title="BINOP_2:func.5">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">M2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">k2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>,<font color="Maroon" title="c2">j</font></span>)</span></span>)</span></span><br/><div class="add"><a NAME="E2:41_1_1_1_1_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c10">k2</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"><font color="Maroon" title="c8">q2</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c8">q2</font> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c6">i2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K3" title="BINOP_2:func.3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">M1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c1">i</font>,<span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c6">i2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K5" title="BINOP_2:func.5">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">M2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c6">i2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>,<font color="Maroon" title="c2">j</font></span>)</span></span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:41_1_1_1_1"><span class="lab"><font color="Green" title="E12">A6</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E3:41_1_1_1_1_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c10">k2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">q2</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c8">q2</font> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c6">i2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K3" title="BINOP_2:func.3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">M1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c1">i</font>,<span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c6">i2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K5" title="BINOP_2:func.5">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">M2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c6">i2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>,<font color="Maroon" title="c2">j</font></span>)</span></span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a></span>;<br/><a NAME="E4:41_1_1_1_1_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c10">k2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">q2</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T56" title="FINSEQ_1:th.56">FINSEQ_1:56</a></span>;<br/><a NAME="E5:41_1_1_1_1_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c10">k2</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="c8">q2</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/><a NAME="E6:41_1_1_1_1_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c10">k2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c8">q2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:41_1_1_1_1_2"><span class="lab"><font color="Green" title="E14">A8</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/><a NAME="E7:41_1_1_1_1_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E18">A12</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">q2</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c8">q2</font> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c6">i2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K3" title="BINOP_2:func.3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">M1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c1">i</font>,<span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c6">i2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K5" title="BINOP_2:func.5">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">M2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c6">i2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>,<font color="Maroon" title="c2">j</font></span>)</span></span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Maroon" title="c10">k2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">q2</font> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Maroon" title="c10">k2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></span>;<br/><a NAME="E8:41_1_1_1_1_2_1_1"/>
<font color="Maroon" title="c10">k2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">i2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:41_1_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E17">A11</font></span></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/><a NAME="E9:41_1_1_1_1_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c10">k2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">q2</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c8">q2</font> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c6">i2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K3" title="BINOP_2:func.3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">M1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c1">i</font>,<span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c6">i2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K5" title="BINOP_2:func.5">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">M2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c6">i2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>,<font color="Maroon" title="c2">j</font></span>)</span></span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:41_1_1_1_1"><span class="lab"><font color="Green" title="E12">A6</font></span></a>, <a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a></span>;<br/><a NAME="E10:41_1_1_1_1_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c10">k2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">q2</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T56" title="FINSEQ_1:th.56">FINSEQ_1:56</a></span>;<br/><a NAME="E11:41_1_1_1_1_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E19">A13</font></span>: 
<font color="Maroon" title="c10">k2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</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="c8">q2</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/><a NAME="E12:41_1_1_1_1_2_1_1"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">k2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:41_1_1_1_1_2"><span class="lab"><font color="Green" title="E14">A8</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E13:41_1_1_1_1_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c10">k2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</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> <font color="Maroon" title="c8">q2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:41_1_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E19">A13</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/><a NAME="E14:41_1_1_1_1_2_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c8">q2</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c8">q2</font> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c6">i2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K3" title="BINOP_2:func.3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">M1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c1">i</font>,<span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c6">i2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K5" title="BINOP_2:func.5">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">M2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c6">i2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>,<font color="Maroon" title="c2">j</font></span>)</span></span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">q2</font> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></span>;<br/><span class="kw">hence </span><a NAME="E15:41_1_1_1_1_2_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c8">q2</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c8">q2</font> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c6">i2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K3" title="BINOP_2:func.3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">M1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c1">i</font>,<span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c6">i2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K5" title="BINOP_2:func.5">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">M2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c6">i2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>,<font color="Maroon" title="c2">j</font></span>)</span></span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">q2</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p3"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c8">q2</font> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <span class="p0">(<span class="default"><font color="Maroon" title="c6">i2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K3" title="BINOP_2:func.3">+</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">M1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c1">i</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">i2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K5" title="BINOP_2:func.5">*</a> <span class="p0">(<span class="default"><font color="Maroon" title="c4">M2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">i2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>,<font color="Maroon" title="c2">j</font></span>)</span></span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Maroon" title="c10">k2</font></span>)</span> <a href="binop_2.html#K3" title="BINOP_2:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">M1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c1">i</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c10">k2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K5" title="BINOP_2:func.5">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">M2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">k2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>,<font color="Maroon" title="c2">j</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:41_1"><span class="lab"><font color="Green" title="E9">A3</font></span></a>, <a class="txt" href="#E2:41_1_1"><span class="lab"><font color="Green" title="E10">A4</font></span></a>, <a class="txt" href="#E1:41_1_1_1_1"><span class="lab"><font color="Green" title="E11">A5</font></span></a>, <a class="txt" href="#E1:41_1_1_1_1_2"><span class="lab"><font color="Green" title="E14">A8</font></span></a>, <a class="txt" href="#E1:41_1_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E17">A11</font></span></a>, <a class="txt" href="#E7:41_1_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E18">A12</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_1_1_1_1_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:41_1_1_1_1_2_1_2"/><span class="lab"><font color="Green" title="E17">A14</font></span>: 
<font color="Maroon" title="c10">k2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">i2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c8">q2</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c8">q2</font> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c6">i2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K3" title="BINOP_2:func.3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">M1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c1">i</font>,<span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c6">i2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K5" title="BINOP_2:func.5">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">M2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c6">i2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>,<font color="Maroon" title="c2">j</font></span>)</span></span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">q2</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p3"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c8">q2</font> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <span class="p0">(<span class="default"><font color="Maroon" title="c6">i2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K3" title="BINOP_2:func.3">+</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">M1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c1">i</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">i2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K5" title="BINOP_2:func.5">*</a> <span class="p0">(<span class="default"><font color="Maroon" title="c4">M2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">i2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>,<font color="Maroon" title="c2">j</font></span>)</span></span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Maroon" title="c10">k2</font></span>)</span> <a href="binop_2.html#K3" title="BINOP_2:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">M1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c1">i</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c10">k2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K5" title="BINOP_2:func.5">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">M2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">k2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>,<font color="Maroon" title="c2">j</font></span>)</span></span>)</span></span><br/><div class="add"><a NAME="E2:41_1_1_1_1_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c10">k2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">i2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</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="E3:41_1_1_1_1_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c10">k2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">q2</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c8">q2</font> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c6">i2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K3" title="BINOP_2:func.3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">M1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c1">i</font>,<span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c6">i2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K5" title="BINOP_2:func.5">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">M2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c6">i2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>,<font color="Maroon" title="c2">j</font></span>)</span></span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:41_1_1_1_1"><span class="lab"><font color="Green" title="E12">A6</font></span></a>, <a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a></span>;<br/><a NAME="E4:41_1_1_1_1_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c10">k2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">q2</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T56" title="FINSEQ_1:th.56">FINSEQ_1:56</a></span>;<br/><a NAME="E5:41_1_1_1_1_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c10">k2</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="c8">q2</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/><a NAME="E6:41_1_1_1_1_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c10">k2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c8">q2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:41_1_1_1_1_2"><span class="lab"><font color="Green" title="E14">A8</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/><a NAME="E7:41_1_1_1_1_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E18">A15</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">q2</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c8">q2</font> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c6">i2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K3" title="BINOP_2:func.3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">M1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c1">i</font>,<span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c6">i2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K5" title="BINOP_2:func.5">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">M2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c6">i2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>,<font color="Maroon" title="c2">j</font></span>)</span></span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Maroon" title="c10">k2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">q2</font> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Maroon" title="c10">k2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></span>;<br/><a NAME="E8:41_1_1_1_1_2_1_2"/>
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T5" title="FINSEQ_1:th.5">FINSEQ_1:5</a></span>;<br/><a NAME="E9:41_1_1_1_1_2_1_2"/><span class="kw">then </span>
( <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c8">q2</font> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">i2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K3" title="BINOP_2:func.3">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">M1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c1">i</font>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c6">i2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K5" title="BINOP_2:func.5">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">M2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c6">i2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>,<font color="Maroon" title="c2">j</font></span>)</span></span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">q2</font> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">i2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K3" title="BINOP_2:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">M1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c1">i</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c6">i2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K5" title="BINOP_2:func.5">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">M2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c6">i2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>,<font color="Maroon" title="c2">j</font></span>)</span></span>)</span> &amp; 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"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c8">q2</font> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">i2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K3" title="BINOP_2:func.3">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">M1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c1">i</font>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c6">i2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K5" title="BINOP_2:func.5">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">M2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c6">i2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>,<font color="Maroon" title="c2">j</font></span>)</span></span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D8" title="FINSEQ_1:def.8">FINSEQ_1:def 8</a></span>;<br/><span class="kw">hence </span><a NAME="E10:41_1_1_1_1_2_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c8">q2</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c8">q2</font> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c6">i2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K3" title="BINOP_2:func.3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">M1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c1">i</font>,<span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c6">i2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K5" title="BINOP_2:func.5">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">M2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c6">i2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>,<font color="Maroon" title="c2">j</font></span>)</span></span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">q2</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p3"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c8">q2</font> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <span class="p0">(<span class="default"><font color="Maroon" title="c6">i2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K3" title="BINOP_2:func.3">+</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">M1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c1">i</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">i2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K5" title="BINOP_2:func.5">*</a> <span class="p0">(<span class="default"><font color="Maroon" title="c4">M2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">i2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>,<font color="Maroon" title="c2">j</font></span>)</span></span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Maroon" title="c10">k2</font></span>)</span> <a href="binop_2.html#K3" title="BINOP_2:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">M1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c1">i</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c10">k2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K5" title="BINOP_2:func.5">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">M2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">k2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>,<font color="Maroon" title="c2">j</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:41_1"><span class="lab"><font color="Green" title="E9">A3</font></span></a>, <a class="txt" href="#E2:41_1_1"><span class="lab"><font color="Green" title="E10">A4</font></span></a>, <a class="txt" href="#E1:41_1_1_1_1"><span class="lab"><font color="Green" title="E11">A5</font></span></a>, <a class="txt" href="#E1:41_1_1_1_1_2_1_2"><span class="lab"><font color="Green" title="E17">A14</font></span></a>, <a class="txt" href="#E7:41_1_1_1_1_2_1_2"><span class="lab"><font color="Green" title="E18">A15</font></span></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div><a NAME="E4:41_1_1_1_1"/>
1 <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"><font color="Maroon" title="c8">q2</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c8">q2</font> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c6">i2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K3" title="BINOP_2:func.3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">M1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c1">i</font>,<span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c6">i2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K5" title="BINOP_2:func.5">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">M2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c6">i2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>,<font color="Maroon" title="c2">j</font></span>)</span></span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:41_1_1_1_1"><span class="lab"><font color="Green" title="E11">A5</font></span></a>, <a class="txt" href="#E2:41_1_1_1_1"><span class="lab"><font color="Green" title="E12">A6</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E5:41_1_1_1_1"/><span class="kw">then </span>
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">q2</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c8">q2</font> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c6">i2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K3" title="BINOP_2:func.3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">M1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c1">i</font>,<span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c6">i2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K5" title="BINOP_2:func.5">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">M2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c6">i2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>,<font color="Maroon" title="c2">j</font></span>)</span></span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a></span>;<br/><a NAME="E6:41_1_1_1_1"/><span class="kw">then </span>
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">q2</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T56" title="FINSEQ_1:th.56">FINSEQ_1:56</a></span>;<br/><a NAME="E7:41_1_1_1_1"/><span class="kw">then </span>
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="c8">q2</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/><a NAME="E8:41_1_1_1_1"/><span class="kw">then </span>
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> <font color="Maroon" title="c8">q2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/><span class="kw">hence </span><a NAME="E9:41_1_1_1_1"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">q2</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c8">q2</font> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c6">i2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K3" title="BINOP_2:func.3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">M1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c1">i</font>,<span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c6">i2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K5" title="BINOP_2:func.5">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">M2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c6">i2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>,<font color="Maroon" title="c2">j</font></span>)</span></span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">i2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c8">q2</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c8">q2</font> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c6">i2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K3" title="BINOP_2:func.3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">M1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c1">i</font>,<span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c6">i2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K5" title="BINOP_2:func.5">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">M2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c6">i2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>,<font color="Maroon" title="c2">j</font></span>)</span></span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">M1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c1">i</font>,1</span>)</span> <a href="binop_2.html#K5" title="BINOP_2:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">M2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c2">j</font></span>)</span> &amp; ( for <font color="Olive" title="b1">k</font> being   <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>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">i2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c8">q2</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c8">q2</font> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c6">i2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K3" title="BINOP_2:func.3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">M1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c1">i</font>,<span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c6">i2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K5" title="BINOP_2:func.5">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">M2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c6">i2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>,<font color="Maroon" title="c2">j</font></span>)</span></span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">q2</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p3"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c8">q2</font> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <span class="p0">(<span class="default"><font color="Maroon" title="c6">i2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K3" title="BINOP_2:func.3">+</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">M1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c1">i</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">i2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K5" title="BINOP_2:func.5">*</a> <span class="p0">(<span class="default"><font color="Maroon" title="c4">M2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">i2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>,<font color="Maroon" title="c2">j</font></span>)</span></span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="binop_2.html#K3" title="BINOP_2:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">M1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c1">i</font>,<span class="p3">(<span class="default"><font color="Olive" title="b1">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K5" title="BINOP_2:func.5">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">M2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>,<font color="Maroon" title="c2">j</font></span>)</span></span>)</span> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:41_1"><span class="lab"><font color="Green" title="E9">A3</font></span></a>, <a class="txt" href="#E2:41_1_1"><span class="lab"><font color="Green" title="E10">A4</font></span></a>, <a class="txt" href="#E1:41_1_1_1_1"><span class="lab"><font color="Green" title="E11">A5</font></span></a>, <a class="txt" href="#E2:41_1_1_1_1"><span class="lab"><font color="Green" title="E12">A6</font></span></a>, <a class="txt" href="#E3:41_1_1_1_1"><span class="lab"><font color="Green" title="E13">A7</font></span></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E5:41_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">i2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E3:41"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">M1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c1">i</font>,1</span>)</span> <a href="binop_2.html#K5" title="BINOP_2:func.5">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">M2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c2">j</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">M1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c1">i</font>,1</span>)</span> <a href="binop_2.html#K5" title="BINOP_2:func.5">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">M2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c2">j</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">M1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c1">i</font>,1</span>)</span> <a href="binop_2.html#K5" title="BINOP_2:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">M2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c2">j</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T56" title="FINSEQ_1:th.56">FINSEQ_1:56</a>, <a class="ref" href="finseq_1.html#D8" title="FINSEQ_1:def.8">FINSEQ_1:def 8</a></span>;<br/>
<a NAME="E4:41"/><span class="kw">then </span><span class="lab"><font color="Green" title="E9">A16</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:41"><span class="lab"><font color="Green" title="E7">A1</font></span></a></span>;<br/>
<a NAME="E5:41"/><span class="lab"><font color="Green" title="E10">A17</font></span>: 
for <font color="Olive" title="b1">j</font> being   <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>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">j</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="#E4:41"><span class="lab"><font color="Green" title="E9">A16</font></span></a>, <a class="txt" href="#E2:41"><span class="lab"><font color="Green" title="E8">A2</font></span></a>);<br/></span>
<span class="kw">assume </span><a NAME="E6:41"/><span class="lab"><font color="Green" title="E11">A18</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">M2</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 class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">s</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a>  st <br/>(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">s</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="c4">M2</font> &amp; <font color="Olive" title="b1">s</font> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">M1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c1">i</font>,1</span>)</span> <a href="binop_2.html#K5" title="BINOP_2:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">M2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c2">j</font></span>)</span> &amp; ( for <font color="Olive" title="b2">k</font> being   <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>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">k</font> &amp; <font color="Olive" title="b2">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> <font color="Maroon" title="c4">M2</font> holds <br/><font color="Olive" title="b1">s</font> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">s</font> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Olive" title="b2">k</font></span>)</span> <a href="binop_2.html#K3" title="BINOP_2:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">M1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c1">i</font>,<span class="p3">(<span class="default"><font color="Olive" title="b2">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K5" title="BINOP_2:func.5">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">M2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p3">(<span class="default"><font color="Olive" title="b2">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>,<font color="Maroon" title="c2">j</font></span>)</span></span>)</span> ) )</span><br/>

<a NAME="E7:41"/><span class="kw">then </span>
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="binop_2.html#K23" title="BINOP_2:func.23">+</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="c4">M2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T8" title="NAT_1:th.8">NAT_1:8</a></span>;<br/>
<a NAME="E8:41"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">M2</font></span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></span>;<br/>
<a NAME="E9:41"/><span class="kw">then </span><span class="lab"><font color="Green" title="E12">A19</font></span>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">M2</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">M2</font></span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_0.html#D2" title="XREAL_0:def.2">XREAL_0:def 2</a></span>;<br/>
<a NAME="E10:41"/><span class="kw">then </span>
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">M2</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:41"><span class="lab"><font color="Green" title="E11">A18</font></span></a>, <a class="ref" href="nat_1.html#T8" title="NAT_1:th.8">NAT_1:8</a></span>;<br/>
<span class="kw">hence </span><a NAME="E11:41"/>
 ex <font color="Olive" title="b1">s</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a>  st <br/>(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">s</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="c4">M2</font> &amp; <font color="Olive" title="b1">s</font> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">M1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c1">i</font>,1</span>)</span> <a href="binop_2.html#K5" title="BINOP_2:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">M2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c2">j</font></span>)</span> &amp; ( for <font color="Olive" title="b2">k</font> being   <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>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">k</font> &amp; <font color="Olive" title="b2">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> <font color="Maroon" title="c4">M2</font> holds <br/><font color="Olive" title="b1">s</font> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">s</font> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Olive" title="b2">k</font></span>)</span> <a href="binop_2.html#K3" title="BINOP_2:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">M1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c1">i</font>,<span class="p3">(<span class="default"><font color="Olive" title="b2">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K5" title="BINOP_2:func.5">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">M2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p3">(<span class="default"><font color="Olive" title="b2">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>,<font color="Maroon" title="c2">j</font></span>)</span></span>)</span> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:41"><span class="lab"><font color="Green" title="E12">A19</font></span></a>, <a class="txt" href="#E5:41"><span class="lab"><font color="Green" title="E10">A17</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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