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

<b>set </b><font color="Maroon" title="c4">p</font> = <font color="Maroon" title="c2">n</font> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <font color="Maroon" title="c3">a</font>;<br/>
<a NAME="E1:16_1_1"/><i><font color="Green" title="E12">A2</font></i>: 
(  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c2">n</font> <a href="matrix_2.html#K2" title="MATRIX_2:func.2">--&gt;</a> <font color="Maroon" title="c3">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">n</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c2">n</font> <a href="matrix_2.html#K2" title="MATRIX_2:func.2">--&gt;</a> <font color="Maroon" title="c3">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">n</font> &amp;  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c2">n</font> <a href="matrix_2.html#K2" title="MATRIX_2:func.2">--&gt;</a> <font color="Maroon" title="c3">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">n</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> )
 
<b>by </b><i><a class="ref" href="matrix_1.html#T25" title="MATRIX_1:th.25">MATRIX_1:25</a></i>;<br/>
<a NAME="E2:16_1_1"/><i><font color="Green" title="E13">A3</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <font color="Maroon" title="c3">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">n</font>
 
<b>by </b><i><a class="ref" href="finseq_2.html#T69" title="FINSEQ_2:th.69">FINSEQ_2:69</a></i>;<br/>
<a NAME="E3:16_1_1"/>
for <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c2">n</font> <a href="matrix_2.html#K2" title="MATRIX_2:func.2">--&gt;</a> <font color="Maroon" title="c3">a</font></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c2">n</font> <a href="matrix_2.html#K2" title="MATRIX_2:func.2">--&gt;</a> <font color="Maroon" title="c3">a</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <font color="Maroon" title="c3">a</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive" title="b2">j</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <font color="Maroon" title="c3">a</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">i</font>, <font color="Maroon" title="c6">j</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c6">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c2">n</font> <a href="matrix_2.html#K2" title="MATRIX_2:func.2">--&gt;</a> <font color="Maroon" title="c3">a</font></span>)</span> implies <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c2">n</font> <a href="matrix_2.html#K2" title="MATRIX_2:func.2">--&gt;</a> <font color="Maroon" title="c3">a</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c6">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <font color="Maroon" title="c3">a</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">j</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c5">i</font></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <font color="Maroon" title="c3">a</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> )</span><br/>

<b>assume </b><a NAME="E1:16_1_1_1"/><i><font color="Green" title="E14">B1</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c6">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c2">n</font> <a href="matrix_2.html#K2" title="MATRIX_2:func.2">--&gt;</a> <font color="Maroon" title="c3">a</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c2">n</font> <a href="matrix_2.html#K2" title="MATRIX_2:func.2">--&gt;</a> <font color="Maroon" title="c3">a</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c6">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <font color="Maroon" title="c3">a</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">j</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c5">i</font></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <font color="Maroon" title="c3">a</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span></span><br/>

<a NAME="E2:16_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">j</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c5">i</font></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 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> <font color="Maroon" title="c2">n</font>
 
<b>by </b><i><a class="txt" href="#E1:16_1_1"><i><font color="Green" title="E12">A2</font></i></a>, <a class="txt" href="matrix16.html#E3"><i><font color="Green" title="E3">Lm2</font></i></a></i>;<br/>
<a NAME="E3:16_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">j</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c5">i</font></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <font color="Maroon" title="c3">a</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 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> <font color="Maroon" title="c2">n</font>
 
<b>by </b><i><a class="ref" href="finseq_2.html#T69" title="FINSEQ_2:th.69">FINSEQ_2:69</a></i>;<br/>
<a NAME="E4:16_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <font color="Maroon" title="c3">a</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">j</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c5">i</font></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <font color="Maroon" title="c3">a</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font>
 
<b>by </b><i><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>;<br/>
<b>hence </b><a NAME="E5:16_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c2">n</font> <a href="matrix_2.html#K2" title="MATRIX_2:func.2">--&gt;</a> <font color="Maroon" title="c3">a</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c6">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <font color="Maroon" title="c3">a</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">j</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c5">i</font></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <font color="Maroon" title="c3">a</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="#E1:16_1_1_1"><i><font color="Green" title="E14">B1</font></i></a>, <a class="ref" href="matrix_2.html#T1" title="MATRIX_2:th.1">MATRIX_2:1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E4:16_1_1"/><b>then </b>
<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c2">n</font> <a href="matrix_2.html#K2" title="MATRIX_2:func.2">--&gt;</a> <font color="Maroon" title="c3">a</font> <a href="matrix16.html#R1" title="MATRIX16:pred.1">is_line_circulant_about</a> <font color="Maroon" title="c2">n</font> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <font color="Maroon" title="c3">a</font>
 
<b>by </b><i><a class="txt" href="#E1:16_1_1"><i><font color="Green" title="E12">A2</font></i></a>, <a class="txt" href="#E2:16_1_1"><i><font color="Green" title="E13">A3</font></i></a>, <a class="ref" href="matrix16.html#D1" target="_self" title="MATRIX16:def.1">Def1</a></i>;<br/>
<b>hence </b><a NAME="E5:16_1_1"/>
<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c2">n</font> <a href="matrix_2.html#K2" title="MATRIX_2:func.2">--&gt;</a> <font color="Maroon" title="c3">a</font> is  <a href="matrix16.html#V1" title="MATRIX16:attr.1">line_circulant</a> <a href="matrix_1.html#NM5" title="MATRIX_1:NM.5">Matrix</a> of <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">K</font>
 <b>by </b><i><a class="ref" href="matrix16.html#D2" target="_self" title="MATRIX16:def.2">Def2</a>, <a class="txt" href="#E1:16_1_1"><i><font color="Green" title="E12">A2</font></i></a>, <a class="txt" href="#E2:16_1_1"><i><font color="Green" title="E13">A3</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
