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<b>let </b><font color="Maroon" title="c1">m</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:46"/><i><font color="Green" title="E43">A1</font></i>: 
<font color="Maroon" title="c1">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 ;<br/>

<b>let </b><font color="Maroon" title="c2">L</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="struct_0.html#L2" title="STRUCT_0:struct.2">ZeroStr</a> ;<br/><b>let </b><font color="Maroon" title="c3">a</font> be   <a href="algseq_1.html#NM1" title="ALGSEQ_1:NM.1">AlgSequence</a> of <font color="Maroon" title="c2">L</font>;<br/><b>let </b><font color="Maroon" title="c4">M</font> be   <a href="matrix_1.html#NM6" title="MATRIX_1:NM.6">Matrix</a> of <font color="Maroon" title="c1">m</font>,1,<font color="Maroon" title="c2">L</font>;<br/>





<b>assume </b><a NAME="E2:46"/><i><font color="Green" title="E44">A2</font></i>: 
for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">m</font> holds <br/><font color="Maroon" title="c4">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span>,1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font>
 ;<br/>

<i><font color="Green" title="E45">A3</font></i>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="polynom8.html#K4" title="POLYNOM8:func.4">mConv</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c1">m</font></span>)</span> = 
<font color="Maroon" title="c1">m</font>
<b>by </b><i><a class="txt" href="polynom8.html#E1:46"><i><font color="Green" title="E43">A1</font></i></a>, <a class="ref" href="polynom8.html#T28" target="_self" title="POLYNOM8:th.28">Th28</a></i>
<br/>.= 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">M</font>
<b>by </b><i><a class="txt" href="polynom8.html#E1:46"><i><font color="Green" title="E43">A1</font></i></a>, <a class="ref" href="matrix_1.html#T24" title="MATRIX_1:th.24">MATRIX_1:24</a></i>
;<br/>
<i><font color="Green" title="E46">A4</font></i>:  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="polynom8.html#K4" title="POLYNOM8:func.4">mConv</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c1">m</font></span>)</span> = 
1
<b>by </b><i><a class="txt" href="polynom8.html#E1:46"><i><font color="Green" title="E43">A1</font></i></a>, <a class="ref" href="polynom8.html#T28" target="_self" title="POLYNOM8:th.28">Th28</a></i>
<br/>.= 
 <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c4">M</font>
<b>by </b><i><a class="txt" href="polynom8.html#E1:46"><i><font color="Green" title="E43">A1</font></i></a>, <a class="ref" href="matrix_1.html#T24" title="MATRIX_1:th.24">MATRIX_1:24</a></i>
;<br/>
<a NAME="E5:46"/>
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"><a href="polynom8.html#K4" title="POLYNOM8:func.4">mConv</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c1">m</font></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="polynom8.html#K4" title="POLYNOM8:func.4">mConv</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c1">m</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> <font color="Maroon" title="c4">M</font> <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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_3"><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>;<br/>

<b>assume </b><a NAME="E1:46_3"/><i><font color="Green" title="E47">A5</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"><a href="polynom8.html#K4" title="POLYNOM8:func.4">mConv</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c1">m</font></span>)</span>
 ;<br/>

<a NAME="E2:46_3"/><b>then </b><i><font color="Green" title="E48">A6</font></i>: 
<font color="Maroon" title="c5">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="polynom8.html#K4" title="POLYNOM8:func.4">mConv</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c1">m</font></span>)</span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E3:46_3"/><i><font color="Green" title="E49">A7</font></i>: 
<font color="Maroon" title="c6">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p2">(<span class="default"><a href="polynom8.html#K4" title="POLYNOM8:func.4">mConv</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c1">m</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="polynom8.html#E1:46_3"><i><font color="Green" title="E47">A5</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E4:46_3"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="polynom8.html#K4" title="POLYNOM8:func.4">mConv</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c1">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">m</font>
 
<b>by </b><i><a class="txt" href="polynom8.html#E1:46"><i><font color="Green" title="E43">A1</font></i></a>, <a class="ref" href="polynom8.html#T28" target="_self" title="POLYNOM8:th.28">Th28</a></i>;<br/>
<a NAME="E5:46_3"/><b>then </b><i><font color="Green" title="E50">A8</font></i>: 
<font color="Maroon" title="c5">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">m</font>
 
<b>by </b><i><a class="txt" href="polynom8.html#E2:46_3"><i><font color="Green" title="E48">A6</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E6:46_3"/><b>then </b><i><font color="Green" title="E51">A9</font></i>: 
0 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">i</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E7:46_3"/>
<font color="Maroon" title="c6">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 1
 
<b>by </b><i><a class="txt" href="polynom8.html#E3:46_3"><i><font color="Green" title="E49">A7</font></i></a>, <a class="txt" href="polynom8.html#E1:46"><i><font color="Green" title="E43">A1</font></i></a>, <a class="ref" href="polynom8.html#T28" target="_self" title="POLYNOM8:th.28">Th28</a></i>;<br/>
<a NAME="E8:46_3"/><b>then </b><i><font color="Green" title="E52">A10</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">j</font> &amp; <font color="Maroon" title="c6">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">k</font> = <font color="Maroon" title="c5">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 as   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> <b>by </b><i><a class="txt" href="polynom8.html#E6:46_3"><i><font color="Green" title="E51">A9</font></i></a>, <a class="ref" href="nat_1.html#T20" title="NAT_1:th.20">NAT_1:20</a></i>;<br/>
<a NAME="E10:46_3"/>
<font color="Maroon" title="c7">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">m</font>
 
<b>by </b><i><a class="txt" href="polynom8.html#E5:46_3"><i><font color="Green" title="E50">A8</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E11:46_3"/><b>then </b><i><font color="Green" title="E53">A11</font></i>: 
<font color="Maroon" title="c7">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">m</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<b>then </b><font color="Maroon" title="c4">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span>,1 = 
<font color="Maroon" title="c3">a</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">k</font>
<b>by </b><i><a class="txt" href="polynom8.html#E2:46"><i><font color="Green" title="E44">A2</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="polynom8.html#K4" title="POLYNOM8:func.4">mConv</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c1">m</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span>,1
<b>by </b><i><a class="txt" href="polynom8.html#E11:46_3"><i><font color="Green" title="E53">A11</font></i></a>, <a class="ref" href="polynom8.html#T28" target="_self" title="POLYNOM8:th.28">Th28</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="polynom8.html#K4" title="POLYNOM8:func.4">mConv</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c1">m</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>
<b>by </b><i><a class="txt" href="polynom8.html#E8:46_3"><i><font color="Green" title="E52">A10</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>
;<br/>
<b>hence </b><a NAME="E13:46_3"/>
<span class="p1">(<span class="default"><a href="polynom8.html#K4" title="POLYNOM8:func.4">mConv</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c1">m</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> <font color="Maroon" title="c4">M</font> <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>
 <b>by </b><i><a class="txt" href="polynom8.html#E8:46_3"><i><font color="Green" title="E52">A10</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:46"/>
 <a href="polynom8.html#K4" title="POLYNOM8:func.4">mConv</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">M</font>
 <b>by </b><i><a class="txt" href="polynom8.html#E3:46"><i><font color="Green" title="E45">A3</font></i></a>, <a class="txt" href="polynom8.html#E4:46"><i><font color="Green" title="E46">A4</font></i></a>, <a class="ref" href="matrix_1.html#T21" title="MATRIX_1:th.21">MATRIX_1:21</a></i>;<br/>


</div>
