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

<b>let </b><font color="Maroon">m</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:46"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">m</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/>

<b>let </b><font color="Maroon">L</font> be  non <a href="struct_0.html#V3">empty</a>  <a href="struct_0.html#L2">ZeroStr</a> ;<br/><b>let </b><font color="Maroon">a</font> be   <a href="algseq_1.html#NM1">AlgSequence</a> of <font color="Maroon">L</font>;<br/><b>let </b><font color="Maroon">M</font> be   <a href="matrix_1.html#NM6">Matrix</a> of <font color="Maroon">m</font>,1,<font color="Maroon">L</font>;<br/>





<b>assume </b><a NAME="E2:46"/><i><font color="Green">E38</font></i>: 
for <font color="Olive">i</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m</font> holds <br/><font color="Maroon">M</font> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Olive">i</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span>,1 <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font>
 ;<br/>

<i><font color="Green">E39</font></i>:  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="polynom8.html#K4" target="_self">mConv</a> <font color="Maroon">a</font>,<font color="Maroon">m</font></span>)</span> = 
<font color="Maroon">m</font>
<b>by </b><i><a class="ref" href="polynom8.html#T2" target="_self">Th2</a>, </i>
<br/>.= 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">M</font>
<b>by </b><i><a class="ref" href="polynom8.html#T2" target="_self">Th2</a>, <a class="ref" href="matrix_1.html#T24">MATRIX_1:24</a></i>
;<br/>
<i><font color="Green">E42</font></i>:  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="polynom8.html#K4" target="_self">mConv</a> <font color="Maroon">a</font>,<font color="Maroon">m</font></span>)</span> = 
1
<b>by </b><i><a class="ref" href="polynom8.html#T2" target="_self">Th2</a>, </i>
<br/>.= 
 <a href="matrix_1.html#K1">width</a> <font color="Maroon">M</font>
<b>by </b><i><a class="ref" href="polynom8.html#T2" target="_self">Th2</a>, <a class="ref" href="matrix_1.html#T24">MATRIX_1:24</a></i>
;<br/>
<a NAME="E5:46"/>
for <font color="Olive">i</font>, <font color="Olive">j</font> being  <a href="nat_1.html#NM1">Nat</a>  st <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">i</font>,<font color="Olive">j</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="polynom8.html#K4" target="_self">mConv</a> <font color="Maroon">a</font>,<font color="Maroon">m</font></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="polynom8.html#K4" target="_self">mConv</a> <font color="Maroon">a</font>,<font color="Maroon">m</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,<font color="Olive">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,<font color="Olive">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">i</font> be   <a href="nat_1.html#NM1">Nat</a>, <font color="Maroon">j</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:46_3"/><i><font color="Green">E51</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i</font>,<font color="Maroon">j</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="polynom8.html#K4" target="_self">mConv</a> <font color="Maroon">a</font>,<font color="Maroon">m</font></span>)</span>
 ;<br/>

<a NAME="E2:46_3"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="polynom8.html#K4" target="_self">mConv</a> <font color="Maroon">a</font>,<font color="Maroon">m</font></span>)</span>
 
<b>by </b><i><a class="ref" href="polynom8.html#T4" target="_self">Th4</a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E3:46_3"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p2">(<span class="default"><a href="polynom8.html#K4" target="_self">mConv</a> <font color="Maroon">a</font>,<font color="Maroon">m</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="polynom8.html#T4" target="_self">Th4</a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E4:46_3"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="polynom8.html#K4" target="_self">mConv</a> <font color="Maroon">a</font>,<font color="Maroon">m</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font>
 
<b>by </b><i><a class="ref" href="polynom8.html#T2" target="_self">Th2</a>, </i>;<br/>
<a NAME="E5:46_3"/><b>then </b><i><font color="Green">E56</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m</font>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E6:46_3"/><b>then </b><i><font color="Green">E58</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E7:46_3"/>
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> 1
 
<b>by </b><i>, <a class="ref" href="polynom8.html#T2" target="_self">Th2</a>, </i>;<br/>
<a NAME="E8:46_3"/><b>then </b><i><font color="Green">E60</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font> &amp; <font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">k</font> = <font color="Maroon">i</font> <a href="xcmplx_0.html#K6">-</a> 1 as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="txt" href="polynom8.html#E1:46_3"><i><font color="Green">E51</font></i></a>, <a class="ref" href="nat_1.html#T60">NAT_1:60</a></i>;<br/>
<a NAME="E10:46_3"/>
<font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E11:46_3"/><b>then </b><i><font color="Green">E61</font></i>: 
<font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<b>then </b><font color="Maroon">M</font> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span>,1 = 
<font color="Maroon">a</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>
<b>by </b><i><a class="ref" href="polynom8.html#T3" target="_self">Th3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="polynom8.html#K4" target="_self">mConv</a> <font color="Maroon">a</font>,<font color="Maroon">m</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span>,1
<b>by </b><i><a class="txt" href="polynom8.html#E2:46_3"><i><font color="Green">E53</font></i></a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><a href="polynom8.html#K4" target="_self">mConv</a> <font color="Maroon">a</font>,<font color="Maroon">m</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font>
<b>by </b><i>, <a class="ref" href="xreal_1.html#T1">XREAL_1:1</a></i>
;<br/>
<b>hence </b><a NAME="E13:46_3"/>
<span class="p1">(<span class="default"><a href="polynom8.html#K4" target="_self">mConv</a> <font color="Maroon">a</font>,<font color="Maroon">m</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T1">XREAL_1:1</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:46"/>
 <a href="polynom8.html#K4" target="_self">mConv</a> <font color="Maroon">a</font>,<font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M</font>
 <b>by </b><i>, , <a class="ref" href="matrix_1.html#T21">MATRIX_1:21</a></i>;<br/>


</div>
