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

<b>let </b><font color="Maroon">L</font> be   <a href="vectsp_1.html#NM2">Field</a>;<br/><b>let </b><font color="Maroon">m</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">n</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">k</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>



<b>assume </b><a NAME="E1:10"/><i><font color="Green">E31</font></i>: 
( <font color="Maroon">m</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 ;<br/>

<b>let </b><font color="Maroon">M1</font> be   <a href="matrix_1.html#NM6">Matrix</a> of <font color="Maroon">m</font>,<font color="Maroon">n</font>,<font color="Maroon">L</font>;<br/><b>let </b><font color="Maroon">M2</font> be   <a href="matrix_1.html#NM6">Matrix</a> of <font color="Maroon">n</font>,<font color="Maroon">k</font>,<font color="Maroon">L</font>;<br/>



<a NAME="E2:10"/><i><font color="Green">E38</font></i>: 
 <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#K1" target="_self">emb</a> <font color="Maroon">m</font>,<font color="Maroon">L</font></span>)</span> <a href="matrix_3.html#K6">*</a> <font color="Maroon">M1</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">M2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1"><b>proof </b></a><div class="add">

 <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#K1" target="_self">emb</a> <font color="Maroon">m</font>,<font color="Maroon">L</font></span>)</span> <a href="matrix_3.html#K6">*</a> <font color="Maroon">M1</font></span>)</span> = 
 <a href="matrix_1.html#K1">width</a> <font color="Maroon">M1</font>
<b>by </b><i><a class="ref" href="matrix_3.html#D5">MATRIX_3:def 5</a></i>
<br/>.= 
<font color="Maroon">n</font>
<b>by </b><i><a class="ref" href="matrix_1.html#T24">MATRIX_1:24</a>, </i>
<br/>.= 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">M2</font>
<b>by </b><i><a class="ref" href="matrix_1.html#T24">MATRIX_1:24</a>, </i>
;<br/>
<b>hence </b><a NAME="E2:10_1"/>
 <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#K1" target="_self">emb</a> <font color="Maroon">m</font>,<font color="Maroon">L</font></span>)</span> <a href="matrix_3.html#K6">*</a> <font color="Maroon">M1</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">M2</font>
 ;<br/>


</div><b>end;</b></div>
<i><font color="Green">E39</font></i>:  <a href="matrix_1.html#K1">width</a> <font color="Maroon">M1</font> = 
<font color="Maroon">n</font>
<b>by </b><i><a class="ref" href="matrix_1.html#T24">MATRIX_1:24</a>, </i>
<br/>.= 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">M2</font>
<b>by </b><i><a class="ref" href="matrix_1.html#T24">MATRIX_1:24</a>, </i>
;<br/>
<i><font color="Green">E42</font></i>:  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom8.html#K1" target="_self">emb</a> <font color="Maroon">m</font>,<font color="Maroon">L</font></span>)</span> <a href="matrix_3.html#K6">*</a> <font color="Maroon">M1</font></span>)</span> <a href="matrix_3.html#K4">*</a> <font color="Maroon">M2</font></span>)</span> = 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#K1" target="_self">emb</a> <font color="Maroon">m</font>,<font color="Maroon">L</font></span>)</span> <a href="matrix_3.html#K6">*</a> <font color="Maroon">M1</font></span>)</span>
<b>by </b><i>, <a class="ref" href="matrix_3.html#D4">MATRIX_3:def 4</a></i>
<br/>.= 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">M1</font>
<b>by </b><i><a class="ref" href="matrix_3.html#D5">MATRIX_3:def 5</a></i>
;<br/>
<i><font color="Green">E51</font></i>:  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#K1" target="_self">emb</a> <font color="Maroon">m</font>,<font color="Maroon">L</font></span>)</span> <a href="matrix_3.html#K6">*</a> <span class="p2">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_3.html#K4">*</a> <font color="Maroon">M2</font></span>)</span></span>)</span> = 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_3.html#K4">*</a> <font color="Maroon">M2</font></span>)</span>
<b>by </b><i><a class="ref" href="matrix_3.html#D5">MATRIX_3:def 5</a></i>
<br/>.= 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">M1</font>
<b>by </b><i>, <a class="ref" href="matrix_3.html#D4">MATRIX_3:def 4</a></i>
;<br/>
 <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#K1" target="_self">emb</a> <font color="Maroon">m</font>,<font color="Maroon">L</font></span>)</span> <a href="matrix_3.html#K6">*</a> <span class="p2">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_3.html#K4">*</a> <font color="Maroon">M2</font></span>)</span></span>)</span> = 
 <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_3.html#K4">*</a> <font color="Maroon">M2</font></span>)</span>
<b>by </b><i><a class="ref" href="matrix_3.html#D5">MATRIX_3:def 5</a></i>
<br/>.= 
 <a href="matrix_1.html#K1">width</a> <font color="Maroon">M2</font>
<b>by </b><i>, <a class="ref" href="matrix_3.html#D4">MATRIX_3:def 4</a></i>
;<br/>
<a NAME="E7:10"/><b>then </b><i><font color="Green">E53</font></i>: 
 <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom8.html#K1" target="_self">emb</a> <font color="Maroon">m</font>,<font color="Maroon">L</font></span>)</span> <a href="matrix_3.html#K6">*</a> <font color="Maroon">M1</font></span>)</span> <a href="matrix_3.html#K4">*</a> <font color="Maroon">M2</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#K1" target="_self">emb</a> <font color="Maroon">m</font>,<font color="Maroon">L</font></span>)</span> <a href="matrix_3.html#K6">*</a> <span class="p2">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_3.html#K4">*</a> <font color="Maroon">M2</font></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="matrix_3.html#D4">MATRIX_3:def 4</a></i>;<br/>
<a NAME="E8:10"/><i><font color="Green">E55</font></i>: 
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"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom8.html#K1" target="_self">emb</a> <font color="Maroon">m</font>,<font color="Maroon">L</font></span>)</span> <a href="matrix_3.html#K6">*</a> <font color="Maroon">M1</font></span>)</span> <a href="matrix_3.html#K4">*</a> <font color="Maroon">M2</font></span>)</span> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom8.html#K1" target="_self">emb</a> <font color="Maroon">m</font>,<font color="Maroon">L</font></span>)</span> <a href="matrix_3.html#K6">*</a> <font color="Maroon">M1</font></span>)</span> <a href="matrix_3.html#K4">*</a> <font color="Maroon">M2</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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#K1" target="_self">emb</a> <font color="Maroon">m</font>,<font color="Maroon">L</font></span>)</span> <a href="matrix_3.html#K6">*</a> <span class="p2">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_3.html#K4">*</a> <font color="Maroon">M2</font></span>)</span></span>)</span> <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="10_6"><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="E1:10_6"/><i><font color="Green">E56</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"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom8.html#K1" target="_self">emb</a> <font color="Maroon">m</font>,<font color="Maroon">L</font></span>)</span> <a href="matrix_3.html#K6">*</a> <font color="Maroon">M1</font></span>)</span> <a href="matrix_3.html#K4">*</a> <font color="Maroon">M2</font></span>)</span>
 ;<br/>

<i><font color="Green">E58</font></i>:  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom8.html#K1" target="_self">emb</a> <font color="Maroon">m</font>,<font color="Maroon">L</font></span>)</span> <a href="matrix_3.html#K6">*</a> <font color="Maroon">M1</font></span>)</span> <a href="matrix_3.html#K4">*</a> <font color="Maroon">M2</font></span>)</span> = 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#K1" target="_self">emb</a> <font color="Maroon">m</font>,<font color="Maroon">L</font></span>)</span> <a href="matrix_3.html#K6">*</a> <font color="Maroon">M1</font></span>)</span>
<b>by </b><i>, <a class="ref" href="matrix_3.html#D4">MATRIX_3:def 4</a></i>
<br/>.= 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">M1</font>
<b>by </b><i><a class="ref" href="matrix_3.html#D5">MATRIX_3:def 5</a></i>
;<br/>
<a NAME="E3:10_6"/><i><font color="Green">E60</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">M1</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_3.html#K4">*</a> <font color="Maroon">M2</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="matrix_3.html#D4">MATRIX_3:def 4</a></i>;<br/>
<a NAME="E4:10_6"/><i><font color="Green">E61</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"><font color="Maroon">M1</font> <a href="matrix_3.html#K4">*</a> <font color="Maroon">M2</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_6_2"><b>proof </b></a><div class="add">

<i><font color="Green">E62</font></i>:  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom8.html#K1" target="_self">emb</a> <font color="Maroon">m</font>,<font color="Maroon">L</font></span>)</span> <a href="matrix_3.html#K6">*</a> <font color="Maroon">M1</font></span>)</span> <a href="matrix_3.html#K4">*</a> <font color="Maroon">M2</font></span>)</span> = 
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">M1</font></span>)</span>
<b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
<br/>.= 
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_3.html#K4">*</a> <font color="Maroon">M2</font></span>)</span>
<b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
;<br/>
 <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"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="polynom8.html#K1" target="_self">emb</a> <font color="Maroon">m</font>,<font color="Maroon">L</font></span>)</span> <a href="matrix_3.html#K6">*</a> <font color="Maroon">M1</font></span>)</span> <a href="matrix_3.html#K4">*</a> <font color="Maroon">M2</font></span>)</span></span>)</span> = 
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">M2</font></span>)</span>
<b>by </b><i>, <a class="ref" href="matrix_3.html#D4">MATRIX_3:def 4</a></i>
<br/>.= 
 <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"><font color="Maroon">M1</font> <a href="matrix_3.html#K4">*</a> <font color="Maroon">M2</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="matrix_3.html#D4">MATRIX_3:def 4</a></i>
;<br/>
<b>hence </b><a NAME="E3:10_6_2"/>
<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"><font color="Maroon">M1</font> <a href="matrix_3.html#K4">*</a> <font color="Maroon">M2</font></span>)</span>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<i><font color="Green">E63</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#K1" target="_self">emb</a> <font color="Maroon">m</font>,<font color="Maroon">L</font></span>)</span> <a href="matrix_3.html#K6">*</a> <span class="p2">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_3.html#K4">*</a> <font color="Maroon">M2</font></span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font> = 
<span class="p1">(<span class="default"><a href="polynom8.html#K1" target="_self">emb</a> <font color="Maroon">m</font>,<font color="Maroon">L</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_3.html#K4">*</a> <font color="Maroon">M2</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span>
<b>by </b><i>, <a class="ref" href="matrix_3.html#D5">MATRIX_3:def 5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="polynom8.html#K1" target="_self">emb</a> <font color="Maroon">m</font>,<font color="Maroon">L</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">M1</font>,<font color="Maroon">i</font></span>)</span> <a href="fvsum_1.html#K13">"*"</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">M2</font>,<font color="Maroon">j</font></span>)</span></span>)</span>
<b>by </b><i>, , <a class="ref" href="matrix_3.html#D4">MATRIX_3:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="polynom8.html#K1" target="_self">emb</a> <font color="Maroon">m</font>,<font color="Maroon">L</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <span class="p2">(<span class="default"><a href="fvsum_1.html#K11">mlt</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">M1</font>,<font color="Maroon">i</font></span>)</span>,<span class="p3">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">M2</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="fvsum_1.html#D10">FVSUM_1:def 10</a></i>
;<br/>
<i><font color="Green">E64</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom8.html#K1" target="_self">emb</a> <font color="Maroon">m</font>,<font color="Maroon">L</font></span>)</span> <a href="matrix_3.html#K6">*</a> <font color="Maroon">M1</font></span>)</span> <a href="matrix_3.html#K4">*</a> <font color="Maroon">M2</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font> = 
<span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom8.html#K1" target="_self">emb</a> <font color="Maroon">m</font>,<font color="Maroon">L</font></span>)</span> <a href="matrix_3.html#K6">*</a> <font color="Maroon">M1</font></span>)</span>,<font color="Maroon">i</font></span>)</span> <a href="fvsum_1.html#K13">"*"</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">M2</font>,<font color="Maroon">j</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="matrix_3.html#D4">MATRIX_3:def 4</a></i>
<br/>.= 
 <a href="rlvect_1.html#K9">Sum</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K11">mlt</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="polynom8.html#K1" target="_self">emb</a> <font color="Maroon">m</font>,<font color="Maroon">L</font></span>)</span> <a href="matrix_3.html#K6">*</a> <font color="Maroon">M1</font></span>)</span>,<font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">M2</font>,<font color="Maroon">j</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="fvsum_1.html#D10">FVSUM_1:def 10</a></i>
;<br/>
<a NAME="E7:10_6"/>
<span class="p1">(<span class="default"><a href="polynom8.html#K1" target="_self">emb</a> <font color="Maroon">m</font>,<font color="Maroon">L</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <span class="p2">(<span class="default"><a href="fvsum_1.html#K11">mlt</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">M1</font>,<font color="Maroon">i</font></span>)</span>,<span class="p3">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">M2</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K9">Sum</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K11">mlt</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="polynom8.html#K1" target="_self">emb</a> <font color="Maroon">m</font>,<font color="Maroon">L</font></span>)</span> <a href="matrix_3.html#K6">*</a> <font color="Maroon">M1</font></span>)</span>,<font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">M2</font>,<font color="Maroon">j</font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_6_5"><b>proof </b></a><div class="add">

<a NAME="E1:10_6_5"/>
<span class="p1">(<span class="default"><a href="polynom8.html#K1" target="_self">emb</a> <font color="Maroon">m</font>,<font color="Maroon">L</font></span>)</span> <a href="fvsum_1.html#K9">*</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K11">mlt</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">M1</font>,<font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">M2</font>,<font color="Maroon">j</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="fvsum_1.html#K11">mlt</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom8.html#K1" target="_self">emb</a> <font color="Maroon">m</font>,<font color="Maroon">L</font></span>)</span> <a href="matrix_3.html#K6">*</a> <font color="Maroon">M1</font></span>)</span>,<font color="Maroon">i</font></span>)</span>,<span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">M2</font>,<font color="Maroon">j</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_6_5_1"><b>proof </b></a><div class="add">

<a NAME="E1:10_6_5_1"/><i><font color="Green">E65</font></i>: 
 <a href="matrix_1.html#K8">Line</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#K1" target="_self">emb</a> <font color="Maroon">m</font>,<font color="Maroon">L</font></span>)</span> <a href="matrix_3.html#K6">*</a> <font color="Maroon">M1</font></span>)</span>,<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="polynom8.html#K1" target="_self">emb</a> <font color="Maroon">m</font>,<font color="Maroon">L</font></span>)</span> <a href="fvsum_1.html#K10">*</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">M1</font>,<font color="Maroon">i</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_6_5_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:10_6_5_1_1"/><i><font color="Green">E66</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom8.html#K1" target="_self">emb</a> <font color="Maroon">m</font>,<font color="Maroon">L</font></span>)</span> <a href="matrix_3.html#K6">*</a> <font color="Maroon">M1</font></span>)</span>,<font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#K1" target="_self">emb</a> <font color="Maroon">m</font>,<font color="Maroon">L</font></span>)</span> <a href="fvsum_1.html#K10">*</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">M1</font>,<font color="Maroon">i</font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_6_5_1_1_1"><b>proof </b></a><div class="add">

 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom8.html#K1" target="_self">emb</a> <font color="Maroon">m</font>,<font color="Maroon">L</font></span>)</span> <a href="matrix_3.html#K6">*</a> <font color="Maroon">M1</font></span>)</span>,<font color="Maroon">i</font></span>)</span> = 
 <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#K1" target="_self">emb</a> <font color="Maroon">m</font>,<font color="Maroon">L</font></span>)</span> <a href="matrix_3.html#K6">*</a> <font color="Maroon">M1</font></span>)</span>
<b>by </b><i><a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>
<br/>.= 
 <a href="matrix_1.html#K1">width</a> <font color="Maroon">M1</font>
<b>by </b><i><a class="ref" href="matrix_3.html#D5">MATRIX_3:def 5</a></i>
;<br/>
<a NAME="E2:10_6_5_1_1_1"/><b>then </b><i><font color="Green">E67</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom8.html#K1" target="_self">emb</a> <font color="Maroon">m</font>,<font color="Maroon">L</font></span>)</span> <a href="matrix_3.html#K6">*</a> <font color="Maroon">M1</font></span>)</span>,<font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">M1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom8.html#K1" target="_self">emb</a> <font color="Maroon">m</font>,<font color="Maroon">L</font></span>)</span> <a href="fvsum_1.html#K10">*</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">M1</font>,<font color="Maroon">i</font></span>)</span></span>)</span></span>)</span> = 
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#K1" target="_self">emb</a> <font color="Maroon">m</font>,<font color="Maroon">L</font></span>)</span> <a href="fvsum_1.html#K10">*</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">M1</font>,<font color="Maroon">i</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
<br/>.= 
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">M1</font>,<font color="Maroon">i</font></span>)</span>
<b>by </b><i><a class="ref" href="polynom1.html#D2">POLYNOM1:def 2</a></i>
<br/>.= 
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">M1</font>,<font color="Maroon">i</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
;<br/>
<b>then </b> <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#K1" target="_self">emb</a> <font color="Maroon">m</font>,<font color="Maroon">L</font></span>)</span> <a href="fvsum_1.html#K10">*</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">M1</font>,<font color="Maroon">i</font></span>)</span></span>)</span> = 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">M1</font>,<font color="Maroon">i</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T8">FINSEQ_1:8</a></i>
<br/>.= 
 <a href="matrix_1.html#K1">width</a> <font color="Maroon">M1</font>
<b>by </b><i><a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>
;<br/>
<b>hence </b><a NAME="E5:10_6_5_1_1_1"/>
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom8.html#K1" target="_self">emb</a> <font color="Maroon">m</font>,<font color="Maroon">L</font></span>)</span> <a href="matrix_3.html#K6">*</a> <font color="Maroon">M1</font></span>)</span>,<font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#K1" target="_self">emb</a> <font color="Maroon">m</font>,<font color="Maroon">L</font></span>)</span> <a href="fvsum_1.html#K10">*</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">M1</font>,<font color="Maroon">i</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:10_6_5_1_1"/>
for <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom8.html#K1" target="_self">emb</a> <font color="Maroon">m</font>,<font color="Maroon">L</font></span>)</span> <a href="matrix_3.html#K6">*</a> <font color="Maroon">M1</font></span>)</span>,<font color="Maroon">i</font></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom8.html#K1" target="_self">emb</a> <font color="Maroon">m</font>,<font color="Maroon">L</font></span>)</span> <a href="matrix_3.html#K6">*</a> <font color="Maroon">M1</font></span>)</span>,<font color="Maroon">i</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#K1" target="_self">emb</a> <font color="Maroon">m</font>,<font color="Maroon">L</font></span>)</span> <a href="fvsum_1.html#K10">*</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">M1</font>,<font color="Maroon">i</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_6_5_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:10_6_5_1_1_2"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom8.html#K1" target="_self">emb</a> <font color="Maroon">m</font>,<font color="Maroon">L</font></span>)</span> <a href="matrix_3.html#K6">*</a> <font color="Maroon">M1</font></span>)</span>,<font color="Maroon">i</font></span>)</span>
 ;<br/>

<i><font color="Green">E69</font></i>:  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom8.html#K1" target="_self">emb</a> <font color="Maroon">m</font>,<font color="Maroon">L</font></span>)</span> <a href="matrix_3.html#K6">*</a> <font color="Maroon">M1</font></span>)</span>,<font color="Maroon">i</font></span>)</span> = 
 <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#K1" target="_self">emb</a> <font color="Maroon">m</font>,<font color="Maroon">L</font></span>)</span> <a href="matrix_3.html#K6">*</a> <font color="Maroon">M1</font></span>)</span>
<b>by </b><i><a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>
<br/>.= 
 <a href="matrix_1.html#K1">width</a> <font color="Maroon">M1</font>
<b>by </b><i><a class="ref" href="matrix_3.html#D5">MATRIX_3:def 5</a></i>
;<br/>
<a NAME="E3:10_6_5_1_1_2"/>
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom8.html#K1" target="_self">emb</a> <font color="Maroon">m</font>,<font color="Maroon">L</font></span>)</span> <a href="matrix_3.html#K6">*</a> <font color="Maroon">M1</font></span>)</span>,<font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">M1</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E4:10_6_5_1_1_2"/><b>then </b><i><font color="Green">E70</font></i>: 
<font color="Maroon">k</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"><span class="p3">(<span class="default"><a href="polynom8.html#K1" target="_self">emb</a> <font color="Maroon">m</font>,<font color="Maroon">L</font></span>)</span> <a href="matrix_3.html#K6">*</a> <font color="Maroon">M1</font></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="matrix_3.html#D5">MATRIX_3:def 5</a></i>;<br/>
<a NAME="E5:10_6_5_1_1_2"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">k</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> <font color="Maroon">M1</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a>, </i>;<br/>
<a NAME="E6:10_6_5_1_1_2"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">M1</font>,<font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">M1</font>
 
<b>by </b><i><a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>;<br/>
<a NAME="E7:10_6_5_1_1_2"/><b>then </b>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">M1</font>,<font color="Maroon">i</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E8:10_6_5_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">M1</font>,<font color="Maroon">i</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">M1</font>,<font color="Maroon">i</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font>
 
<b>by </b><i><a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">a</font> = <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">M1</font>,<font color="Maroon">i</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> ;<br/>
<a NAME="E10:10_6_5_1_1_2"/><i><font color="Green">E72</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#K1" target="_self">emb</a> <font color="Maroon">m</font>,<font color="Maroon">L</font></span>)</span> <a href="fvsum_1.html#K10">*</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">M1</font>,<font color="Maroon">i</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="polynom8.html#K1" target="_self">emb</a> <font color="Maroon">m</font>,<font color="Maroon">L</font></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">a</font>
 
<b>by </b><i>, , <a class="ref" href="fvsum_1.html#T62">FVSUM_1:62</a></i>;<br/>
<a NAME="E11:10_6_5_1_1_2"/><i><font color="Green">E73</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i</font>,<font color="Maroon">k</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <font color="Maroon">M1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_6_5_1_1_2_2"><b>proof </b></a><div class="add">

<a NAME="E1:10_6_5_1_1_2_2"/><i><font color="Green">E74</font></i>: 
 <a href="matrix_1.html#K2">Indices</a> <font color="Maroon">M1</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="mcart_1.html#K12">[:</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m</font></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span></span><a href="mcart_1.html#K12">:]</a></span>
 
<b>by </b><i><a class="ref" href="matrix_1.html#T24">MATRIX_1:24</a>, </i>;<br/>
<a NAME="E2:10_6_5_1_1_2_2"/><i><font color="Green">E75</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"><font color="Maroon">M1</font> <a href="matrix_3.html#K4">*</a> <font color="Maroon">M2</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_3.html#K4">*</a> <font color="Maroon">M2</font></span>)</span> = 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">M1</font>
<b>by </b><i>, <a class="ref" href="matrix_3.html#D4">MATRIX_3:def 4</a></i>
<br/>.= 
<font color="Maroon">m</font>
<b>by </b><i><a class="ref" href="matrix_1.html#T24">MATRIX_1:24</a>, </i>
;<br/>
<a NAME="E4:10_6_5_1_1_2_2"/><b>then </b><i><font color="Green">E77</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_3.html#K4">*</a> <font color="Maroon">M2</font></span>)</span> <a href="hidden.html#R1">=</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="E5:10_6_5_1_1_2_2"/>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font>
 
<b>by </b><i>, <a class="ref" href="matrix_1.html#T24">MATRIX_1:24</a>, </i>;<br/>
<b>hence </b><a NAME="E6:10_6_5_1_1_2_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i</font>,<font color="Maroon">k</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <font color="Maroon">M1</font>
 <b>by </b><i>, <a class="txt" href="polynom8.html#E1:10"><i><font color="Green">E31</font></i></a>, , <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E12:10_6_5_1_1_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M1</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">k</font>
 
<b>by </b><i>, <a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>;<br/>
<a NAME="E13:10_6_5_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="polynom8.html#K1" target="_self">emb</a> <font color="Maroon">m</font>,<font color="Maroon">L</font></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#K1" target="_self">emb</a> <font color="Maroon">m</font>,<font color="Maroon">L</font></span>)</span> <a href="matrix_3.html#K6">*</a> <font color="Maroon">M1</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">k</font>
 
<b>by </b><i><a class="txt" href="polynom8.html#E1"><i><font color="Green">Lemma29</font></i></a>, <a class="ref" href="matrix_3.html#D5">MATRIX_3:def 5</a></i>;<br/>
<b>hence </b><a NAME="E14:10_6_5_1_1_2"/>
<span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom8.html#K1" target="_self">emb</a> <font color="Maroon">m</font>,<font color="Maroon">L</font></span>)</span> <a href="matrix_3.html#K6">*</a> <font color="Maroon">M1</font></span>)</span>,<font color="Maroon">i</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#K1" target="_self">emb</a> <font color="Maroon">m</font>,<font color="Maroon">L</font></span>)</span> <a href="fvsum_1.html#K10">*</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">M1</font>,<font color="Maroon">i</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>
 <b>by </b><i>, <a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a>, </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:10_6_5_1_1"/>
 <a href="matrix_1.html#K8">Line</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#K1" target="_self">emb</a> <font color="Maroon">m</font>,<font color="Maroon">L</font></span>)</span> <a href="matrix_3.html#K6">*</a> <font color="Maroon">M1</font></span>)</span>,<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="polynom8.html#K1" target="_self">emb</a> <font color="Maroon">m</font>,<font color="Maroon">L</font></span>)</span> <a href="fvsum_1.html#K10">*</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">M1</font>,<font color="Maroon">i</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T17">FINSEQ_1:17</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:10_6_5_1"/>
<span class="p1">(<span class="default"><a href="polynom8.html#K1" target="_self">emb</a> <font color="Maroon">m</font>,<font color="Maroon">L</font></span>)</span> <a href="fvsum_1.html#K9">*</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K11">mlt</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">M1</font>,<font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">M2</font>,<font color="Maroon">j</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="fvsum_1.html#K11">mlt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#K1" target="_self">emb</a> <font color="Maroon">m</font>,<font color="Maroon">L</font></span>)</span> <a href="fvsum_1.html#K10">*</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">M1</font>,<font color="Maroon">i</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">M2</font>,<font color="Maroon">j</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_6_5_1_2"><b>proof </b></a><div class="add">

<a NAME="E1:10_6_5_1_2"/><i><font color="Green">E78</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K11">mlt</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom8.html#K1" target="_self">emb</a> <font color="Maroon">m</font>,<font color="Maroon">L</font></span>)</span> <a href="fvsum_1.html#K10">*</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">M1</font>,<font color="Maroon">i</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">M2</font>,<font color="Maroon">j</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#K1" target="_self">emb</a> <font color="Maroon">m</font>,<font color="Maroon">L</font></span>)</span> <a href="fvsum_1.html#K9">*</a> <span class="p2">(<span class="default"><a href="fvsum_1.html#K11">mlt</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">M1</font>,<font color="Maroon">i</font></span>)</span>,<span class="p3">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">M2</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_6_5_1_2_1"><b>proof </b></a><div class="add">

 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom8.html#K1" target="_self">emb</a> <font color="Maroon">m</font>,<font color="Maroon">L</font></span>)</span> <a href="fvsum_1.html#K10">*</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">M1</font>,<font color="Maroon">i</font></span>)</span></span>)</span></span>)</span> = 
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#K1" target="_self">emb</a> <font color="Maroon">m</font>,<font color="Maroon">L</font></span>)</span> <a href="fvsum_1.html#K10">*</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">M1</font>,<font color="Maroon">i</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
<br/>.= 
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">M1</font>,<font color="Maroon">i</font></span>)</span>
<b>by </b><i><a class="ref" href="polynom1.html#D2">POLYNOM1:def 2</a></i>
<br/>.= 
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">M1</font>,<font color="Maroon">i</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
;<br/>
<b>then </b><i><font color="Green">E79</font></i>:  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#K1" target="_self">emb</a> <font color="Maroon">m</font>,<font color="Maroon">L</font></span>)</span> <a href="fvsum_1.html#K10">*</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">M1</font>,<font color="Maroon">i</font></span>)</span></span>)</span> = 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">M1</font>,<font color="Maroon">i</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T8">FINSEQ_1:8</a></i>
<br/>.= 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">M2</font>
<b>by </b><i>, <a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>
<br/>.= 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">M2</font>,<font color="Maroon">j</font></span>)</span>
<b>by </b><i><a class="ref" href="matrix_1.html#D9">MATRIX_1:def 9</a></i>
;<br/>
<i><font color="Green">E80</font></i>:  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#K1" target="_self">emb</a> <font color="Maroon">m</font>,<font color="Maroon">L</font></span>)</span> <a href="fvsum_1.html#K10">*</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">M1</font>,<font color="Maroon">i</font></span>)</span></span>)</span> = 
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom8.html#K1" target="_self">emb</a> <font color="Maroon">m</font>,<font color="Maroon">L</font></span>)</span> <a href="fvsum_1.html#K10">*</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">M1</font>,<font color="Maroon">i</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
<br/>.= 
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="fvsum_1.html#K11">mlt</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="polynom8.html#K1" target="_self">emb</a> <font color="Maroon">m</font>,<font color="Maroon">L</font></span>)</span> <a href="fvsum_1.html#K10">*</a> <span class="p4">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">M1</font>,<font color="Maroon">i</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">M2</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="matrix_3.html#T8">MATRIX_3:8</a></i>
<br/>.= 
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K11">mlt</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom8.html#K1" target="_self">emb</a> <font color="Maroon">m</font>,<font color="Maroon">L</font></span>)</span> <a href="fvsum_1.html#K10">*</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">M1</font>,<font color="Maroon">i</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">M2</font>,<font color="Maroon">j</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
;<br/>
<i><font color="Green">E91</font></i>:  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">M1</font>,<font color="Maroon">i</font></span>)</span> = 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">M2</font>
<b>by </b><i>, <a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>
<br/>.= 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">M2</font>,<font color="Maroon">j</font></span>)</span>
<b>by </b><i><a class="ref" href="matrix_1.html#D9">MATRIX_1:def 9</a></i>
;<br/>
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#K1" target="_self">emb</a> <font color="Maroon">m</font>,<font color="Maroon">L</font></span>)</span> <a href="fvsum_1.html#K10">*</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">M1</font>,<font color="Maroon">i</font></span>)</span></span>)</span> = 
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">M1</font>,<font color="Maroon">i</font></span>)</span>
<b>by </b><i><a class="ref" href="polynom1.html#D2">POLYNOM1:def 2</a></i>
<br/>.= 
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">M1</font>,<font color="Maroon">i</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
<br/>.= 
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="fvsum_1.html#K11">mlt</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">M1</font>,<font color="Maroon">i</font></span>)</span>,<span class="p3">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">M2</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="matrix_3.html#T8">MATRIX_3:8</a></i>
<br/>.= 
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K11">mlt</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">M1</font>,<font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">M2</font>,<font color="Maroon">j</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
;<br/>
<b>hence </b><a NAME="E6:10_6_5_1_2_1"/>
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K11">mlt</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom8.html#K1" target="_self">emb</a> <font color="Maroon">m</font>,<font color="Maroon">L</font></span>)</span> <a href="fvsum_1.html#K10">*</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">M1</font>,<font color="Maroon">i</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">M2</font>,<font color="Maroon">j</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#K1" target="_self">emb</a> <font color="Maroon">m</font>,<font color="Maroon">L</font></span>)</span> <a href="fvsum_1.html#K9">*</a> <span class="p2">(<span class="default"><a href="fvsum_1.html#K11">mlt</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">M1</font>,<font color="Maroon">i</font></span>)</span>,<span class="p3">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">M2</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="polynom1.html#D2">POLYNOM1:def 2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:10_6_5_1_2"/>
for <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K11">mlt</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom8.html#K1" target="_self">emb</a> <font color="Maroon">m</font>,<font color="Maroon">L</font></span>)</span> <a href="fvsum_1.html#K10">*</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">M1</font>,<font color="Maroon">i</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">M2</font>,<font color="Maroon">j</font></span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="fvsum_1.html#K11">mlt</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom8.html#K1" target="_self">emb</a> <font color="Maroon">m</font>,<font color="Maroon">L</font></span>)</span> <a href="fvsum_1.html#K10">*</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">M1</font>,<font color="Maroon">i</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">M2</font>,<font color="Maroon">j</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#K1" target="_self">emb</a> <font color="Maroon">m</font>,<font color="Maroon">L</font></span>)</span> <a href="fvsum_1.html#K9">*</a> <span class="p2">(<span class="default"><a href="fvsum_1.html#K11">mlt</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">M1</font>,<font color="Maroon">i</font></span>)</span>,<span class="p3">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">M2</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_6_5_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:10_6_5_1_2_2"/><i><font color="Green">E92</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K11">mlt</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom8.html#K1" target="_self">emb</a> <font color="Maroon">m</font>,<font color="Maroon">L</font></span>)</span> <a href="fvsum_1.html#K10">*</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">M1</font>,<font color="Maroon">i</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">M2</font>,<font color="Maroon">j</font></span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:10_6_5_1_2_2"/><i><font color="Green">E93</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#K1" target="_self">emb</a> <font color="Maroon">m</font>,<font color="Maroon">L</font></span>)</span> <a href="fvsum_1.html#K9">*</a> <span class="p2">(<span class="default"><a href="fvsum_1.html#K11">mlt</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">M1</font>,<font color="Maroon">i</font></span>)</span>,<span class="p3">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">M2</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K11">mlt</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">M1</font>,<font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">M2</font>,<font color="Maroon">j</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="polynom1.html#D2">POLYNOM1:def 2</a></i>;<br/>
<a NAME="E3:10_6_5_1_2_2"/><i><font color="Green">E94</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K11">mlt</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">M1</font>,<font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">M2</font>,<font color="Maroon">j</font></span>)</span></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="polynom1.html#D2">POLYNOM1:def 2</a></i>;<br/>
<i><font color="Green">E95</font></i>:  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">M1</font>,<font color="Maroon">i</font></span>)</span> = 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">M2</font>
<b>by </b><i>, <a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>
<br/>.= 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">M2</font>,<font color="Maroon">j</font></span>)</span>
<b>by </b><i><a class="ref" href="matrix_1.html#D9">MATRIX_1:def 9</a></i>
;<br/>
<i><font color="Green">E96</font></i>:  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">M1</font>,<font color="Maroon">i</font></span>)</span> = 
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">M1</font>,<font color="Maroon">i</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
<br/>.= 
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="fvsum_1.html#K11">mlt</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">M1</font>,<font color="Maroon">i</font></span>)</span>,<span class="p3">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">M2</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="matrix_3.html#T8">MATRIX_3:8</a></i>
<br/>.= 
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K11">mlt</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">M1</font>,<font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">M2</font>,<font color="Maroon">j</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
;<br/>
<a NAME="E6:10_6_5_1_2_2"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">M1</font>,<font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">M1</font>
 
<b>by </b><i><a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>;<br/>
<a NAME="E7:10_6_5_1_2_2"/><b>then </b><i><font color="Green">E97</font></i>: 
<font color="Maroon">k</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> <font color="Maroon">M1</font></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E8:10_6_5_1_2_2"/><b>then </b><i><font color="Green">E98</font></i>: 
<span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">M1</font>,<font color="Maroon">i</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M1</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">k</font>
 
<b>by </b><i><a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>;<br/>
<a NAME="E9:10_6_5_1_2_2"/>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">M2</font>
 
<b>by </b><i>, <a class="txt" href="polynom8.html#E1"><i><font color="Green">Lemma29</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E10:10_6_5_1_2_2"/><b>then </b><i><font color="Green">E99</font></i>: 
<span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">M2</font>,<font color="Maroon">j</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M2</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">k</font>,<font color="Maroon">j</font>
 
<b>by </b><i><a class="ref" href="matrix_1.html#D9">MATRIX_1:def 9</a></i>;<br/>
<a NAME="E11:10_6_5_1_2_2"/><i><font color="Green">E100</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#K1" target="_self">emb</a> <font color="Maroon">m</font>,<font color="Maroon">L</font></span>)</span> <a href="fvsum_1.html#K10">*</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">M1</font>,<font color="Maroon">i</font></span>)</span></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="polynom1.html#D2">POLYNOM1:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c</font> = <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">M2</font>,<font color="Maroon">j</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>, <font color="Maroon">d</font> = <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">M1</font>,<font color="Maroon">i</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> <b>by </b><i>, <a class="txt" href="polynom8.html#E1:10"><i><font color="Green">E31</font></i></a></i>;<br/>
<a NAME="E13:10_6_5_1_2_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#K1" target="_self">emb</a> <font color="Maroon">m</font>,<font color="Maroon">L</font></span>)</span> <a href="fvsum_1.html#K10">*</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">M1</font>,<font color="Maroon">i</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="polynom8.html#K1" target="_self">emb</a> <font color="Maroon">m</font>,<font color="Maroon">L</font></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">d</font>
 
<b>by </b><i>, <a class="ref" href="fvsum_1.html#T62">FVSUM_1:62</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">b</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#K1" target="_self">emb</a> <font color="Maroon">m</font>,<font color="Maroon">L</font></span>)</span> <a href="fvsum_1.html#K10">*</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">M1</font>,<font color="Maroon">i</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> ;<br/>
<a NAME="E15:10_6_5_1_2_2"/>
<span class="p1">(<span class="default"><a href="fvsum_1.html#K11">mlt</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">M1</font>,<font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">M2</font>,<font color="Maroon">j</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> <a href="group_1.html#K10">*</a> <font color="Maroon">d</font>
 
<b>by </b><i>, <a class="ref" href="fvsum_1.html#T73">FVSUM_1:73</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">a</font> = <span class="p1">(<span class="default"><a href="fvsum_1.html#K11">mlt</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">M1</font>,<font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">M2</font>,<font color="Maroon">j</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> ;<br/>
<a NAME="E17:10_6_5_1_2_2"/><i><font color="Green">E104</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#K1" target="_self">emb</a> <font color="Maroon">m</font>,<font color="Maroon">L</font></span>)</span> <a href="fvsum_1.html#K9">*</a> <span class="p2">(<span class="default"><a href="fvsum_1.html#K11">mlt</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">M1</font>,<font color="Maroon">i</font></span>)</span>,<span class="p3">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">M2</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="polynom8.html#K1" target="_self">emb</a> <font color="Maroon">m</font>,<font color="Maroon">L</font></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">a</font>
 
<b>by </b><i>, , <a class="ref" href="fvsum_1.html#T62">FVSUM_1:62</a></i>;<br/>
<font color="Maroon">b</font> <a href="group_1.html#K10">*</a> <font color="Maroon">c</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#K1" target="_self">emb</a> <font color="Maroon">m</font>,<font color="Maroon">L</font></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">d</font></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">c</font>
<b>by </b><i>, <a class="ref" href="fvsum_1.html#T62">FVSUM_1:62</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="polynom8.html#K1" target="_self">emb</a> <font color="Maroon">m</font>,<font color="Maroon">L</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="group_1.html#K10">*</a> <font color="Maroon">c</font></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#D4">GROUP_1:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="polynom8.html#K1" target="_self">emb</a> <font color="Maroon">m</font>,<font color="Maroon">L</font></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">a</font>
<b>by </b><i>, , , <a class="ref" href="fvsum_1.html#T73">FVSUM_1:73</a></i>
;<br/>
<b>hence </b><a NAME="E19:10_6_5_1_2_2"/>
<span class="p1">(<span class="default"><a href="fvsum_1.html#K11">mlt</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom8.html#K1" target="_self">emb</a> <font color="Maroon">m</font>,<font color="Maroon">L</font></span>)</span> <a href="fvsum_1.html#K10">*</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">M1</font>,<font color="Maroon">i</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">M2</font>,<font color="Maroon">j</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#K1" target="_self">emb</a> <font color="Maroon">m</font>,<font color="Maroon">L</font></span>)</span> <a href="fvsum_1.html#K9">*</a> <span class="p2">(<span class="default"><a href="fvsum_1.html#K11">mlt</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">M1</font>,<font color="Maroon">i</font></span>)</span>,<span class="p3">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">M2</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="txt" href="polynom8.html#E2"><i><font color="Green">Lemma33</font></i></a>, , <a class="ref" href="fvsum_1.html#T73">FVSUM_1:73</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:10_6_5_1_2"/>
<span class="p1">(<span class="default"><a href="polynom8.html#K1" target="_self">emb</a> <font color="Maroon">m</font>,<font color="Maroon">L</font></span>)</span> <a href="fvsum_1.html#K9">*</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K11">mlt</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">M1</font>,<font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">M2</font>,<font color="Maroon">j</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="fvsum_1.html#K11">mlt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#K1" target="_self">emb</a> <font color="Maroon">m</font>,<font color="Maroon">L</font></span>)</span> <a href="fvsum_1.html#K10">*</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">M1</font>,<font color="Maroon">i</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">M2</font>,<font color="Maroon">j</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T17">FINSEQ_1:17</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:10_6_5_1"/>
<span class="p1">(<span class="default"><a href="polynom8.html#K1" target="_self">emb</a> <font color="Maroon">m</font>,<font color="Maroon">L</font></span>)</span> <a href="fvsum_1.html#K9">*</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K11">mlt</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">M1</font>,<font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">M2</font>,<font color="Maroon">j</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="fvsum_1.html#K11">mlt</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom8.html#K1" target="_self">emb</a> <font color="Maroon">m</font>,<font color="Maroon">L</font></span>)</span> <a href="matrix_3.html#K6">*</a> <font color="Maroon">M1</font></span>)</span>,<font color="Maroon">i</font></span>)</span>,<span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">M2</font>,<font color="Maroon">j</font></span>)</span>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:10_6_5"/>
<span class="p1">(<span class="default"><a href="polynom8.html#K1" target="_self">emb</a> <font color="Maroon">m</font>,<font color="Maroon">L</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <span class="p2">(<span class="default"><a href="fvsum_1.html#K11">mlt</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">M1</font>,<font color="Maroon">i</font></span>)</span>,<span class="p3">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">M2</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K9">Sum</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K11">mlt</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="polynom8.html#K1" target="_self">emb</a> <font color="Maroon">m</font>,<font color="Maroon">L</font></span>)</span> <a href="matrix_3.html#K6">*</a> <font color="Maroon">M1</font></span>)</span>,<font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">M2</font>,<font color="Maroon">j</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="fvsum_1.html#T92">FVSUM_1:92</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E8:10_6"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom8.html#K1" target="_self">emb</a> <font color="Maroon">m</font>,<font color="Maroon">L</font></span>)</span> <a href="matrix_3.html#K6">*</a> <font color="Maroon">M1</font></span>)</span> <a href="matrix_3.html#K4">*</a> <font color="Maroon">M2</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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#K1" target="_self">emb</a> <font color="Maroon">m</font>,<font color="Maroon">L</font></span>)</span> <a href="matrix_3.html#K6">*</a> <span class="p2">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_3.html#K4">*</a> <font color="Maroon">M2</font></span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<b>thus </b><a NAME="E9:10"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#K1" target="_self">emb</a> <font color="Maroon">m</font>,<font color="Maroon">L</font></span>)</span> <a href="matrix_3.html#K6">*</a> <font color="Maroon">M1</font></span>)</span> <a href="matrix_3.html#K4">*</a> <font color="Maroon">M2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="polynom8.html#K1" target="_self">emb</a> <font color="Maroon">m</font>,<font color="Maroon">L</font></span>)</span> <a href="matrix_3.html#K6">*</a> <span class="p1">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_3.html#K4">*</a> <font color="Maroon">M2</font></span>)</span>
 <b>by </b><i>, , , , <a class="ref" href="matrix_1.html#T21">MATRIX_1:21</a></i>;<br/>


</div>
