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

<b>let </b><font color="Maroon" title="c1">L</font> be   <a href="vectsp_1.html#NM3" title="VECTSP_1:NM.3">Field</a>;<br/><b>let </b><font color="Maroon" title="c2">m</font>, <font color="Maroon" title="c3">n</font>, <font color="Maroon" title="c4">k</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>



<b>assume </b><a NAME="E1:10"/><i><font color="Green" title="E9">A1</font></i>: 
( <font color="Maroon" title="c2">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 &amp; <font color="Maroon" title="c3">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 )
 ;<br/>

<b>let </b><font color="Maroon" title="c5">M1</font> be   <a href="matrix_1.html#NM6" title="MATRIX_1:NM.6">Matrix</a> of <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c1">L</font>;<br/><b>let </b><font color="Maroon" title="c6">M2</font> be   <a href="matrix_1.html#NM6" title="MATRIX_1:NM.6">Matrix</a> of <font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c4">k</font>,<font color="Maroon" title="c1">L</font>;<br/>



<a NAME="E2:10"/><i><font color="Green" title="E10">A2</font></i>: 
 <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">L</font></span>)</span> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <font color="Maroon" title="c5">M1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">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" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">L</font></span>)</span> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <font color="Maroon" title="c5">M1</font></span>)</span> = 
 <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">M1</font>
<b>by </b><i><a class="ref" href="matrix_3.html#D5" title="MATRIX_3:def.5">MATRIX_3:def 5</a></i>
<br/>.= 
<font color="Maroon" title="c3">n</font>
<b>by </b><i><a class="ref" href="matrix_1.html#T24" title="MATRIX_1:th.24">MATRIX_1:24</a>, <a class="txt" href="polynom8.html#E1:10"><i><font color="Green" title="E9">A1</font></i></a></i>
<br/>.= 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">M2</font>
<b>by </b><i><a class="ref" href="matrix_1.html#T24" title="MATRIX_1:th.24">MATRIX_1:24</a>, <a class="txt" href="polynom8.html#E1:10"><i><font color="Green" title="E9">A1</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E2:10_1"/>
 <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">L</font></span>)</span> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <font color="Maroon" title="c5">M1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">M2</font>
 ;<br/>


</div><b>end;</b></div>
<i><font color="Green" title="E11">A3</font></i>:  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">M1</font> = 
<font color="Maroon" title="c3">n</font>
<b>by </b><i><a class="ref" href="matrix_1.html#T24" title="MATRIX_1:th.24">MATRIX_1:24</a>, <a class="txt" href="polynom8.html#E1:10"><i><font color="Green" title="E9">A1</font></i></a></i>
<br/>.= 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">M2</font>
<b>by </b><i><a class="ref" href="matrix_1.html#T24" title="MATRIX_1:th.24">MATRIX_1:24</a>, <a class="txt" href="polynom8.html#E1:10"><i><font color="Green" title="E9">A1</font></i></a></i>
;<br/>
<i><font color="Green" title="E12">A4</font></i>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">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" title="POLYNOM8:func.1">emb</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">L</font></span>)</span> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <font color="Maroon" title="c5">M1</font></span>)</span> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <font color="Maroon" title="c6">M2</font></span>)</span> = 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">L</font></span>)</span> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <font color="Maroon" title="c5">M1</font></span>)</span>
<b>by </b><i><a class="txt" href="polynom8.html#E2:10"><i><font color="Green" title="E10">A2</font></i></a>, <a class="ref" href="matrix_3.html#D4" title="MATRIX_3:def.4">MATRIX_3:def 4</a></i>
<br/>.= 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">M1</font>
<b>by </b><i><a class="ref" href="matrix_3.html#D5" title="MATRIX_3:def.5">MATRIX_3:def 5</a></i>
;<br/>
<i><font color="Green" title="E13">A5</font></i>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">L</font></span>)</span> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">M1</font> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <font color="Maroon" title="c6">M2</font></span>)</span></span>)</span> = 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">M1</font> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <font color="Maroon" title="c6">M2</font></span>)</span>
<b>by </b><i><a class="ref" href="matrix_3.html#D5" title="MATRIX_3:def.5">MATRIX_3:def 5</a></i>
<br/>.= 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">M1</font>
<b>by </b><i><a class="txt" href="polynom8.html#E3:10"><i><font color="Green" title="E11">A3</font></i></a>, <a class="ref" href="matrix_3.html#D4" title="MATRIX_3:def.4">MATRIX_3:def 4</a></i>
;<br/>
 <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">L</font></span>)</span> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">M1</font> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <font color="Maroon" title="c6">M2</font></span>)</span></span>)</span> = 
 <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">M1</font> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <font color="Maroon" title="c6">M2</font></span>)</span>
<b>by </b><i><a class="ref" href="matrix_3.html#D5" title="MATRIX_3:def.5">MATRIX_3:def 5</a></i>
<br/>.= 
 <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c6">M2</font>
<b>by </b><i><a class="txt" href="polynom8.html#E3:10"><i><font color="Green" title="E11">A3</font></i></a>, <a class="ref" href="matrix_3.html#D4" title="MATRIX_3:def.4">MATRIX_3:def 4</a></i>
;<br/>
<a NAME="E7:10"/><b>then </b><i><font color="Green" title="E14">A6</font></i>: 
 <a href="matrix_1.html#K1" title="MATRIX_1:func.1">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" title="POLYNOM8:func.1">emb</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">L</font></span>)</span> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <font color="Maroon" title="c5">M1</font></span>)</span> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <font color="Maroon" title="c6">M2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">L</font></span>)</span> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">M1</font> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <font color="Maroon" title="c6">M2</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="polynom8.html#E2:10"><i><font color="Green" title="E10">A2</font></i></a>, <a class="ref" href="matrix_3.html#D4" title="MATRIX_3:def.4">MATRIX_3:def 4</a></i>;<br/>
<a NAME="E8:10"/>
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"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">L</font></span>)</span> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <font color="Maroon" title="c5">M1</font></span>)</span> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <font color="Maroon" title="c6">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" title="POLYNOM8:func.1">emb</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">L</font></span>)</span> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <font color="Maroon" title="c5">M1</font></span>)</span> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <font color="Maroon" title="c6">M2</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">L</font></span>)</span> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">M1</font> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <font color="Maroon" title="c6">M2</font></span>)</span></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>
 
<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" title="c7">i</font>, <font color="Maroon" title="c8">j</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:10_6"/><i><font color="Green" title="E15">A7</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c8">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"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">L</font></span>)</span> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <font color="Maroon" title="c5">M1</font></span>)</span> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <font color="Maroon" title="c6">M2</font></span>)</span>
 ;<br/>

<i><font color="Green" title="E16">A8</font></i>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">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" title="POLYNOM8:func.1">emb</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">L</font></span>)</span> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <font color="Maroon" title="c5">M1</font></span>)</span> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <font color="Maroon" title="c6">M2</font></span>)</span> = 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">L</font></span>)</span> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <font color="Maroon" title="c5">M1</font></span>)</span>
<b>by </b><i><a class="txt" href="polynom8.html#E2:10"><i><font color="Green" title="E10">A2</font></i></a>, <a class="ref" href="matrix_3.html#D4" title="MATRIX_3:def.4">MATRIX_3:def 4</a></i>
<br/>.= 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">M1</font>
<b>by </b><i><a class="ref" href="matrix_3.html#D5" title="MATRIX_3:def.5">MATRIX_3:def 5</a></i>
;<br/>
<a NAME="E3:10_6"/><i><font color="Green" title="E17">A9</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">M1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">M1</font> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <font color="Maroon" title="c6">M2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="polynom8.html#E3:10"><i><font color="Green" title="E11">A3</font></i></a>, <a class="ref" href="matrix_3.html#D4" title="MATRIX_3:def.4">MATRIX_3:def 4</a></i>;<br/>
<a NAME="E4:10_6"/><i><font color="Green" title="E18">A10</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c8">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">M1</font> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <font color="Maroon" title="c6">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" title="E19">A11</font></i>:  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">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" title="POLYNOM8:func.1">emb</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">L</font></span>)</span> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <font color="Maroon" title="c5">M1</font></span>)</span> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <font color="Maroon" title="c6">M2</font></span>)</span> = 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">M1</font></span>)</span>
<b>by </b><i><a class="txt" href="polynom8.html#E2:10_6"><i><font color="Green" title="E16">A8</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>
<br/>.= 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">M1</font> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <font color="Maroon" title="c6">M2</font></span>)</span>
<b>by </b><i><a class="txt" href="polynom8.html#E3:10_6"><i><font color="Green" title="E17">A9</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>
;<br/>
 <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"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">L</font></span>)</span> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <font color="Maroon" title="c5">M1</font></span>)</span> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <font color="Maroon" title="c6">M2</font></span>)</span></span>)</span> = 
 <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> <font color="Maroon" title="c6">M2</font></span>)</span>
<b>by </b><i><a class="txt" href="polynom8.html#E2:10"><i><font color="Green" title="E10">A2</font></i></a>, <a class="ref" href="matrix_3.html#D4" title="MATRIX_3:def.4">MATRIX_3:def 4</a></i>
<br/>.= 
 <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"><font color="Maroon" title="c5">M1</font> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <font color="Maroon" title="c6">M2</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="polynom8.html#E3:10"><i><font color="Green" title="E11">A3</font></i></a>, <a class="ref" href="matrix_3.html#D4" title="MATRIX_3:def.4">MATRIX_3:def 4</a></i>
;<br/>
<b>hence </b><a NAME="E3:10_6_2"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c8">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">M1</font> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <font color="Maroon" title="c6">M2</font></span>)</span>
 <b>by </b><i><a class="txt" href="polynom8.html#E1:10_6_2"><i><font color="Green" title="E19">A11</font></i></a>, <a class="txt" href="polynom8.html#E1:10_6"><i><font color="Green" title="E15">A7</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then </b><i><font color="Green" title="E19">A12</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">L</font></span>)</span> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">M1</font> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <font color="Maroon" title="c6">M2</font></span>)</span></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c8">j</font> = 
<span class="p1">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">L</font></span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">M1</font> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <font color="Maroon" title="c6">M2</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c8">j</font></span>)</span>
<b>by </b><i><a class="ref" href="matrix_3.html#D5" title="MATRIX_3:def.5">MATRIX_3:def 5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">L</font></span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">M1</font>,<font color="Maroon" title="c7">i</font></span>)</span> <a href="fvsum_1.html#K13" title="FVSUM_1:func.13">"*"</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c6">M2</font>,<font color="Maroon" title="c8">j</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="polynom8.html#E3:10"><i><font color="Green" title="E11">A3</font></i></a>, <a class="txt" href="polynom8.html#E4:10_6"><i><font color="Green" title="E18">A10</font></i></a>, <a class="ref" href="matrix_3.html#D4" title="MATRIX_3:def.4">MATRIX_3:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">L</font></span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K8" title="RLVECT_1:func.8">Sum</a> <span class="p2">(<span class="default"><a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">M1</font>,<font color="Maroon" title="c7">i</font></span>)</span>,<span class="p3">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c6">M2</font>,<font color="Maroon" title="c8">j</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="fvsum_1.html#D10" title="FVSUM_1:def.10">FVSUM_1:def 10</a></i>
;<br/>
<i><font color="Green" title="E20">A13</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" title="POLYNOM8:func.1">emb</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">L</font></span>)</span> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <font color="Maroon" title="c5">M1</font></span>)</span> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <font color="Maroon" title="c6">M2</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c8">j</font> = 
<span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">L</font></span>)</span> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <font color="Maroon" title="c5">M1</font></span>)</span>,<font color="Maroon" title="c7">i</font></span>)</span> <a href="fvsum_1.html#K13" title="FVSUM_1:func.13">"*"</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c6">M2</font>,<font color="Maroon" title="c8">j</font></span>)</span>
<b>by </b><i><a class="txt" href="polynom8.html#E2:10"><i><font color="Green" title="E10">A2</font></i></a>, <a class="txt" href="polynom8.html#E1:10_6"><i><font color="Green" title="E15">A7</font></i></a>, <a class="ref" href="matrix_3.html#D4" title="MATRIX_3:def.4">MATRIX_3:def 4</a></i>
<br/>.= 
 <a href="rlvect_1.html#K8" title="RLVECT_1:func.8">Sum</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">L</font></span>)</span> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <font color="Maroon" title="c5">M1</font></span>)</span>,<font color="Maroon" title="c7">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c6">M2</font>,<font color="Maroon" title="c8">j</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="fvsum_1.html#D10" title="FVSUM_1:def.10">FVSUM_1:def 10</a></i>
;<br/>
<a NAME="E7:10_6"/>
<span class="p1">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">L</font></span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K8" title="RLVECT_1:func.8">Sum</a> <span class="p2">(<span class="default"><a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">M1</font>,<font color="Maroon" title="c7">i</font></span>)</span>,<span class="p3">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c6">M2</font>,<font color="Maroon" title="c8">j</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K8" title="RLVECT_1:func.8">Sum</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">L</font></span>)</span> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <font color="Maroon" title="c5">M1</font></span>)</span>,<font color="Maroon" title="c7">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c6">M2</font>,<font color="Maroon" title="c8">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" title="POLYNOM8:func.1">emb</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">L</font></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">M1</font>,<font color="Maroon" title="c7">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c6">M2</font>,<font color="Maroon" title="c8">j</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">L</font></span>)</span> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <font color="Maroon" title="c5">M1</font></span>)</span>,<font color="Maroon" title="c7">i</font></span>)</span>,<span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c6">M2</font>,<font color="Maroon" title="c8">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" title="E21">A14</font></i>: 
 <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">L</font></span>)</span> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <font color="Maroon" title="c5">M1</font></span>)</span>,<font color="Maroon" title="c7">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">L</font></span>)</span> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">M1</font>,<font color="Maroon" title="c7">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" title="E22">A15</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">L</font></span>)</span> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <font color="Maroon" title="c5">M1</font></span>)</span>,<font color="Maroon" title="c7">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">L</font></span>)</span> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">M1</font>,<font color="Maroon" title="c7">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" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">L</font></span>)</span> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <font color="Maroon" title="c5">M1</font></span>)</span>,<font color="Maroon" title="c7">i</font></span>)</span> = 
 <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">L</font></span>)</span> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <font color="Maroon" title="c5">M1</font></span>)</span>
<b>by </b><i><a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a></i>
<br/>.= 
 <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">M1</font>
<b>by </b><i><a class="ref" href="matrix_3.html#D5" title="MATRIX_3:def.5">MATRIX_3:def 5</a></i>
;<br/>
<a NAME="E2:10_6_5_1_1_1"/><b>then </b><i><font color="Green" title="E23">A16</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">L</font></span>)</span> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <font color="Maroon" title="c5">M1</font></span>)</span>,<font color="Maroon" title="c7">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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> <font color="Maroon" title="c5">M1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">L</font></span>)</span> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">M1</font>,<font color="Maroon" title="c7">i</font></span>)</span></span>)</span></span>)</span> = 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">L</font></span>)</span> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">M1</font>,<font color="Maroon" title="c7">i</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>
<br/>.= 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">M1</font>,<font color="Maroon" title="c7">i</font></span>)</span>
<b>by </b><i><a class="ref" href="polynom1.html#D2" title="POLYNOM1:def.2">POLYNOM1:def 2</a></i>
<br/>.= 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">M1</font>,<font color="Maroon" title="c7">i</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>
;<br/>
<b>then </b> <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">L</font></span>)</span> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">M1</font>,<font color="Maroon" title="c7">i</font></span>)</span></span>)</span> = 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">M1</font>,<font color="Maroon" title="c7">i</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T8" title="FINSEQ_1:th.8">FINSEQ_1:8</a></i>
<br/>.= 
 <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">M1</font>
<b>by </b><i><a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">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" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">L</font></span>)</span> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <font color="Maroon" title="c5">M1</font></span>)</span>,<font color="Maroon" title="c7">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">L</font></span>)</span> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">M1</font>,<font color="Maroon" title="c7">i</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="polynom8.html#E2:10_6_5_1_1_1"><i><font color="Green" title="E23">A16</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:10_6_5_1_1"/>
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">k</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="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">L</font></span>)</span> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <font color="Maroon" title="c5">M1</font></span>)</span>,<font color="Maroon" title="c7">i</font></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">L</font></span>)</span> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <font color="Maroon" title="c5">M1</font></span>)</span>,<font color="Maroon" title="c7">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">L</font></span>)</span> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">M1</font>,<font color="Maroon" title="c7">i</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">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" title="c9">k</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:10_6_5_1_1_2"/><i><font color="Green" title="E23">A17</font></i>: 
<font color="Maroon" title="c9">k</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="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">L</font></span>)</span> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <font color="Maroon" title="c5">M1</font></span>)</span>,<font color="Maroon" title="c7">i</font></span>)</span>
 ;<br/>

<i><font color="Green" title="E24">A18</font></i>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">L</font></span>)</span> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <font color="Maroon" title="c5">M1</font></span>)</span>,<font color="Maroon" title="c7">i</font></span>)</span> = 
 <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">L</font></span>)</span> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <font color="Maroon" title="c5">M1</font></span>)</span>
<b>by </b><i><a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a></i>
<br/>.= 
 <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">M1</font>
<b>by </b><i><a class="ref" href="matrix_3.html#D5" title="MATRIX_3:def.5">MATRIX_3:def 5</a></i>
;<br/>
<a NAME="E3:10_6_5_1_1_2"/><b>then </b>
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">L</font></span>)</span> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <font color="Maroon" title="c5">M1</font></span>)</span>,<font color="Maroon" title="c7">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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> <font color="Maroon" title="c5">M1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E4:10_6_5_1_1_2"/><b>then </b><i><font color="Green" title="E25">A19</font></i>: 
<font color="Maroon" title="c9">k</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"><span class="p3">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">L</font></span>)</span> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <font color="Maroon" title="c5">M1</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="polynom8.html#E1:10_6_5_1_1_2"><i><font color="Green" title="E23">A17</font></i></a>, <a class="ref" href="matrix_3.html#D5" title="MATRIX_3:def.5">MATRIX_3:def 5</a></i>;<br/>
<a NAME="E5:10_6_5_1_1_2"/><i><font color="Green" title="E26">A20</font></i>: 
<font color="Maroon" title="c9">k</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> <font color="Maroon" title="c5">M1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="polynom8.html#E2:10_6_5_1_1_2"><i><font color="Green" title="E24">A18</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a>, <a class="txt" href="polynom8.html#E1:10_6_5_1_1_2"><i><font color="Green" title="E23">A17</font></i></a></i>;<br/>
<a NAME="E6:10_6_5_1_1_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">M1</font>,<font color="Maroon" title="c7">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">M1</font>
 
<b>by </b><i><a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a></i>;<br/>
<a NAME="E7:10_6_5_1_1_2"/><b>then </b>
<font color="Maroon" title="c9">k</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="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">M1</font>,<font color="Maroon" title="c7">i</font></span>)</span>
 
<b>by </b><i><a class="txt" href="polynom8.html#E5:10_6_5_1_1_2"><i><font color="Green" title="E26">A20</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="E8:10_6_5_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">M1</font>,<font color="Maroon" title="c7">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">M1</font>,<font color="Maroon" title="c7">i</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c9">k</font>
 
<b>by </b><i><a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c10">a</font> = <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">M1</font>,<font color="Maroon" title="c7">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> ;<br/>
<a NAME="E10:10_6_5_1_1_2"/><i><font color="Green" title="E27">A21</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">L</font></span>)</span> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">M1</font>,<font color="Maroon" title="c7">i</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">L</font></span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <font color="Maroon" title="c10">a</font>
 
<b>by </b><i><a class="txt" href="polynom8.html#E1:10_6_5_1_1_2"><i><font color="Green" title="E23">A17</font></i></a>, <a class="txt" href="polynom8.html#E1:10_6_5_1_1"><i><font color="Green" title="E22">A15</font></i></a>, <a class="ref" href="fvsum_1.html#T62" title="FVSUM_1:th.62">FVSUM_1:62</a></i>;<br/>
<a NAME="E11:10_6_5_1_1_2"/><i><font color="Green" title="E28">A22</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c9">k</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> <font color="Maroon" title="c5">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" title="E29">A23</font></i>: 
 <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c5">M1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">m</font></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c3">n</font></span>)</span></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span>
 
<b>by </b><i><a class="ref" href="matrix_1.html#T24" title="MATRIX_1:th.24">MATRIX_1:24</a>, <a class="txt" href="polynom8.html#E1:10"><i><font color="Green" title="E9">A1</font></i></a></i>;<br/>
<a NAME="E2:10_6_5_1_1_2_2"/><i><font color="Green" title="E30">A24</font></i>: 
<font color="Maroon" title="c7">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"><font color="Maroon" title="c5">M1</font> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <font color="Maroon" title="c6">M2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="polynom8.html#E4:10_6"><i><font color="Green" title="E18">A10</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">M1</font> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <font color="Maroon" title="c6">M2</font></span>)</span> = 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">M1</font>
<b>by </b><i><a class="txt" href="polynom8.html#E3:10"><i><font color="Green" title="E11">A3</font></i></a>, <a class="ref" href="matrix_3.html#D4" title="MATRIX_3:def.4">MATRIX_3:def 4</a></i>
<br/>.= 
<font color="Maroon" title="c2">m</font>
<b>by </b><i><a class="ref" href="matrix_1.html#T24" title="MATRIX_1:th.24">MATRIX_1:24</a>, <a class="txt" href="polynom8.html#E1:10"><i><font color="Green" title="E9">A1</font></i></a></i>
;<br/>
<a NAME="E4:10_6_5_1_1_2_2"/><b>then </b><i><font color="Green" title="E31">A25</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">M1</font> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <font color="Maroon" title="c6">M2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">m</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E5:10_6_5_1_1_2_2"/>
<font color="Maroon" title="c9">k</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="c3">n</font>
 
<b>by </b><i><a class="txt" href="polynom8.html#E5:10_6_5_1_1_2"><i><font color="Green" title="E26">A20</font></i></a>, <a class="ref" href="matrix_1.html#T24" title="MATRIX_1:th.24">MATRIX_1:24</a>, <a class="txt" href="polynom8.html#E1:10"><i><font color="Green" title="E9">A1</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:10_6_5_1_1_2_2"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c9">k</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> <font color="Maroon" title="c5">M1</font>
 <b>by </b><i><a class="txt" href="polynom8.html#E1:10_6_5_1_1_2_2"><i><font color="Green" title="E29">A23</font></i></a>, <a class="txt" href="polynom8.html#E2:10_6_5_1_1_2_2"><i><font color="Green" title="E30">A24</font></i></a>, <a class="txt" href="polynom8.html#E4:10_6_5_1_1_2_2"><i><font color="Green" title="E31">A25</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E12:10_6_5_1_1_2"/>
<font color="Maroon" title="c10">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">M1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c9">k</font>
 
<b>by </b><i><a class="txt" href="polynom8.html#E5:10_6_5_1_1_2"><i><font color="Green" title="E26">A20</font></i></a>, <a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">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" title="POLYNOM8:func.1">emb</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">L</font></span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <font color="Maroon" title="c10">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">L</font></span>)</span> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <font color="Maroon" title="c5">M1</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c9">k</font>
 
<b>by </b><i><a class="txt" href="polynom8.html#E11:10_6_5_1_1_2"><i><font color="Green" title="E28">A22</font></i></a>, <a class="ref" href="matrix_3.html#D5" title="MATRIX_3:def.5">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" title="MATRIX_1:func.8">Line</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">L</font></span>)</span> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <font color="Maroon" title="c5">M1</font></span>)</span>,<font color="Maroon" title="c7">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">L</font></span>)</span> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">M1</font>,<font color="Maroon" title="c7">i</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font>
 <b>by </b><i><a class="txt" href="polynom8.html#E4:10_6_5_1_1_2"><i><font color="Green" title="E25">A19</font></i></a>, <a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a>, <a class="txt" href="polynom8.html#E10:10_6_5_1_1_2"><i><font color="Green" title="E27">A21</font></i></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" title="MATRIX_1:func.8">Line</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">L</font></span>)</span> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <font color="Maroon" title="c5">M1</font></span>)</span>,<font color="Maroon" title="c7">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">L</font></span>)</span> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">M1</font>,<font color="Maroon" title="c7">i</font></span>)</span>
 <b>by </b><i><a class="txt" href="polynom8.html#E1:10_6_5_1_1"><i><font color="Green" title="E22">A15</font></i></a>, <a class="ref" href="finseq_1.html#T17" title="FINSEQ_1:th.17">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" title="POLYNOM8:func.1">emb</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">L</font></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">M1</font>,<font color="Maroon" title="c7">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c6">M2</font>,<font color="Maroon" title="c8">j</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">L</font></span>)</span> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">M1</font>,<font color="Maroon" title="c7">i</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c6">M2</font>,<font color="Maroon" title="c8">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" title="E22">A26</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">L</font></span>)</span> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">M1</font>,<font color="Maroon" title="c7">i</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c6">M2</font>,<font color="Maroon" title="c8">j</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">L</font></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p2">(<span class="default"><a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">M1</font>,<font color="Maroon" title="c7">i</font></span>)</span>,<span class="p3">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c6">M2</font>,<font color="Maroon" title="c8">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" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">L</font></span>)</span> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">M1</font>,<font color="Maroon" title="c7">i</font></span>)</span></span>)</span></span>)</span> = 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">L</font></span>)</span> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">M1</font>,<font color="Maroon" title="c7">i</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>
<br/>.= 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">M1</font>,<font color="Maroon" title="c7">i</font></span>)</span>
<b>by </b><i><a class="ref" href="polynom1.html#D2" title="POLYNOM1:def.2">POLYNOM1:def 2</a></i>
<br/>.= 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">M1</font>,<font color="Maroon" title="c7">i</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>
;<br/>
<b>then </b><i><font color="Green" title="E23">A27</font></i>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">L</font></span>)</span> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">M1</font>,<font color="Maroon" title="c7">i</font></span>)</span></span>)</span> = 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">M1</font>,<font color="Maroon" title="c7">i</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T8" title="FINSEQ_1:th.8">FINSEQ_1:8</a></i>
<br/>.= 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">M2</font>
<b>by </b><i><a class="txt" href="polynom8.html#E3:10"><i><font color="Green" title="E11">A3</font></i></a>, <a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a></i>
<br/>.= 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c6">M2</font>,<font color="Maroon" title="c8">j</font></span>)</span>
<b>by </b><i><a class="ref" href="matrix_1.html#D9" title="MATRIX_1:def.9">MATRIX_1:def 9</a></i>
;<br/>
<i><font color="Green" title="E24">A28</font></i>:  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">L</font></span>)</span> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">M1</font>,<font color="Maroon" title="c7">i</font></span>)</span></span>)</span> = 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">L</font></span>)</span> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">M1</font>,<font color="Maroon" title="c7">i</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>
<br/>.= 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">L</font></span>)</span> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p4">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">M1</font>,<font color="Maroon" title="c7">i</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c6">M2</font>,<font color="Maroon" title="c8">j</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="polynom8.html#E2:10_6_5_1_2_1"><i><font color="Green" title="E23">A27</font></i></a>, <a class="ref" href="matrix_3.html#T8" title="MATRIX_3:th.8">MATRIX_3:8</a></i>
<br/>.= 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">L</font></span>)</span> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">M1</font>,<font color="Maroon" title="c7">i</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c6">M2</font>,<font color="Maroon" title="c8">j</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>
;<br/>
<i><font color="Green" title="E25">A29</font></i>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">M1</font>,<font color="Maroon" title="c7">i</font></span>)</span> = 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">M2</font>
<b>by </b><i><a class="txt" href="polynom8.html#E3:10"><i><font color="Green" title="E11">A3</font></i></a>, <a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a></i>
<br/>.= 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c6">M2</font>,<font color="Maroon" title="c8">j</font></span>)</span>
<b>by </b><i><a class="ref" href="matrix_1.html#D9" title="MATRIX_1:def.9">MATRIX_1:def 9</a></i>
;<br/>
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">L</font></span>)</span> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">M1</font>,<font color="Maroon" title="c7">i</font></span>)</span></span>)</span> = 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">M1</font>,<font color="Maroon" title="c7">i</font></span>)</span>
<b>by </b><i><a class="ref" href="polynom1.html#D2" title="POLYNOM1:def.2">POLYNOM1:def 2</a></i>
<br/>.= 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">M1</font>,<font color="Maroon" title="c7">i</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>
<br/>.= 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">M1</font>,<font color="Maroon" title="c7">i</font></span>)</span>,<span class="p3">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c6">M2</font>,<font color="Maroon" title="c8">j</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="polynom8.html#E4:10_6_5_1_2_1"><i><font color="Green" title="E25">A29</font></i></a>, <a class="ref" href="matrix_3.html#T8" title="MATRIX_3:th.8">MATRIX_3:8</a></i>
<br/>.= 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">M1</font>,<font color="Maroon" title="c7">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c6">M2</font>,<font color="Maroon" title="c8">j</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">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" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">L</font></span>)</span> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">M1</font>,<font color="Maroon" title="c7">i</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c6">M2</font>,<font color="Maroon" title="c8">j</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">L</font></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p2">(<span class="default"><a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">M1</font>,<font color="Maroon" title="c7">i</font></span>)</span>,<span class="p3">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c6">M2</font>,<font color="Maroon" title="c8">j</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="polynom8.html#E3:10_6_5_1_2_1"><i><font color="Green" title="E24">A28</font></i></a>, <a class="ref" href="polynom1.html#D2" title="POLYNOM1:def.2">POLYNOM1:def 2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:10_6_5_1_2"/>
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">k</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="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">L</font></span>)</span> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">M1</font>,<font color="Maroon" title="c7">i</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c6">M2</font>,<font color="Maroon" title="c8">j</font></span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">L</font></span>)</span> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">M1</font>,<font color="Maroon" title="c7">i</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c6">M2</font>,<font color="Maroon" title="c8">j</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">L</font></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p2">(<span class="default"><a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">M1</font>,<font color="Maroon" title="c7">i</font></span>)</span>,<span class="p3">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c6">M2</font>,<font color="Maroon" title="c8">j</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">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" title="c9">k</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:10_6_5_1_2_2"/><i><font color="Green" title="E23">A30</font></i>: 
<font color="Maroon" title="c9">k</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="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">L</font></span>)</span> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">M1</font>,<font color="Maroon" title="c7">i</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c6">M2</font>,<font color="Maroon" title="c8">j</font></span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:10_6_5_1_2_2"/><i><font color="Green" title="E24">A31</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">L</font></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p2">(<span class="default"><a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">M1</font>,<font color="Maroon" title="c7">i</font></span>)</span>,<span class="p3">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c6">M2</font>,<font color="Maroon" title="c8">j</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">M1</font>,<font color="Maroon" title="c7">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c6">M2</font>,<font color="Maroon" title="c8">j</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="polynom1.html#D2" title="POLYNOM1:def.2">POLYNOM1:def 2</a></i>;<br/>
<a NAME="E3:10_6_5_1_2_2"/><i><font color="Green" title="E25">A32</font></i>: 
<font color="Maroon" title="c9">k</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="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">M1</font>,<font color="Maroon" title="c7">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c6">M2</font>,<font color="Maroon" title="c8">j</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="polynom8.html#E1:10_6_5_1_2"><i><font color="Green" title="E22">A26</font></i></a>, <a class="txt" href="polynom8.html#E1:10_6_5_1_2_2"><i><font color="Green" title="E23">A30</font></i></a>, <a class="ref" href="polynom1.html#D2" title="POLYNOM1:def.2">POLYNOM1:def 2</a></i>;<br/>
<i><font color="Green" title="E26">A33</font></i>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">M1</font>,<font color="Maroon" title="c7">i</font></span>)</span> = 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">M2</font>
<b>by </b><i><a class="txt" href="polynom8.html#E3:10"><i><font color="Green" title="E11">A3</font></i></a>, <a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a></i>
<br/>.= 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c6">M2</font>,<font color="Maroon" title="c8">j</font></span>)</span>
<b>by </b><i><a class="ref" href="matrix_1.html#D9" title="MATRIX_1:def.9">MATRIX_1:def 9</a></i>
;<br/>
<i><font color="Green" title="E27">A34</font></i>:  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">M1</font>,<font color="Maroon" title="c7">i</font></span>)</span> = 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">M1</font>,<font color="Maroon" title="c7">i</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>
<br/>.= 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">M1</font>,<font color="Maroon" title="c7">i</font></span>)</span>,<span class="p3">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c6">M2</font>,<font color="Maroon" title="c8">j</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="polynom8.html#E4:10_6_5_1_2_2"><i><font color="Green" title="E26">A33</font></i></a>, <a class="ref" href="matrix_3.html#T8" title="MATRIX_3:th.8">MATRIX_3:8</a></i>
<br/>.= 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">M1</font>,<font color="Maroon" title="c7">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c6">M2</font>,<font color="Maroon" title="c8">j</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>
;<br/>
<a NAME="E6:10_6_5_1_2_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">M1</font>,<font color="Maroon" title="c7">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">M1</font>
 
<b>by </b><i><a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a></i>;<br/>
<a NAME="E7:10_6_5_1_2_2"/><b>then </b><i><font color="Green" title="E28">A35</font></i>: 
<font color="Maroon" title="c9">k</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> <font color="Maroon" title="c5">M1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="polynom8.html#E3:10_6_5_1_2_2"><i><font color="Green" title="E25">A32</font></i></a>, <a class="txt" href="polynom8.html#E5:10_6_5_1_2_2"><i><font color="Green" title="E27">A34</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="E8:10_6_5_1_2_2"/><b>then </b><i><font color="Green" title="E29">A36</font></i>: 
<span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">M1</font>,<font color="Maroon" title="c7">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">M1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c9">k</font>
 
<b>by </b><i><a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a></i>;<br/>
<a NAME="E9:10_6_5_1_2_2"/>
<font color="Maroon" title="c9">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">M2</font>
 
<b>by </b><i><a class="txt" href="polynom8.html#E3:10"><i><font color="Green" title="E11">A3</font></i></a>, <a class="txt" href="polynom8.html#E7:10_6_5_1_2_2"><i><font color="Green" title="E28">A35</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="E10:10_6_5_1_2_2"/><b>then </b><i><font color="Green" title="E30">A37</font></i>: 
<span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c6">M2</font>,<font color="Maroon" title="c8">j</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">M2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">k</font>,<font color="Maroon" title="c8">j</font>
 
<b>by </b><i><a class="ref" href="matrix_1.html#D9" title="MATRIX_1:def.9">MATRIX_1:def 9</a></i>;<br/>
<a NAME="E11:10_6_5_1_2_2"/><i><font color="Green" title="E31">A38</font></i>: 
<font color="Maroon" title="c9">k</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"><span class="p2">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">L</font></span>)</span> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">M1</font>,<font color="Maroon" title="c7">i</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="polynom8.html#E3:10_6_5_1_2_2"><i><font color="Green" title="E25">A32</font></i></a>, <a class="txt" href="polynom8.html#E5:10_6_5_1_2_2"><i><font color="Green" title="E27">A34</font></i></a>, <a class="ref" href="polynom1.html#D2" title="POLYNOM1:def.2">POLYNOM1:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c10">c</font> = <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c6">M2</font>,<font color="Maroon" title="c8">j</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font>, <font color="Maroon" title="c11">d</font> = <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">M1</font>,<font color="Maroon" title="c7">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> <b>by </b><i><a class="txt" href="polynom8.html#E8:10_6_5_1_2_2"><i><font color="Green" title="E29">A36</font></i></a>, <a class="txt" href="polynom8.html#E10:10_6_5_1_2_2"><i><font color="Green" title="E30">A37</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" title="POLYNOM8:func.1">emb</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">L</font></span>)</span> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">M1</font>,<font color="Maroon" title="c7">i</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">L</font></span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <font color="Maroon" title="c11">d</font>
 
<b>by </b><i><a class="txt" href="polynom8.html#E11:10_6_5_1_2_2"><i><font color="Green" title="E31">A38</font></i></a>, <a class="ref" href="fvsum_1.html#T62" title="FVSUM_1:th.62">FVSUM_1:62</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c12">b</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">L</font></span>)</span> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">M1</font>,<font color="Maroon" title="c7">i</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> ;<br/>
<a NAME="E15:10_6_5_1_2_2"/>
<span class="p1">(<span class="default"><a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">M1</font>,<font color="Maroon" title="c7">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c6">M2</font>,<font color="Maroon" title="c8">j</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">c</font> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <font color="Maroon" title="c11">d</font>
 
<b>by </b><i><a class="txt" href="polynom8.html#E3:10_6_5_1_2_2"><i><font color="Green" title="E25">A32</font></i></a>, <a class="ref" href="fvsum_1.html#T73" title="FVSUM_1:th.73">FVSUM_1:73</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c13">a</font> = <span class="p1">(<span class="default"><a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">M1</font>,<font color="Maroon" title="c7">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c6">M2</font>,<font color="Maroon" title="c8">j</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> ;<br/>
<a NAME="E17:10_6_5_1_2_2"/><i><font color="Green" title="E32">A39</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">L</font></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p2">(<span class="default"><a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">M1</font>,<font color="Maroon" title="c7">i</font></span>)</span>,<span class="p3">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c6">M2</font>,<font color="Maroon" title="c8">j</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">L</font></span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <font color="Maroon" title="c13">a</font>
 
<b>by </b><i><a class="txt" href="polynom8.html#E1:10_6_5_1_2"><i><font color="Green" title="E22">A26</font></i></a>, <a class="txt" href="polynom8.html#E1:10_6_5_1_2_2"><i><font color="Green" title="E23">A30</font></i></a>, <a class="ref" href="fvsum_1.html#T62" title="FVSUM_1:th.62">FVSUM_1:62</a></i>;<br/>
<font color="Maroon" title="c12">b</font> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <font color="Maroon" title="c10">c</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">L</font></span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <font color="Maroon" title="c11">d</font></span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <font color="Maroon" title="c10">c</font>
<b>by </b><i><a class="txt" href="polynom8.html#E11:10_6_5_1_2_2"><i><font color="Green" title="E31">A38</font></i></a>, <a class="ref" href="fvsum_1.html#T62" title="FVSUM_1:th.62">FVSUM_1:62</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">L</font></span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">d</font> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <font color="Maroon" title="c10">c</font></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#D4" title="GROUP_1:def.4">GROUP_1:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">L</font></span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <font color="Maroon" title="c13">a</font>
<b>by </b><i><a class="txt" href="polynom8.html#E1:10_6_5_1_2_2"><i><font color="Green" title="E23">A30</font></i></a>, <a class="txt" href="polynom8.html#E2:10_6_5_1_2_2"><i><font color="Green" title="E24">A31</font></i></a>, <a class="txt" href="polynom8.html#E1:10_6_5_1_2"><i><font color="Green" title="E22">A26</font></i></a>, <a class="ref" href="fvsum_1.html#T73" title="FVSUM_1:th.73">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" title="FVSUM_1:func.11">mlt</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">L</font></span>)</span> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">M1</font>,<font color="Maroon" title="c7">i</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c6">M2</font>,<font color="Maroon" title="c8">j</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">L</font></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p2">(<span class="default"><a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">M1</font>,<font color="Maroon" title="c7">i</font></span>)</span>,<span class="p3">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c6">M2</font>,<font color="Maroon" title="c8">j</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font>
 <b>by </b><i><a class="txt" href="polynom8.html#E17:10_6_5_1_2_2"><i><font color="Green" title="E32">A39</font></i></a>, <a class="txt" href="polynom8.html#E1:10_6_5_1_2_2"><i><font color="Green" title="E23">A30</font></i></a>, <a class="ref" href="fvsum_1.html#T73" title="FVSUM_1:th.73">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" title="POLYNOM8:func.1">emb</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">L</font></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">M1</font>,<font color="Maroon" title="c7">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c6">M2</font>,<font color="Maroon" title="c8">j</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">L</font></span>)</span> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">M1</font>,<font color="Maroon" title="c7">i</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c6">M2</font>,<font color="Maroon" title="c8">j</font></span>)</span>
 <b>by </b><i><a class="txt" href="polynom8.html#E1:10_6_5_1_2"><i><font color="Green" title="E22">A26</font></i></a>, <a class="ref" href="finseq_1.html#T17" title="FINSEQ_1:th.17">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" title="POLYNOM8:func.1">emb</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">L</font></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">M1</font>,<font color="Maroon" title="c7">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c6">M2</font>,<font color="Maroon" title="c8">j</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">L</font></span>)</span> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <font color="Maroon" title="c5">M1</font></span>)</span>,<font color="Maroon" title="c7">i</font></span>)</span>,<span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c6">M2</font>,<font color="Maroon" title="c8">j</font></span>)</span>
 <b>by </b><i><a class="txt" href="polynom8.html#E1:10_6_5_1"><i><font color="Green" title="E21">A14</font></i></a></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" title="POLYNOM8:func.1">emb</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">L</font></span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K8" title="RLVECT_1:func.8">Sum</a> <span class="p2">(<span class="default"><a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">M1</font>,<font color="Maroon" title="c7">i</font></span>)</span>,<span class="p3">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c6">M2</font>,<font color="Maroon" title="c8">j</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K8" title="RLVECT_1:func.8">Sum</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">L</font></span>)</span> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <font color="Maroon" title="c5">M1</font></span>)</span>,<font color="Maroon" title="c7">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c6">M2</font>,<font color="Maroon" title="c8">j</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="fvsum_1.html#T92" title="FVSUM_1:th.92">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" title="POLYNOM8:func.1">emb</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">L</font></span>)</span> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <font color="Maroon" title="c5">M1</font></span>)</span> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <font color="Maroon" title="c6">M2</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c8">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">L</font></span>)</span> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">M1</font> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <font color="Maroon" title="c6">M2</font></span>)</span></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c8">j</font>
 <b>by </b><i><a class="txt" href="polynom8.html#E5:10_6"><i><font color="Green" title="E19">A12</font></i></a>, <a class="txt" href="polynom8.html#E6:10_6"><i><font color="Green" title="E20">A13</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E9:10"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">L</font></span>)</span> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <font color="Maroon" title="c5">M1</font></span>)</span> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <font color="Maroon" title="c6">M2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polynom8.html#K1" title="POLYNOM8:func.1">emb</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">L</font></span>)</span> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">M1</font> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <font color="Maroon" title="c6">M2</font></span>)</span>
 <b>by </b><i><a class="txt" href="polynom8.html#E5:10"><i><font color="Green" title="E13">A5</font></i></a>, <a class="txt" href="polynom8.html#E4:10"><i><font color="Green" title="E12">A4</font></i></a>, <a class="txt" href="polynom8.html#E7:10"><i><font color="Green" title="E14">A6</font></i></a>, <a class="ref" href="matrix_1.html#T21" title="MATRIX_1:th.21">MATRIX_1:21</a></i>;<br/>


</div>
