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

<b>let </b><font color="Maroon">K</font> be   <a href="vectsp_1.html#NM2">Field</a>;<br/><b>let </b><font color="Maroon">V1</font> be  <a href="matrlin.html#V2" target="_self">finite-dimensional</a> <a href="vectsp_1.html#NM6">VectSp</a> of <font color="Maroon">K</font>;<br/><b>let </b><font color="Maroon">V2</font> be  <a href="matrlin.html#V2" target="_self">finite-dimensional</a> <a href="vectsp_1.html#NM6">VectSp</a> of <font color="Maroon">K</font>;<br/><b>let </b><font color="Maroon">f1</font> be   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">V1</font>,<font color="Maroon">V2</font>;<br/><b>let </b><font color="Maroon">f2</font> be   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">V1</font>,<font color="Maroon">V2</font>;<br/><b>let </b><font color="Maroon">b1</font> be    <a href="matrlin.html#M1" target="_self">OrdBasis</a> of <font color="Maroon">V1</font>;<br/><b>let </b><font color="Maroon">b2</font> be    <a href="matrlin.html#M1" target="_self">OrdBasis</a> of <font color="Maroon">V2</font>;<br/>













 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrlin.html#K12" target="_self">AutMt</a> <font color="Maroon">f1</font>,<font color="Maroon">b1</font>,<font color="Maroon">b2</font></span>)</span> = 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">b1</font>
<b>by </b><i><a class="ref" href="matrlin.html#T3" target="_self">Th3</a></i>
<br/>.= 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrlin.html#K12" target="_self">AutMt</a> <font color="Maroon">f2</font>,<font color="Maroon">b1</font>,<font color="Maroon">b2</font></span>)</span>
<b>by </b><i><a class="ref" href="matrlin.html#T3" target="_self">Th3</a></i>
;<br/>
<a NAME="E2:64"/><b>then </b><i><font color="Green">E43</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="matrlin.html#K12" target="_self">AutMt</a> <font color="Maroon">f1</font>,<font color="Maroon">b1</font>,<font color="Maroon">b2</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="matrlin.html#K12" target="_self">AutMt</a> <font color="Maroon">f2</font>,<font color="Maroon">b1</font>,<font color="Maroon">b2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/>
<a NAME="E3:64"/><i><font color="Green">E44</font></i>: 
 <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="matrlin.html#K12" target="_self">AutMt</a> <font color="Maroon">f1</font>,<font color="Maroon">b1</font>,<font color="Maroon">b2</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="matrlin.html#K12" target="_self">AutMt</a> <font color="Maroon">f2</font>,<font color="Maroon">b1</font>,<font color="Maroon">b2</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_2"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:64_2_1"/>
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">b1</font> <a href="hidden.html#R1">=</a> 0 or  <a href="finseq_1.html#K3">len</a> <font color="Maroon">b1</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_2_1_1"><b>suppose </b></a><a NAME="E1:64_2_1_1"/><i><font color="Green">E50</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">b1</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>then </b> <a href="matrlin.html#K12" target="_self">AutMt</a> <font color="Maroon">f1</font>,<font color="Maroon">b1</font>,<font color="Maroon">b2</font> = 
 <a href="xboole_0.html#K1">{}</a> 
<b>by </b><i/>
<br/>.= 
 <a href="matrlin.html#K12" target="_self">AutMt</a> <font color="Maroon">f2</font>,<font color="Maroon">b1</font>,<font color="Maroon">b2</font>
<b>by </b><i>, </i>
;<br/><b>hence </b><a NAME="E3:64_2_1_1"/>
 <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="matrlin.html#K12" target="_self">AutMt</a> <font color="Maroon">f1</font>,<font color="Maroon">b1</font>,<font color="Maroon">b2</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="matrlin.html#K12" target="_self">AutMt</a> <font color="Maroon">f2</font>,<font color="Maroon">b1</font>,<font color="Maroon">b2</font></span>)</span>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_2_1_2"><b>suppose </b></a><a NAME="E1:64_2_1_2"/><i><font color="Green">E52</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">b1</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><div class="add"><b>hence </b> <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="matrlin.html#K12" target="_self">AutMt</a> <font color="Maroon">f1</font>,<font color="Maroon">b1</font>,<font color="Maroon">b2</font></span>)</span> = 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">b2</font>
<b>by </b><i/>
<br/>.= 
 <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="matrlin.html#K12" target="_self">AutMt</a> <font color="Maroon">f2</font>,<font color="Maroon">b1</font>,<font color="Maroon">b2</font></span>)</span>
<b>by </b><i><a class="txt" href="matrlin.html#E1:64_2_1_2"><i><font color="Green">E52</font></i></a>, </i>
;<br/><br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<i><font color="Green">E54</font></i>:  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrlin.html#K12" target="_self">AutMt</a> <span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="matrlin.html#K5" target="_self">+</a> <font color="Maroon">f2</font></span>)</span>,<font color="Maroon">b1</font>,<font color="Maroon">b2</font></span>)</span> = 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">b1</font>
<b>by </b><i><a class="ref" href="matrlin.html#T3" target="_self">Th3</a></i>
<br/>.= 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrlin.html#K12" target="_self">AutMt</a> <font color="Maroon">f1</font>,<font color="Maroon">b1</font>,<font color="Maroon">b2</font></span>)</span>
<b>by </b><i><a class="ref" href="matrlin.html#T3" target="_self">Th3</a></i>
;<br/>
<a NAME="E5:64"/><b>then </b><i><font color="Green">E55</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrlin.html#K12" target="_self">AutMt</a> <span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="matrlin.html#K5" target="_self">+</a> <font color="Maroon">f2</font></span>)</span>,<font color="Maroon">b1</font>,<font color="Maroon">b2</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrlin.html#K12" target="_self">AutMt</a> <font color="Maroon">f1</font>,<font color="Maroon">b1</font>,<font color="Maroon">b2</font></span>)</span> <a href="matrix_3.html#K3">+</a> <span class="p2">(<span class="default"><a href="matrlin.html#K12" target="_self">AutMt</a> <font color="Maroon">f2</font>,<font color="Maroon">b1</font>,<font color="Maroon">b2</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="matrix_3.html#D3">MATRIX_3:def 3</a></i>;<br/>
<a NAME="E6:64"/><i><font color="Green">E57</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="matrlin.html#K12" target="_self">AutMt</a> <span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="matrlin.html#K5" target="_self">+</a> <font color="Maroon">f2</font></span>)</span>,<font color="Maroon">b1</font>,<font color="Maroon">b2</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="matrlin.html#K12" target="_self">AutMt</a> <font color="Maroon">f1</font>,<font color="Maroon">b1</font>,<font color="Maroon">b2</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/>
<a NAME="E7:64"/><i><font color="Green">E58</font></i>: 
 <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="matrlin.html#K12" target="_self">AutMt</a> <span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="matrlin.html#K5" target="_self">+</a> <font color="Maroon">f2</font></span>)</span>,<font color="Maroon">b1</font>,<font color="Maroon">b2</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="matrlin.html#K12" target="_self">AutMt</a> <font color="Maroon">f1</font>,<font color="Maroon">b1</font>,<font color="Maroon">b2</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_4"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_4_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:64_4_1"/>
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">b1</font> <a href="hidden.html#R1">=</a> 0 or  <a href="finseq_1.html#K3">len</a> <font color="Maroon">b1</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_4_1_1"><b>suppose </b></a><a NAME="E1:64_4_1_1"/><i><font color="Green">E65</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">b1</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>then </b> <a href="matrlin.html#K12" target="_self">AutMt</a> <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="matrlin.html#K5" target="_self">+</a> <font color="Maroon">f2</font></span>)</span>,<font color="Maroon">b1</font>,<font color="Maroon">b2</font> = 
 <a href="xboole_0.html#K1">{}</a> 
<b>by </b><i/>
<br/>.= 
 <a href="matrlin.html#K12" target="_self">AutMt</a> <font color="Maroon">f1</font>,<font color="Maroon">b1</font>,<font color="Maroon">b2</font>
<b>by </b><i>, </i>
;<br/><b>hence </b><a NAME="E3:64_4_1_1"/>
 <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="matrlin.html#K12" target="_self">AutMt</a> <span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="matrlin.html#K5" target="_self">+</a> <font color="Maroon">f2</font></span>)</span>,<font color="Maroon">b1</font>,<font color="Maroon">b2</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="matrlin.html#K12" target="_self">AutMt</a> <font color="Maroon">f1</font>,<font color="Maroon">b1</font>,<font color="Maroon">b2</font></span>)</span>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_4_1_2"><b>suppose </b></a><a NAME="E1:64_4_1_2"/><i><font color="Green">E66</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">b1</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><div class="add"><b>hence </b> <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="matrlin.html#K12" target="_self">AutMt</a> <span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="matrlin.html#K5" target="_self">+</a> <font color="Maroon">f2</font></span>)</span>,<font color="Maroon">b1</font>,<font color="Maroon">b2</font></span>)</span> = 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">b2</font>
<b>by </b><i/>
<br/>.= 
 <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="matrlin.html#K12" target="_self">AutMt</a> <font color="Maroon">f1</font>,<font color="Maroon">b1</font>,<font color="Maroon">b2</font></span>)</span>
<b>by </b><i>, </i>
;<br/><br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E8:64"/><b>then </b><i><font color="Green">E67</font></i>: 
 <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="matrlin.html#K12" target="_self">AutMt</a> <span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="matrlin.html#K5" target="_self">+</a> <font color="Maroon">f2</font></span>)</span>,<font color="Maroon">b1</font>,<font color="Maroon">b2</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrlin.html#K12" target="_self">AutMt</a> <font color="Maroon">f1</font>,<font color="Maroon">b1</font>,<font color="Maroon">b2</font></span>)</span> <a href="matrix_3.html#K3">+</a> <span class="p2">(<span class="default"><a href="matrlin.html#K12" target="_self">AutMt</a> <font color="Maroon">f2</font>,<font color="Maroon">b1</font>,<font color="Maroon">b2</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="matrix_3.html#D3">MATRIX_3:def 3</a></i>;<br/>
<a NAME="E9:64"/>
for <font color="Olive">i</font>, <font color="Olive">j</font> being  <a href="nat_1.html#NM1">Nat</a>  st <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">i</font>,<font color="Olive">j</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="matrlin.html#K12" target="_self">AutMt</a> <span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="matrlin.html#K5" target="_self">+</a> <font color="Maroon">f2</font></span>)</span>,<font color="Maroon">b1</font>,<font color="Maroon">b2</font></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="matrlin.html#K12" target="_self">AutMt</a> <span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="matrlin.html#K5" target="_self">+</a> <font color="Maroon">f2</font></span>)</span>,<font color="Maroon">b1</font>,<font color="Maroon">b2</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,<font color="Olive">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrlin.html#K12" target="_self">AutMt</a> <font color="Maroon">f1</font>,<font color="Maroon">b1</font>,<font color="Maroon">b2</font></span>)</span> <a href="matrix_3.html#K3">+</a> <span class="p2">(<span class="default"><a href="matrlin.html#K12" target="_self">AutMt</a> <font color="Maroon">f2</font>,<font color="Maroon">b1</font>,<font color="Maroon">b2</font></span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,<font color="Olive">j</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_5"><b>proof </b></a><div class="add">

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



<b>assume </b><a NAME="E1:64_5"/><i><font color="Green">E68</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i</font>,<font color="Maroon">j</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="matrlin.html#K12" target="_self">AutMt</a> <span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="matrlin.html#K5" target="_self">+</a> <font color="Maroon">f2</font></span>)</span>,<font color="Maroon">b1</font>,<font color="Maroon">b2</font></span>)</span>
 ;<br/>

<a NAME="E2:64_5"/><b>then </b><i><font color="Green">E69</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i</font>,<font color="Maroon">j</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p3">(<span class="default"><a href="matrlin.html#K12" target="_self">AutMt</a> <span class="p4">(<span class="default"><font color="Maroon">f1</font> <a href="matrlin.html#K5" target="_self">+</a> <font color="Maroon">f2</font></span>)</span>,<font color="Maroon">b1</font>,<font color="Maroon">b2</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p4">(<span class="default"><a href="matrlin.html#K12" target="_self">AutMt</a> <span class="p5">(<span class="default"><font color="Maroon">f1</font> <a href="matrlin.html#K5" target="_self">+</a> <font color="Maroon">f2</font></span>)</span>,<font color="Maroon">b1</font>,<font color="Maroon">b2</font></span>)</span></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="matrix_1.html#D5">MATRIX_1:def 5</a></i>;<br/>
<a NAME="E3:64_5"/><b>then </b><i><font color="Green">E111</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i</font>,<font color="Maroon">j</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="matrlin.html#K12" target="_self">AutMt</a> <font color="Maroon">f1</font>,<font color="Maroon">b1</font>,<font color="Maroon">b2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="matrlin.html#E4:64"><i><font color="Green">E54</font></i></a>, <a class="txt" href="matrlin.html#E5:64"><i><font color="Green">E55</font></i></a>, <a class="ref" href="matrix_1.html#D5">MATRIX_1:def 5</a></i>;<br/>
<a NAME="E4:64_5"/><i><font color="Green">E112</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i</font>,<font color="Maroon">j</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="matrlin.html#K12" target="_self">AutMt</a> <font color="Maroon">f2</font>,<font color="Maroon">b1</font>,<font color="Maroon">b2</font></span>)</span>
 
<b>by </b><i>, , <a class="txt" href="matrlin.html#E4:64"><i><font color="Green">E54</font></i></a>, <a class="txt" href="matrlin.html#E5:64"><i><font color="Green">E55</font></i></a>, <a class="txt" href="matrlin.html#E7:64"><i><font color="Green">E58</font></i></a>, <a class="ref" href="matrix_1.html#D5">MATRIX_1:def 5</a></i>;<br/>
<a NAME="E5:64_5"/>
<span class="p1">(<span class="default"><a href="matrlin.html#K12" target="_self">AutMt</a> <span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="matrlin.html#K5" target="_self">+</a> <font color="Maroon">f2</font></span>)</span>,<font color="Maroon">b1</font>,<font color="Maroon">b2</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrlin.html#K12" target="_self">AutMt</a> <font color="Maroon">f1</font>,<font color="Maroon">b1</font>,<font color="Maroon">b2</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrlin.html#K12" target="_self">AutMt</a> <font color="Maroon">f2</font>,<font color="Maroon">b1</font>,<font color="Maroon">b2</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_5_1"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon">p1</font> being   <a href="struct_0.html#NM7">FinSequence</a> of <font color="Maroon">K</font><b> such that </b><br/><a NAME="E2:64_5_1"/><i><font color="Green">E113</font></i>: 
( <font color="Maroon">p1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="matrlin.html#K12" target="_self">AutMt</a> <span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="matrlin.html#K5" target="_self">+</a> <font color="Maroon">f2</font></span>)</span>,<font color="Maroon">b1</font>,<font color="Maroon">b2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> &amp; <span class="p1">(<span class="default"><a href="matrlin.html#K12" target="_self">AutMt</a> <span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="matrlin.html#K5" target="_self">+</a> <font color="Maroon">f2</font></span>)</span>,<font color="Maroon">b1</font>,<font color="Maroon">b2</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> )
 <b>by </b><i><a class="txt" href="matrlin.html#E6:64"><i><font color="Green">E57</font></i></a>, <a class="ref" href="matrix_1.html#D6">MATRIX_1:def 6</a></i>;<br/>
<b>consider </b><font color="Maroon">p2</font> being   <a href="struct_0.html#NM7">FinSequence</a> of <font color="Maroon">K</font><b> such that </b><br/><a NAME="E4:64_5_1"/><i><font color="Green">E134</font></i>: 
( <font color="Maroon">p2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="matrlin.html#K12" target="_self">AutMt</a> <font color="Maroon">f1</font>,<font color="Maroon">b1</font>,<font color="Maroon">b2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> &amp; <span class="p1">(<span class="default"><a href="matrlin.html#K12" target="_self">AutMt</a> <font color="Maroon">f1</font>,<font color="Maroon">b1</font>,<font color="Maroon">b2</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> )
 <b>by </b><i>, <a class="ref" href="matrix_1.html#D6">MATRIX_1:def 6</a></i>;<br/>
<b>consider </b><font color="Maroon">p3</font> being   <a href="struct_0.html#NM7">FinSequence</a> of <font color="Maroon">K</font><b> such that </b><br/><a NAME="E6:64_5_1"/><i><font color="Green">E135</font></i>: 
( <font color="Maroon">p3</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="matrlin.html#K12" target="_self">AutMt</a> <font color="Maroon">f2</font>,<font color="Maroon">b1</font>,<font color="Maroon">b2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> &amp; <span class="p1">(<span class="default"><a href="matrlin.html#K12" target="_self">AutMt</a> <font color="Maroon">f2</font>,<font color="Maroon">b1</font>,<font color="Maroon">b2</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p3</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> )
 <b>by </b><i><a class="ref" href="matrlin.html#T5" target="_self">Th5</a>, <a class="ref" href="matrix_1.html#D6">MATRIX_1:def 6</a></i>;<br/>
<a NAME="E7:64_5_1"/><i><font color="Green">E136</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p2">(<span class="default"><a href="matrlin.html#K12" target="_self">AutMt</a> <span class="p3">(<span class="default"><font color="Maroon">f1</font> <a href="matrlin.html#K5" target="_self">+</a> <font color="Maroon">f2</font></span>)</span>,<font color="Maroon">b1</font>,<font color="Maroon">b2</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="matrlin.html#E7:64"><i><font color="Green">E58</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E8:64_5_1"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">b1</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrlin.html#K12" target="_self">AutMt</a> <span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="matrlin.html#K5" target="_self">+</a> <font color="Maroon">f2</font></span>)</span>,<font color="Maroon">b1</font>,<font color="Maroon">b2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="matrlin.html#T3" target="_self">Th3</a></i>;<br/>
<a NAME="E9:64_5_1"/><b>then </b>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">b1</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="matrlin.html#K12" target="_self">AutMt</a> <span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="matrlin.html#K5" target="_self">+</a> <font color="Maroon">f2</font></span>)</span>,<font color="Maroon">b1</font>,<font color="Maroon">b2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/>
<a NAME="E10:64_5_1"/><b>then </b>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">b1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="txt" href="matrlin.html#E7:64"><i><font color="Green">E58</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E11:64_5_1"/><b>then </b>
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b1</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E12:64_5_1"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">b1</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a>, <a class="ref" href="nat_1.html#T19">NAT_1:19</a></i>;<br/>
<a NAME="E13:64_5_1"/><b>then </b><i><font color="Green">E137</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">b2</font></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E14:64_5_1"/><b>then </b><i><font color="Green">E154</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font> &amp; <font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">b2</font> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E15:64_5_1"/><i><font color="Green">E155</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">b2</font>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E16:64_5_1"/><i><font color="Green">E156</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="matrlin.html#K12" target="_self">AutMt</a> <span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="matrlin.html#K5" target="_self">+</a> <font color="Maroon">f2</font></span>)</span>,<font color="Maroon">b1</font>,<font color="Maroon">b2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="matrlin.html#E7:64"><i><font color="Green">E58</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E17:64_5_1"/><b>then </b><i><font color="Green">E158</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">b1</font>
 
<b>by </b><i><a class="ref" href="matrlin.html#T4" target="_self">Th4</a></i>;<br/>
<a NAME="E18:64_5_1"/><b>then </b><i><font color="Green">E160</font></i>: 
( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="matrlin.html#K12" target="_self">AutMt</a> <font color="Maroon">f1</font>,<font color="Maroon">b1</font>,<font color="Maroon">b2</font></span>)</span> &amp; <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="matrlin.html#K12" target="_self">AutMt</a> <font color="Maroon">f2</font>,<font color="Maroon">b1</font>,<font color="Maroon">b2</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="matrlin.html#T4" target="_self">Th4</a></i>;<br/>
<a NAME="E19:64_5_1"/><i><font color="Green">E161</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font> &amp; <font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f1</font> <a href="matrlin.html#K5" target="_self">+</a> <font color="Maroon">f2</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">b1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="matrlin.html#K11" target="_self">|--</a> <font color="Maroon">b2</font></span>)</span> &amp; <font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">b1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="matrlin.html#K11" target="_self">|--</a> <font color="Maroon">b2</font></span>)</span> &amp; <font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f2</font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">b1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="matrlin.html#K11" target="_self">|--</a> <font color="Maroon">b2</font></span>)</span> )
 
<b>by </b><i>, </i>;<br/>
<i><font color="Green">E162</font></i>: <font color="Maroon">p1</font> = 
<span class="p1">(<span class="default"><a href="matrlin.html#K12" target="_self">AutMt</a> <span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="matrlin.html#K5" target="_self">+</a> <font color="Maroon">f2</font></span>)</span>,<font color="Maroon">b1</font>,<font color="Maroon">b2</font></span>)</span> <a href="matrlin.html#K1" target="_self">/.</a> <font color="Maroon">i</font>
<b>by </b><i><a class="ref" href="matrlin.html#T6" target="_self">Th6</a>, <a class="txt" href="matrlin.html#E1:64_5"><i><font color="Green">E68</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="matrlin.html#K5" target="_self">+</a> <font color="Maroon">f2</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">b1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="matrlin.html#K11" target="_self">|--</a> <font color="Maroon">b2</font>
<b>by </b><i><a class="txt" href="matrlin.html#E2:64_5"><i><font color="Green">E69</font></i></a>, <a class="ref" href="matrlin.html#T3" target="_self">Th3</a></i>
;<br/>
<i><font color="Green">E163</font></i>: <font color="Maroon">p2</font> = 
<span class="p1">(<span class="default"><a href="matrlin.html#K12" target="_self">AutMt</a> <font color="Maroon">f1</font>,<font color="Maroon">b1</font>,<font color="Maroon">b2</font></span>)</span> <a href="matrlin.html#K1" target="_self">/.</a> <font color="Maroon">i</font>
<b>by </b><i>, , <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">b1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="matrlin.html#K11" target="_self">|--</a> <font color="Maroon">b2</font>
<b>by </b><i><a class="txt" href="matrlin.html#E2:64_5"><i><font color="Green">E69</font></i></a>, <a class="ref" href="matrlin.html#T3" target="_self">Th3</a></i>
;<br/>
<i><font color="Green">E164</font></i>: <font color="Maroon">p3</font> = 
<span class="p1">(<span class="default"><a href="matrlin.html#K12" target="_self">AutMt</a> <font color="Maroon">f2</font>,<font color="Maroon">b1</font>,<font color="Maroon">b2</font></span>)</span> <a href="matrlin.html#K1" target="_self">/.</a> <font color="Maroon">i</font>
<b>by </b><i>, , <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">b1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="matrlin.html#K11" target="_self">|--</a> <font color="Maroon">b2</font>
<b>by </b><i><a class="txt" href="matrlin.html#E2:64_5"><i><font color="Green">E69</font></i></a>, <a class="ref" href="matrlin.html#T3" target="_self">Th3</a></i>
;<br/>
<a NAME="E23:64_5_1"/><i><font color="Green">E165</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">p1</font>
 
<b>by </b><i><a class="txt" href="matrlin.html#E8:64"><i><font color="Green">E67</font></i></a>, , </i>;<br/>
<a NAME="E24:64_5_1"/><i><font color="Green">E166</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">p2</font>
 
<b>by </b><i><a class="txt" href="matrlin.html#E8:64"><i><font color="Green">E67</font></i></a>, , </i>;<br/>
<a NAME="E25:64_5_1"/><i><font color="Green">E167</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">p3</font>
 
<b>by </b><i><a class="txt" href="matrlin.html#E8:64"><i><font color="Green">E67</font></i></a>, , </i>;<br/>
<a NAME="E26:64_5_1"/><i><font color="Green">E168</font></i>: 
<span class="p1">(<span class="default"><a href="matrlin.html#K12" target="_self">AutMt</a> <font color="Maroon">f1</font>,<font color="Maroon">b1</font>,<font color="Maroon">b2</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j</font>
 
<b>by </b><i>, , <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/>
<a NAME="E27:64_5_1"/><i><font color="Green">E169</font></i>: 
<span class="p1">(<span class="default"><a href="matrlin.html#K12" target="_self">AutMt</a> <font color="Maroon">f2</font>,<font color="Maroon">b1</font>,<font color="Maroon">b2</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p3</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j</font>
 
<b>by </b><i>, , <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">KL1</font> being    <a href="vectsp_6.html#M1">Linear_Combination</a> of <font color="Maroon">V2</font><b> such that </b><br/><a NAME="E29:64_5_1"/><i><font color="Green">E170</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="matrlin.html#K5" target="_self">+</a> <font color="Maroon">f2</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="vectsp_6.html#K4">Sum</a> <font color="Maroon">KL1</font>
 <b>and </b><br/><a NAME="E30:64_5_1"/><i><font color="Green">E171</font></i>: 
 <a href="vectsp_6.html#K1">Carrier</a> <font color="Maroon">KL1</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">b2</font>
 <b>and </b><br/><a NAME="E31:64_5_1"/><i><font color="Green">E172</font></i>: 
for <font color="Olive">t</font> being  <a href="nat_1.html#NM1">Nat</a>  st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">t</font> &amp; <font color="Olive">t</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f1</font> <a href="matrlin.html#K5" target="_self">+</a> <font color="Maroon">f2</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">b1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="matrlin.html#K11" target="_self">|--</a> <font color="Maroon">b2</font></span>)</span> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f1</font> <a href="matrlin.html#K5" target="_self">+</a> <font color="Maroon">f2</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">b1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="matrlin.html#K11" target="_self">|--</a> <font color="Maroon">b2</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Olive">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">KL1</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">b2</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">t</font></span>)</span>
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">KL2</font> being    <a href="vectsp_6.html#M1">Linear_Combination</a> of <font color="Maroon">V2</font><b> such that </b><br/><a NAME="E33:64_5_1"/><i><font color="Green">E174</font></i>: 
<font color="Maroon">f1</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="vectsp_6.html#K4">Sum</a> <font color="Maroon">KL2</font>
 <b>and </b><br/><a NAME="E34:64_5_1"/><i><font color="Green">E175</font></i>: 
 <a href="vectsp_6.html#K1">Carrier</a> <font color="Maroon">KL2</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">b2</font>
 <b>and </b><br/><a NAME="E35:64_5_1"/><i><font color="Green">E178</font></i>: 
for <font color="Olive">t</font> being  <a href="nat_1.html#NM1">Nat</a>  st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">t</font> &amp; <font color="Olive">t</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">b1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="matrlin.html#K11" target="_self">|--</a> <font color="Maroon">b2</font></span>)</span> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">b1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="matrlin.html#K11" target="_self">|--</a> <font color="Maroon">b2</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Olive">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">KL2</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">b2</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">t</font></span>)</span>
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">KL3</font> being    <a href="vectsp_6.html#M1">Linear_Combination</a> of <font color="Maroon">V2</font><b> such that </b><br/><a NAME="E37:64_5_1"/><i><font color="Green">E179</font></i>: 
<font color="Maroon">f2</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="vectsp_6.html#K4">Sum</a> <font color="Maroon">KL3</font>
 <b>and </b><br/><a NAME="E38:64_5_1"/><i><font color="Green">E180</font></i>: 
 <a href="vectsp_6.html#K1">Carrier</a> <font color="Maroon">KL3</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">b2</font>
 <b>and </b><br/><a NAME="E39:64_5_1"/><i><font color="Green">E181</font></i>: 
for <font color="Olive">t</font> being  <a href="nat_1.html#NM1">Nat</a>  st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">t</font> &amp; <font color="Olive">t</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f2</font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">b1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="matrlin.html#K11" target="_self">|--</a> <font color="Maroon">b2</font></span>)</span> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f2</font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">b1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="matrlin.html#K11" target="_self">|--</a> <font color="Maroon">b2</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Olive">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">KL3</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">b2</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">t</font></span>)</span>
 <b>by </b><i/>;<br/>
<a NAME="E40:64_5_1"/><i><font color="Green">E182</font></i>: 
<font color="Maroon">p1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">KL1</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">b2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j</font></span>)</span>
 
<b>by </b><i><a class="ref" href="matrlin.html#D1" target="_self">Def1</a>, , </i>;<br/>
<a NAME="E41:64_5_1"/><i><font color="Green">E183</font></i>: 
<font color="Maroon">p2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">KL2</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">b2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j</font></span>)</span>
 
<b>by </b><i><a class="ref" href="matrlin.html#D1" target="_self">Def1</a>, , </i>;<br/>
<a NAME="E42:64_5_1"/><i><font color="Green">E184</font></i>: 
<font color="Maroon">p3</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">KL3</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">b2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j</font></span>)</span>
 
<b>by </b><i><a class="ref" href="matrlin.html#D1" target="_self">Def1</a>, , <a class="ref" href="matrlin.html#T9" target="_self">Th9</a></i>;<br/>
<b>reconsider </b><font color="Maroon">b4</font> =  <a href="relat_1.html#K2">rng</a> <font color="Maroon">b2</font> as    <a href="vectsp_7.html#M1">Basis</a> of <font color="Maroon">V2</font> <b>by </b><i/>;<br/>
<a NAME="E44:64_5_1"/><i><font color="Green">E185</font></i>: 
<font color="Maroon">b4</font> is <a href="vectsp_7.html#V1">linearly-independent</a>
 
<b>by </b><i><a class="ref" href="vectsp_7.html#D3">VECTSP_7:def 3</a></i>;<br/>
<a NAME="E45:64_5_1"/>
<span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="matrlin.html#K5" target="_self">+</a> <font color="Maroon">f2</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">b1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">b1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span></span>)</span>
 
<b>by </b><i/>;<br/>
<b>then </b><font color="Maroon">KL1</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">b2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">KL2</font> <a href="vectsp_6.html#K5">+</a> <font color="Maroon">KL3</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">b2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j</font></span>)</span>
<b>by </b><i>, , , , , , <a class="ref" href="matrlin.html#T12" target="_self">Th12</a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">KL2</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">b2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">KL3</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">b2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_6.html#D11">VECTSP_6:def 11</a></i>
;<br/>
<b>hence </b><a NAME="E47:64_5_1"/>
<span class="p1">(<span class="default"><a href="matrlin.html#K12" target="_self">AutMt</a> <span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="matrlin.html#K5" target="_self">+</a> <font color="Maroon">f2</font></span>)</span>,<font color="Maroon">b1</font>,<font color="Maroon">b2</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrlin.html#K12" target="_self">AutMt</a> <font color="Maroon">f1</font>,<font color="Maroon">b1</font>,<font color="Maroon">b2</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrlin.html#K12" target="_self">AutMt</a> <font color="Maroon">f2</font>,<font color="Maroon">b1</font>,<font color="Maroon">b2</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span>
 <b>by </b><i><a class="ref" href="matrlin.html#T6" target="_self">Th6</a>, <a class="ref" href="matrlin.html#D2" target="_self">Def2</a>, , , , <a class="ref" href="matrlin.html#T10" target="_self">Th10</a>, <a class="ref" href="matrlin.html#T11" target="_self">Th11</a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:64_5"/>
<span class="p1">(<span class="default"><a href="matrlin.html#K12" target="_self">AutMt</a> <span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="matrlin.html#K5" target="_self">+</a> <font color="Maroon">f2</font></span>)</span>,<font color="Maroon">b1</font>,<font color="Maroon">b2</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrlin.html#K12" target="_self">AutMt</a> <font color="Maroon">f1</font>,<font color="Maroon">b1</font>,<font color="Maroon">b2</font></span>)</span> <a href="matrix_3.html#K3">+</a> <span class="p2">(<span class="default"><a href="matrlin.html#K12" target="_self">AutMt</a> <font color="Maroon">f2</font>,<font color="Maroon">b1</font>,<font color="Maroon">b2</font></span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font>
 <b>by </b><i>, <a class="ref" href="matrix_3.html#D3">MATRIX_3:def 3</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E10:64"/>
 <a href="matrlin.html#K12" target="_self">AutMt</a> <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="matrlin.html#K5" target="_self">+</a> <font color="Maroon">f2</font></span>)</span>,<font color="Maroon">b1</font>,<font color="Maroon">b2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="matrlin.html#K12" target="_self">AutMt</a> <font color="Maroon">f1</font>,<font color="Maroon">b1</font>,<font color="Maroon">b2</font></span>)</span> <a href="matrix_3.html#K3">+</a> <span class="p1">(<span class="default"><a href="matrlin.html#K12" target="_self">AutMt</a> <font color="Maroon">f2</font>,<font color="Maroon">b1</font>,<font color="Maroon">b2</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="matrix_1.html#T21">MATRIX_1:21</a></i>;<br/>


</div>
