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

<b>let </b><font color="Maroon" title="c1">K</font> be   <a href="vectsp_1.html#NM2" title="VECTSP_1:NM.2">Field</a>;<br/><b>let </b><font color="Maroon" title="c2">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">K</font>;<br/><b>let </b><font color="Maroon" title="c3">V1</font>, <font color="Maroon" title="c4">V2</font> be  <a href="matrlin.html#V2" title="MATRLIN:attr.2">finite-dimensional</a> <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">VectSp</a> of <font color="Maroon" title="c1">K</font>;<br/><b>let </b><font color="Maroon" title="c5">f</font> be   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c3">V1</font>,<font color="Maroon" title="c4">V2</font>;<br/><b>let </b><font color="Maroon" title="c6">b1</font> be    <a href="matrlin.html#M1" title="MATRLIN:mode.1">OrdBasis</a> of <font color="Maroon" title="c3">V1</font>;<br/><b>let </b><font color="Maroon" title="c7">b2</font> be    <a href="matrlin.html#M1" title="MATRLIN:mode.1">OrdBasis</a> of <font color="Maroon" title="c4">V2</font>;<br/>













<b>assume </b><a NAME="E1:65"/><i><font color="Green" title="E50">A1</font></i>: 
<font color="Maroon" title="c2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c1">K</font>
 ;<br/>

<i><font color="Green" title="E51">A2</font></i>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrlin.html#K12" title="MATRLIN:func.12">AutMt</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="matrlin.html#K6" title="MATRLIN:func.6">*</a> <font color="Maroon" title="c5">f</font></span>)</span>,<font color="Maroon" title="c6">b1</font>,<font color="Maroon" title="c7">b2</font></span>)</span> = 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">b1</font>
<b>by </b><i><a class="ref" href="matrlin.html#D10" target="_self" title="MATRLIN:def.10">Def10</a></i>
<br/>.= 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrlin.html#K12" title="MATRLIN:func.12">AutMt</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c6">b1</font>,<font color="Maroon" title="c7">b2</font></span>)</span>
<b>by </b><i><a class="ref" href="matrlin.html#D10" target="_self" title="MATRLIN:def.10">Def10</a></i>
;<br/>
<a NAME="E3:65"/><b>then </b><i><font color="Green" title="E52">A3</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrlin.html#K12" title="MATRLIN:func.12">AutMt</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="matrlin.html#K6" title="MATRLIN:func.6">*</a> <font color="Maroon" title="c5">f</font></span>)</span>,<font color="Maroon" title="c6">b1</font>,<font color="Maroon" title="c7">b2</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> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <span class="p2">(<span class="default"><a href="matrlin.html#K12" title="MATRLIN:func.12">AutMt</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c6">b1</font>,<font color="Maroon" title="c7">b2</font></span>)</span></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 NAME="E4:65"/><i><font color="Green" title="E53">A4</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="matrlin.html#K12" title="MATRLIN:func.12">AutMt</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="matrlin.html#K6" title="MATRLIN:func.6">*</a> <font color="Maroon" title="c5">f</font></span>)</span>,<font color="Maroon" title="c6">b1</font>,<font color="Maroon" title="c7">b2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="matrlin.html#K12" title="MATRLIN:func.12">AutMt</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c6">b1</font>,<font color="Maroon" title="c7">b2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="matrlin.html#E2:65"><i><font color="Green" title="E51">A2</font></i></a>, <a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></i>;<br/>
<a NAME="E5:65"/><i><font color="Green" title="E54">A5</font></i>: 
 <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="matrlin.html#K12" title="MATRLIN:func.12">AutMt</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="matrlin.html#K6" title="MATRLIN:func.6">*</a> <font color="Maroon" title="c5">f</font></span>)</span>,<font color="Maroon" title="c6">b1</font>,<font color="Maroon" title="c7">b2</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"><a href="matrlin.html#K12" title="MATRLIN:func.12">AutMt</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c6">b1</font>,<font color="Maroon" title="c7">b2</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="65_2"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="65_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:65_2_1"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 or  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">b1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="65_2_1_1"><b>suppose </b></a><a NAME="E1:65_2_1_1"/><i><font color="Green" title="E55">A6</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 ;<br/><div class="add"><b>then </b> <a href="matrlin.html#K12" title="MATRLIN:func.12">AutMt</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="matrlin.html#K6" title="MATRLIN:func.6">*</a> <font color="Maroon" title="c5">f</font></span>)</span>,<font color="Maroon" title="c6">b1</font>,<font color="Maroon" title="c7">b2</font> = 
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
<b>by </b><i><a class="ref" href="matrlin.html#T43" target="_self" title="MATRLIN:th.43">Th43</a></i>
<br/>.= 
 <a href="matrlin.html#K12" title="MATRLIN:func.12">AutMt</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c6">b1</font>,<font color="Maroon" title="c7">b2</font>
<b>by </b><i><a class="txt" href="matrlin.html#E1:65_2_1_1"><i><font color="Green" title="E55">A6</font></i></a>, <a class="ref" href="matrlin.html#T43" target="_self" title="MATRLIN:th.43">Th43</a></i>
;<br/><b>hence </b><a NAME="E3:65_2_1_1"/>
 <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="matrlin.html#K12" title="MATRLIN:func.12">AutMt</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="matrlin.html#K6" title="MATRLIN:func.6">*</a> <font color="Maroon" title="c5">f</font></span>)</span>,<font color="Maroon" title="c6">b1</font>,<font color="Maroon" title="c7">b2</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"><a href="matrlin.html#K12" title="MATRLIN:func.12">AutMt</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c6">b1</font>,<font color="Maroon" title="c7">b2</font></span>)</span>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="65_2_1_2"><b>suppose </b></a><a NAME="E1:65_2_1_2"/><i><font color="Green" title="E55">A7</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">b1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 ;<br/><div class="add"><b>hence </b> <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="matrlin.html#K12" title="MATRLIN:func.12">AutMt</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="matrlin.html#K6" title="MATRLIN:func.6">*</a> <font color="Maroon" title="c5">f</font></span>)</span>,<font color="Maroon" title="c6">b1</font>,<font color="Maroon" title="c7">b2</font></span>)</span> = 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">b2</font>
<b>by </b><i><a class="ref" href="matrlin.html#T44" target="_self" title="MATRLIN:th.44">Th44</a></i>
<br/>.= 
 <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="matrlin.html#K12" title="MATRLIN:func.12">AutMt</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c6">b1</font>,<font color="Maroon" title="c7">b2</font></span>)</span>
<b>by </b><i><a class="txt" href="matrlin.html#E1:65_2_1_2"><i><font color="Green" title="E55">A7</font></i></a>, <a class="ref" href="matrlin.html#T44" target="_self" title="MATRLIN:th.44">Th44</a></i>
;<br/><br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E6:65"/><b>then </b><i><font color="Green" title="E55">A8</font></i>: 
 <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="matrlin.html#K12" title="MATRLIN:func.12">AutMt</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="matrlin.html#K6" title="MATRLIN:func.6">*</a> <font color="Maroon" title="c5">f</font></span>)</span>,<font color="Maroon" title="c6">b1</font>,<font color="Maroon" title="c7">b2</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"><font color="Maroon" title="c2">a</font> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <span class="p2">(<span class="default"><a href="matrlin.html#K12" title="MATRLIN:func.12">AutMt</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c6">b1</font>,<font color="Maroon" title="c7">b2</font></span>)</span></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 NAME="E7:65"/>
for <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="matrlin.html#K12" title="MATRLIN:func.12">AutMt</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="matrlin.html#K6" title="MATRLIN:func.6">*</a> <font color="Maroon" title="c5">f</font></span>)</span>,<font color="Maroon" title="c6">b1</font>,<font color="Maroon" title="c7">b2</font></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="matrlin.html#K12" title="MATRLIN:func.12">AutMt</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="matrlin.html#K6" title="MATRLIN:func.6">*</a> <font color="Maroon" title="c5">f</font></span>)</span>,<font color="Maroon" title="c6">b1</font>,<font color="Maroon" title="c7">b2</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"><font color="Maroon" title="c2">a</font> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <span class="p2">(<span class="default"><a href="matrlin.html#K12" title="MATRLIN:func.12">AutMt</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c6">b1</font>,<font color="Maroon" title="c7">b2</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="65_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">i</font>, <font color="Maroon" title="c9">j</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>



<b>assume </b><a NAME="E1:65_3"/><i><font color="Green" title="E56">A9</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">i</font>,<font color="Maroon" title="c9">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="matrlin.html#K12" title="MATRLIN:func.12">AutMt</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="matrlin.html#K6" title="MATRLIN:func.6">*</a> <font color="Maroon" title="c5">f</font></span>)</span>,<font color="Maroon" title="c6">b1</font>,<font color="Maroon" title="c7">b2</font></span>)</span>
 ;<br/>

<a NAME="E2:65_3"/><b>then </b><i><font color="Green" title="E57">A10</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">i</font>,<font color="Maroon" title="c9">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> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p3">(<span class="default"><a href="matrlin.html#K12" title="MATRLIN:func.12">AutMt</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="matrlin.html#K6" title="MATRLIN:func.6">*</a> <font color="Maroon" title="c5">f</font></span>)</span>,<font color="Maroon" title="c6">b1</font>,<font color="Maroon" title="c7">b2</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p4">(<span class="default"><a href="matrlin.html#K12" title="MATRLIN:func.12">AutMt</a> <span class="p5">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="matrlin.html#K6" title="MATRLIN:func.6">*</a> <font color="Maroon" title="c5">f</font></span>)</span>,<font color="Maroon" title="c6">b1</font>,<font color="Maroon" title="c7">b2</font></span>)</span></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="ref" href="matrix_1.html#D5" title="MATRIX_1:def.5">MATRIX_1:def 5</a></i>;<br/>
<a NAME="E3:65_3"/><b>then </b><i><font color="Green" title="E58">A11</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">i</font>,<font color="Maroon" title="c9">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="matrlin.html#K12" title="MATRLIN:func.12">AutMt</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c6">b1</font>,<font color="Maroon" title="c7">b2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="matrlin.html#E4:65"><i><font color="Green" title="E53">A4</font></i></a>, <a class="txt" href="matrlin.html#E5:65"><i><font color="Green" title="E54">A5</font></i></a>, <a class="ref" href="matrix_1.html#D5" title="MATRIX_1:def.5">MATRIX_1:def 5</a></i>;<br/>
<a NAME="E4:65_3"/>
<span class="p1">(<span class="default"><a href="matrlin.html#K12" title="MATRLIN:func.12">AutMt</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="matrlin.html#K6" title="MATRLIN:func.6">*</a> <font color="Maroon" title="c5">f</font></span>)</span>,<font color="Maroon" title="c6">b1</font>,<font color="Maroon" title="c7">b2</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c8">i</font>,<font color="Maroon" title="c9">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> <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="matrlin.html#K12" title="MATRLIN:func.12">AutMt</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c6">b1</font>,<font color="Maroon" title="c7">b2</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c8">i</font>,<font color="Maroon" title="c9">j</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="65_3_1"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon" title="c10">p1</font> being   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c1">K</font><b> such that </b><br/><a NAME="E2:65_3_1"/><i><font color="Green" title="E59">A12</font></i>: 
( <font color="Maroon" title="c10">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrlin.html#K12" title="MATRLIN:func.12">AutMt</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="matrlin.html#K6" title="MATRLIN:func.6">*</a> <font color="Maroon" title="c5">f</font></span>)</span>,<font color="Maroon" title="c6">b1</font>,<font color="Maroon" title="c7">b2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font> &amp; <span class="p1">(<span class="default"><a href="matrlin.html#K12" title="MATRLIN:func.12">AutMt</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="matrlin.html#K6" title="MATRLIN:func.6">*</a> <font color="Maroon" title="c5">f</font></span>)</span>,<font color="Maroon" title="c6">b1</font>,<font color="Maroon" title="c7">b2</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c8">i</font>,<font color="Maroon" title="c9">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">p1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">j</font> )
 <b>by </b><i><a class="txt" href="matrlin.html#E1:65_3"><i><font color="Green" title="E56">A9</font></i></a>, <a class="ref" href="matrix_1.html#D6" title="MATRIX_1:def.6">MATRIX_1:def 6</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c11">p2</font> being   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c1">K</font><b> such that </b><br/><a NAME="E4:65_3_1"/><i><font color="Green" title="E60">A13</font></i>: 
( <font color="Maroon" title="c11">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrlin.html#K12" title="MATRLIN:func.12">AutMt</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c6">b1</font>,<font color="Maroon" title="c7">b2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font> &amp; <span class="p1">(<span class="default"><a href="matrlin.html#K12" title="MATRLIN:func.12">AutMt</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c6">b1</font>,<font color="Maroon" title="c7">b2</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c8">i</font>,<font color="Maroon" title="c9">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">p2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">j</font> )
 <b>by </b><i><a class="txt" href="matrlin.html#E3:65_3"><i><font color="Green" title="E58">A11</font></i></a>, <a class="ref" href="matrix_1.html#D6" title="MATRIX_1:def.6">MATRIX_1:def 6</a></i>;<br/>
<a NAME="E5:65_3_1"/><i><font color="Green" title="E61">A14</font></i>: 
<font color="Maroon" title="c9">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p2">(<span class="default"><a href="matrlin.html#K12" title="MATRLIN:func.12">AutMt</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="matrlin.html#K6" title="MATRLIN:func.6">*</a> <font color="Maroon" title="c5">f</font></span>)</span>,<font color="Maroon" title="c6">b1</font>,<font color="Maroon" title="c7">b2</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="matrlin.html#E2:65_3"><i><font color="Green" title="E57">A10</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E6:65_3_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">b1</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"><a href="matrlin.html#K12" title="MATRLIN:func.12">AutMt</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="matrlin.html#K6" title="MATRLIN:func.6">*</a> <font color="Maroon" title="c5">f</font></span>)</span>,<font color="Maroon" title="c6">b1</font>,<font color="Maroon" title="c7">b2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="matrlin.html#D10" target="_self" title="MATRLIN:def.10">Def10</a></i>;<br/>
<a NAME="E7:65_3_1"/><b>then </b>
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c6">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="matrlin.html#K12" title="MATRLIN:func.12">AutMt</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="matrlin.html#K6" title="MATRLIN:func.6">*</a> <font color="Maroon" title="c5">f</font></span>)</span>,<font color="Maroon" title="c6">b1</font>,<font color="Maroon" title="c7">b2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></i>;<br/>
<a NAME="E8:65_3_1"/><b>then </b>
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c6">b1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="txt" href="matrlin.html#E2:65_3"><i><font color="Green" title="E57">A10</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E9:65_3_1"/><b>then </b>
 <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="c6">b1</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<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="E10:65_3_1"/><b>then </b>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">b1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 
<b>by </b><i><a class="ref" href="finseq_1.html#T4" title="FINSEQ_1:th.4">FINSEQ_1:4</a>, <a class="ref" href="nat_1.html#T3" title="NAT_1:th.3">NAT_1:3</a></i>;<br/>
<a NAME="E11:65_3_1"/><b>then </b><i><font color="Green" title="E62">A15</font></i>: 
<font color="Maroon" title="c9">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">b2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="matrlin.html#E5:65_3_1"><i><font color="Green" title="E61">A14</font></i></a>, <a class="ref" href="matrlin.html#T44" target="_self" title="MATRLIN:th.44">Th44</a></i>;<br/>
<a NAME="E12:65_3_1"/><b>then </b><i><font color="Green" title="E63">A16</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">j</font> &amp; <font color="Maroon" title="c9">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">b2</font> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E13:65_3_1"/><i><font color="Green" title="E64">A17</font></i>: 
<font color="Maroon" title="c9">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c7">b2</font>
 
<b>by </b><i><a class="txt" href="matrlin.html#E11:65_3_1"><i><font color="Green" title="E62">A15</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="E14:65_3_1"/><i><font color="Green" title="E65">A18</font></i>: 
<font color="Maroon" title="c8">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="matrlin.html#K12" title="MATRLIN:func.12">AutMt</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="matrlin.html#K6" title="MATRLIN:func.6">*</a> <font color="Maroon" title="c5">f</font></span>)</span>,<font color="Maroon" title="c6">b1</font>,<font color="Maroon" title="c7">b2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="matrlin.html#E2:65_3"><i><font color="Green" title="E57">A10</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E15:65_3_1"/><b>then </b><i><font color="Green" title="E66">A19</font></i>: 
<font color="Maroon" title="c8">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c6">b1</font>
 
<b>by </b><i><a class="txt" href="matrlin.html#E55"><i><font color="Green" title="E47">Lm3</font></i></a></i>;<br/>
<a NAME="E16:65_3_1"/><b>then </b><i><font color="Green" title="E67">A20</font></i>: 
<font color="Maroon" title="c8">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="matrlin.html#K12" title="MATRLIN:func.12">AutMt</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c6">b1</font>,<font color="Maroon" title="c7">b2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="matrlin.html#E55"><i><font color="Green" title="E47">Lm3</font></i></a></i>;<br/>
<a NAME="E17:65_3_1"/><i><font color="Green" title="E68">A21</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">j</font> &amp; <font color="Maroon" title="c9">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <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"><font color="Maroon" title="c2">a</font> <a href="matrlin.html#K6" title="MATRLIN:func.6">*</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">b1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c8">i</font></span>)</span></span>)</span> <a href="matrlin.html#K11" title="MATRLIN:func.11">|--</a> <font color="Maroon" title="c7">b2</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="matrlin.html#E12:65_3_1"><i><font color="Green" title="E63">A16</font></i></a>, <a class="ref" href="matrlin.html#D9" target="_self" title="MATRLIN:def.9">Def9</a></i>;<br/>
<a NAME="E18:65_3_1"/><i><font color="Green" title="E69">A22</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">j</font> &amp; <font color="Maroon" title="c9">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <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"><font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">b1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c8">i</font></span>)</span></span>)</span> <a href="matrlin.html#K11" title="MATRLIN:func.11">|--</a> <font color="Maroon" title="c7">b2</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="matrlin.html#E12:65_3_1"><i><font color="Green" title="E63">A16</font></i></a>, <a class="ref" href="matrlin.html#D9" target="_self" title="MATRLIN:def.9">Def9</a></i>;<br/>
<i><font color="Green" title="E70">A23</font></i>: <font color="Maroon" title="c10">p1</font> = 
<span class="p1">(<span class="default"><a href="matrlin.html#K12" title="MATRLIN:func.12">AutMt</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="matrlin.html#K6" title="MATRLIN:func.6">*</a> <font color="Maroon" title="c5">f</font></span>)</span>,<font color="Maroon" title="c6">b1</font>,<font color="Maroon" title="c7">b2</font></span>)</span> <a href="matrlin.html#K1" title="MATRLIN:func.1">/.</a> <font color="Maroon" title="c8">i</font>
<b>by </b><i><a class="txt" href="matrlin.html#E2:65_3_1"><i><font color="Green" title="E59">A12</font></i></a>, <a class="txt" href="matrlin.html#E14:65_3_1"><i><font color="Green" title="E65">A18</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="matrlin.html#K6" title="MATRLIN:func.6">*</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">b1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c8">i</font></span>)</span></span>)</span> <a href="matrlin.html#K11" title="MATRLIN:func.11">|--</a> <font color="Maroon" title="c7">b2</font>
<b>by </b><i><a class="txt" href="matrlin.html#E15:65_3_1"><i><font color="Green" title="E66">A19</font></i></a>, <a class="ref" href="matrlin.html#D10" target="_self" title="MATRLIN:def.10">Def10</a></i>
;<br/>
<i><font color="Green" title="E71">A24</font></i>: <font color="Maroon" title="c11">p2</font> = 
<span class="p1">(<span class="default"><a href="matrlin.html#K12" title="MATRLIN:func.12">AutMt</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c6">b1</font>,<font color="Maroon" title="c7">b2</font></span>)</span> <a href="matrlin.html#K1" title="MATRLIN:func.1">/.</a> <font color="Maroon" title="c8">i</font>
<b>by </b><i><a class="txt" href="matrlin.html#E4:65_3_1"><i><font color="Green" title="E60">A13</font></i></a>, <a class="txt" href="matrlin.html#E16:65_3_1"><i><font color="Green" title="E67">A20</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">b1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c8">i</font></span>)</span></span>)</span> <a href="matrlin.html#K11" title="MATRLIN:func.11">|--</a> <font color="Maroon" title="c7">b2</font>
<b>by </b><i><a class="txt" href="matrlin.html#E15:65_3_1"><i><font color="Green" title="E66">A19</font></i></a>, <a class="ref" href="matrlin.html#D10" target="_self" title="MATRLIN:def.10">Def10</a></i>
;<br/>
<a NAME="E21:65_3_1"/><i><font color="Green" title="E72">A25</font></i>: 
<font color="Maroon" title="c9">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c10">p1</font>
 
<b>by </b><i><a class="txt" href="matrlin.html#E13:65_3_1"><i><font color="Green" title="E64">A17</font></i></a>, <a class="txt" href="matrlin.html#E19:65_3_1"><i><font color="Green" title="E70">A23</font></i></a>, <a class="txt" href="matrlin.html#E48"><i><font color="Green" title="E40">Lm1</font></i></a></i>;<br/>
<a NAME="E22:65_3_1"/>
<font color="Maroon" title="c9">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">b1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c8">i</font></span>)</span></span>)</span> <a href="matrlin.html#K11" title="MATRLIN:func.11">|--</a> <font color="Maroon" title="c7">b2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="matrlin.html#E13:65_3_1"><i><font color="Green" title="E64">A17</font></i></a>, <a class="txt" href="matrlin.html#E48"><i><font color="Green" title="E40">Lm1</font></i></a></i>;<br/>
<a NAME="E23:65_3_1"/><b>then </b><i><font color="Green" title="E73">A26</font></i>: 
<span class="p1">(<span class="default"><a href="matrlin.html#K12" title="MATRLIN:func.12">AutMt</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c6">b1</font>,<font color="Maroon" title="c7">b2</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c8">i</font>,<font color="Maroon" title="c9">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">p2</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c9">j</font>
 
<b>by </b><i><a class="txt" href="matrlin.html#E4:65_3_1"><i><font color="Green" title="E60">A13</font></i></a>, <a class="txt" href="matrlin.html#E20:65_3_1"><i><font color="Green" title="E71">A24</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c12">KL1</font> being    <a href="vectsp_6.html#M1" title="VECTSP_6:mode.1">Linear_Combination</a> of <font color="Maroon" title="c4">V2</font><b> such that </b><br/><a NAME="E25:65_3_1"/><i><font color="Green" title="E74">A27</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="matrlin.html#K6" title="MATRLIN:func.6">*</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">b1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c8">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_6.html#K4" title="VECTSP_6:func.4">Sum</a> <font color="Maroon" title="c12">KL1</font> &amp;  <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c12">KL1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">b2</font> &amp; ( for <font color="Olive" title="b1">t</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">t</font> &amp; <font color="Olive" title="b1">t</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <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"><font color="Maroon" title="c2">a</font> <a href="matrlin.html#K6" title="MATRLIN:func.6">*</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">b1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c8">i</font></span>)</span></span>)</span> <a href="matrlin.html#K11" title="MATRLIN:func.11">|--</a> <font color="Maroon" title="c7">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" title="c2">a</font> <a href="matrlin.html#K6" title="MATRLIN:func.6">*</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">b1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c8">i</font></span>)</span></span>)</span> <a href="matrlin.html#K11" title="MATRLIN:func.11">|--</a> <font color="Maroon" title="c7">b2</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b1">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">KL1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">b2</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b1">t</font></span>)</span> ) )
 <b>by </b><i><a class="ref" href="matrlin.html#D9" target="_self" title="MATRLIN:def.9">Def9</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c13">KL2</font> being    <a href="vectsp_6.html#M1" title="VECTSP_6:mode.1">Linear_Combination</a> of <font color="Maroon" title="c4">V2</font><b> such that </b><br/><a NAME="E27:65_3_1"/><i><font color="Green" title="E75">A28</font></i>: 
( <font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">b1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c8">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_6.html#K4" title="VECTSP_6:func.4">Sum</a> <font color="Maroon" title="c13">KL2</font> &amp;  <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c13">KL2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">b2</font> &amp; ( for <font color="Olive" title="b1">t</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">t</font> &amp; <font color="Olive" title="b1">t</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <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"><font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">b1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c8">i</font></span>)</span></span>)</span> <a href="matrlin.html#K11" title="MATRLIN:func.11">|--</a> <font color="Maroon" title="c7">b2</font></span>)</span> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">b1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c8">i</font></span>)</span></span>)</span> <a href="matrlin.html#K11" title="MATRLIN:func.11">|--</a> <font color="Maroon" title="c7">b2</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b1">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">KL2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">b2</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b1">t</font></span>)</span> ) )
 <b>by </b><i><a class="ref" href="matrlin.html#D9" target="_self" title="MATRLIN:def.9">Def9</a></i>;<br/>
<a NAME="E28:65_3_1"/><i><font color="Green" title="E76">A29</font></i>: 
<font color="Maroon" title="c10">p1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c9">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">KL1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">b2</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c9">j</font></span>)</span>
 
<b>by </b><i><a class="txt" href="matrlin.html#E17:65_3_1"><i><font color="Green" title="E68">A21</font></i></a>, <a class="txt" href="matrlin.html#E19:65_3_1"><i><font color="Green" title="E70">A23</font></i></a>, <a class="txt" href="matrlin.html#E25:65_3_1"><i><font color="Green" title="E74">A27</font></i></a></i>;<br/>
<a NAME="E29:65_3_1"/><i><font color="Green" title="E77">A30</font></i>: 
<font color="Maroon" title="c11">p2</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c9">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">KL2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">b2</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c9">j</font></span>)</span>
 
<b>by </b><i><a class="txt" href="matrlin.html#E18:65_3_1"><i><font color="Green" title="E69">A22</font></i></a>, <a class="txt" href="matrlin.html#E20:65_3_1"><i><font color="Green" title="E71">A24</font></i></a>, <a class="txt" href="matrlin.html#E27:65_3_1"><i><font color="Green" title="E75">A28</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c14">b4</font> =  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">b2</font> as    <a href="vectsp_7.html#M1" title="VECTSP_7:mode.1">Basis</a> of <font color="Maroon" title="c4">V2</font> <b>by </b><i><a class="ref" href="matrlin.html#D4" target="_self" title="MATRLIN:def.4">Def4</a></i>;<br/>
<a NAME="E31:65_3_1"/><i><font color="Green" title="E78">A31</font></i>: 
<font color="Maroon" title="c14">b4</font> is <a href="vectsp_7.html#V1" title="VECTSP_7:attr.1">linearly-independent</a>
 
<b>by </b><i><a class="ref" href="vectsp_7.html#D3" title="VECTSP_7:def.3">VECTSP_7:def 3</a></i>;<br/>
<a NAME="E32:65_3_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="matrlin.html#K6" title="MATRLIN:func.6">*</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">b1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c8">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">b1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c8">i</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="matrlin.html#D6" target="_self" title="MATRLIN:def.6">Def6</a></i>;<br/>
<b>then </b><font color="Maroon" title="c12">KL1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">b2</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c9">j</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="vectsp_6.html#K6" title="VECTSP_6:func.6">*</a> <font color="Maroon" title="c13">KL2</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">b2</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c9">j</font></span>)</span>
<b>by </b><i><a class="txt" href="matrlin.html#E1:65"><i><font color="Green" title="E50">A1</font></i></a>, <a class="txt" href="matrlin.html#E25:65_3_1"><i><font color="Green" title="E74">A27</font></i></a>, <a class="txt" href="matrlin.html#E27:65_3_1"><i><font color="Green" title="E75">A28</font></i></a>, <a class="txt" href="matrlin.html#E31:65_3_1"><i><font color="Green" title="E78">A31</font></i></a>, <a class="ref" href="matrlin.html#T11" target="_self" title="MATRLIN:th.11">Th11</a></i>
<br/>.= 
<font color="Maroon" title="c2">a</font> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">KL2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">b2</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c9">j</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_6.html#D12" title="VECTSP_6:def.12">VECTSP_6:def 12</a></i>
;<br/>
<b>hence </b><a NAME="E34:65_3_1"/>
<span class="p1">(<span class="default"><a href="matrlin.html#K12" title="MATRLIN:func.12">AutMt</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="matrlin.html#K6" title="MATRLIN:func.6">*</a> <font color="Maroon" title="c5">f</font></span>)</span>,<font color="Maroon" title="c6">b1</font>,<font color="Maroon" title="c7">b2</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c8">i</font>,<font color="Maroon" title="c9">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> <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="matrlin.html#K12" title="MATRLIN:func.12">AutMt</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c6">b1</font>,<font color="Maroon" title="c7">b2</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c8">i</font>,<font color="Maroon" title="c9">j</font></span>)</span>
 <b>by </b><i><a class="txt" href="matrlin.html#E2:65_3_1"><i><font color="Green" title="E59">A12</font></i></a>, <a class="txt" href="matrlin.html#E21:65_3_1"><i><font color="Green" title="E72">A25</font></i></a>, <a class="txt" href="matrlin.html#E23:65_3_1"><i><font color="Green" title="E73">A26</font></i></a>, <a class="txt" href="matrlin.html#E28:65_3_1"><i><font color="Green" title="E76">A29</font></i></a>, <a class="txt" href="matrlin.html#E29:65_3_1"><i><font color="Green" title="E77">A30</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:65_3"/>
<span class="p1">(<span class="default"><a href="matrlin.html#K12" title="MATRLIN:func.12">AutMt</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="matrlin.html#K6" title="MATRLIN:func.6">*</a> <font color="Maroon" title="c5">f</font></span>)</span>,<font color="Maroon" title="c6">b1</font>,<font color="Maroon" title="c7">b2</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c8">i</font>,<font color="Maroon" title="c9">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <span class="p2">(<span class="default"><a href="matrlin.html#K12" title="MATRLIN:func.12">AutMt</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c6">b1</font>,<font color="Maroon" title="c7">b2</font></span>)</span></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c8">i</font>,<font color="Maroon" title="c9">j</font>
 <b>by </b><i><a class="txt" href="matrlin.html#E3:65_3"><i><font color="Green" title="E58">A11</font></i></a>, <a class="ref" href="matrix_3.html#D5" title="MATRIX_3:def.5">MATRIX_3:def 5</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E8:65"/>
 <a href="matrlin.html#K12" title="MATRLIN:func.12">AutMt</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="matrlin.html#K6" title="MATRLIN:func.6">*</a> <font color="Maroon" title="c5">f</font></span>)</span>,<font color="Maroon" title="c6">b1</font>,<font color="Maroon" title="c7">b2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <span class="p1">(<span class="default"><a href="matrlin.html#K12" title="MATRLIN:func.12">AutMt</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c6">b1</font>,<font color="Maroon" title="c7">b2</font></span>)</span>
 <b>by </b><i><a class="txt" href="matrlin.html#E3:65"><i><font color="Green" title="E52">A3</font></i></a>, <a class="txt" href="matrlin.html#E6:65"><i><font color="Green" title="E55">A8</font></i></a>, <a class="ref" href="matrix_1.html#T21" title="MATRIX_1:th.21">MATRIX_1:21</a></i>;<br/>


</div>
