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<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">K</font> be   <a href="vectsp_1.html#NM3" title="VECTSP_1:NM.3">Field</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">a</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">K</font><br/>  for <font color="Olive" title="b2">V1</font>, <font color="Olive" title="b3">V2</font> being   <a href="matrlin.html#V1" title="MATRLIN:attr.1">finite-dimensional</a>  <a href="vectsp_1.html#NM7" title="VECTSP_1:NM.7">VectSp</a> of <font color="Maroon" title="c1">K</font><br/>  for <font color="Olive" title="b4">f</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Olive" title="b2">V1</font>,<font color="Olive" title="b3">V2</font><br/>  for <font color="Olive" title="b5">b1</font> being    <a href="matrlin.html#M1" title="MATRLIN:mode.1">OrdBasis</a> of <font color="Olive" title="b2">V1</font><br/>  for <font color="Olive" title="b6">b2</font> being    <a href="matrlin.html#M1" title="MATRLIN:mode.1">OrdBasis</a> of <font color="Olive" title="b3">V2</font>  st <font color="Olive" title="b1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font> holds <br/> <a href="matrlin.html#K10" title="MATRLIN:func.10">AutMt</a> (<span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</a> <font color="Olive" title="b4">f</font></span>)</span>,<font color="Olive" title="b5">b1</font>,<font color="Olive" title="b6">b2</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <span class="p1">(<span class="default"><a href="matrlin.html#K10" title="MATRLIN:func.10">AutMt</a> (<font color="Olive" title="b4">f</font>,<font color="Olive" title="b5">b1</font>,<font color="Olive" title="b6">b2</font>)</span>)</span></span><br/><span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">V1</font>, <font color="Olive" title="b2">V2</font> being   <a href="matrlin.html#V1" title="MATRLIN:attr.1">finite-dimensional</a>  <a href="vectsp_1.html#NM7" title="VECTSP_1:NM.7">VectSp</a> of <font color="Maroon" title="c1">K</font><br/>  for <font color="Olive" title="b3">f</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Olive" title="b1">V1</font>,<font color="Olive" title="b2">V2</font><br/>  for <font color="Olive" title="b4">b1</font> being    <a href="matrlin.html#M1" title="MATRLIN:mode.1">OrdBasis</a> of <font color="Olive" title="b1">V1</font><br/>  for <font color="Olive" title="b5">b2</font> being    <a href="matrlin.html#M1" title="MATRLIN:mode.1">OrdBasis</a> of <font color="Olive" title="b2">V2</font>  st <font color="Maroon" title="c2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font> holds <br/> <a href="matrlin.html#K10" title="MATRLIN:func.10">AutMt</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</a> <font color="Olive" title="b3">f</font></span>)</span>,<font color="Olive" title="b4">b1</font>,<font color="Olive" title="b5">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#K10" title="MATRLIN:func.10">AutMt</a> (<font color="Olive" title="b3">f</font>,<font color="Olive" title="b4">b1</font>,<font color="Olive" title="b5">b2</font>)</span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c3">V1</font>, <font color="Maroon" title="c4">V2</font> be   <a href="matrlin.html#V1" title="MATRLIN:attr.1">finite-dimensional</a>  <a href="vectsp_1.html#NM7" title="VECTSP_1:NM.7">VectSp</a> of <font color="Maroon" title="c1">K</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">f</font> being   <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/>  for <font color="Olive" title="b2">b1</font> being    <a href="matrlin.html#M1" title="MATRLIN:mode.1">OrdBasis</a> of <font color="Maroon" title="c3">V1</font><br/>  for <font color="Olive" title="b3">b2</font> being    <a href="matrlin.html#M1" title="MATRLIN:mode.1">OrdBasis</a> of <font color="Maroon" title="c4">V2</font>  st <font color="Maroon" title="c2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font> holds <br/> <a href="matrlin.html#K10" title="MATRLIN:func.10">AutMt</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</a> <font color="Olive" title="b1">f</font></span>)</span>,<font color="Olive" title="b2">b1</font>,<font color="Olive" title="b3">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#K10" title="MATRLIN:func.10">AutMt</a> (<font color="Olive" title="b1">f</font>,<font color="Olive" title="b2">b1</font>,<font color="Olive" title="b3">b2</font>)</span>)</span></span><br/><span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">b1</font> being    <a href="matrlin.html#M1" title="MATRLIN:mode.1">OrdBasis</a> of <font color="Maroon" title="c3">V1</font><br/>  for <font color="Olive" title="b2">b2</font> being    <a href="matrlin.html#M1" title="MATRLIN:mode.1">OrdBasis</a> of <font color="Maroon" title="c4">V2</font>  st <font color="Maroon" title="c2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font> holds <br/> <a href="matrlin.html#K10" title="MATRLIN:func.10">AutMt</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</a> <font color="Maroon" title="c5">f</font></span>)</span>,<font color="Olive" title="b1">b1</font>,<font color="Olive" title="b2">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#K10" title="MATRLIN:func.10">AutMt</a> (<font color="Maroon" title="c5">f</font>,<font color="Olive" title="b1">b1</font>,<font color="Olive" title="b2">b2</font>)</span>)</span></span><br/><span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">b2</font> being    <a href="matrlin.html#M1" title="MATRLIN:mode.1">OrdBasis</a> of <font color="Maroon" title="c4">V2</font>  st <font color="Maroon" title="c2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font> holds <br/> <a href="matrlin.html#K10" title="MATRLIN:func.10">AutMt</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</a> <font color="Maroon" title="c5">f</font></span>)</span>,<font color="Maroon" title="c6">b1</font>,<font color="Olive" title="b1">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#K10" title="MATRLIN:func.10">AutMt</a> (<font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c6">b1</font>,<font color="Olive" title="b1">b2</font>)</span>)</span></span><br/><span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font> implies  <a href="matrlin.html#K10" title="MATRLIN:func.10">AutMt</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</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#K10" title="MATRLIN:func.10">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><br/>













<span class="kw">assume </span><a NAME="E1:60"/><span class="lab"><font color="Green" title="E48">A1</font></span>: 
<font color="Maroon" title="c2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="matrlin.html#K10" title="MATRLIN:func.10">AutMt</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</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#K10" title="MATRLIN:func.10">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><br/>

<a NAME="E2:60"/><span class="lab"><font color="Green" title="E49">A2</font></span>: 
 <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="matrlin.html#K10" title="MATRLIN:func.10">AutMt</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</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#K10" title="MATRLIN:func.10">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="60_1"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="60_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:60_1_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="numbers.html#K6" title="NUMBERS:func.6">0</a>  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>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="60_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:60_1_1_1"/><span class="lab"><font color="Green" title="E50">A3</font></span>: 
 <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="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="matrlin.html#K10" title="MATRLIN:func.10">AutMt</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</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#K10" title="MATRLIN:func.10">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><br/><div class="add"><span class="kw">then </span> <a href="matrlin.html#K10" title="MATRLIN:func.10">AutMt</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</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> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrlin.html#T38" target="_self" title="MATRLIN:th.38">Th43</a></span>
<br/>.= 
 <a href="matrlin.html#K10" title="MATRLIN:func.10">AutMt</a> (<font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c6">b1</font>,<font color="Maroon" title="c7">b2</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:60_1_1_1"><span class="lab"><font color="Green" title="E50">A3</font></span></a>, <a class="ref" href="matrlin.html#T38" target="_self" title="MATRLIN:th.38">Th43</a></span>
;<br/><span class="kw">hence </span><a NAME="E3:60_1_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#K10" title="MATRLIN:func.10">AutMt</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</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#K10" title="MATRLIN:func.10">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 class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="60_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:60_1_1_2"/><span class="lab"><font color="Green" title="E50">A4</font></span>: 
 <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>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="matrlin.html#K10" title="MATRLIN:func.10">AutMt</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</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#K10" title="MATRLIN:func.10">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><br/><div class="add"><span class="kw">hence </span> <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="matrlin.html#K10" title="MATRLIN:func.10">AutMt</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrlin.html#T39" target="_self" title="MATRLIN:th.39">Th44</a></span>
<br/>.= 
 <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="matrlin.html#K10" title="MATRLIN:func.10">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 class="kw">by</span> <span class="lab"><a class="txt" href="#E1:60_1_1_2"><span class="lab"><font color="Green" title="E50">A4</font></span></a>, <a class="ref" href="matrlin.html#T39" target="_self" title="MATRLIN:th.39">Th44</a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<a NAME="E3:60"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A5</font></span>: 
 <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="matrlin.html#K10" title="MATRLIN:func.10">AutMt</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</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#K10" title="MATRLIN:func.10">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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_3.html#D5" title="MATRIX_3:def.5">MATRIX_3:def 5</a></span>;<br/>
<span class="lab"><font color="Green" title="E51">A6</font></span>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrlin.html#K10" title="MATRLIN:func.10">AutMt</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrlin.html#D8" target="_self" title="MATRLIN:def.8">Def10</a></span>
<br/>.= 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrlin.html#K10" title="MATRLIN:func.10">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 class="kw">by</span> <span class="lab"><a class="ref" href="matrlin.html#D8" target="_self" title="MATRLIN:def.8">Def10</a></span>
;<br/>
<a NAME="E5:60"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A7</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="matrlin.html#K10" title="MATRLIN:func.10">AutMt</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</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#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="matrlin.html#K10" title="MATRLIN:func.10">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 class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T29" title="FINSEQ_3:th.29">FINSEQ_3:29</a></span>;<br/>
<a NAME="E6:60"/><span class="lab"><font color="Green" title="E53">A8</font></span>: 
 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#K10" title="MATRLIN:func.10">AutMt</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</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#K10" title="MATRLIN:func.10">AutMt</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</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#K10" title="MATRLIN:func.10">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="60_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <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#K10" title="MATRLIN:func.10">AutMt</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</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> implies <span class="p1">(<span class="default"><a href="matrlin.html#K10" title="MATRLIN:func.10">AutMt</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</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#K10" title="MATRLIN:func.10">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>) )</span><br/>



<span class="kw">assume </span><a NAME="E1:60_3"/><span class="lab"><font color="Green" title="E54">A9</font></span>: 
<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#K10" title="MATRLIN:func.10">AutMt</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</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 class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="matrlin.html#K10" title="MATRLIN:func.10">AutMt</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</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#K10" title="MATRLIN:func.10">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>)</span><br/>

<a NAME="E2:60_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E55">A10</font></span>: 
<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#NK2" title="RELAT_1:NK.2">dom</a> <span class="p3">(<span class="default"><a href="matrlin.html#K10" title="MATRLIN:func.10">AutMt</a> (<span class="p4">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</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#K10" title="MATRLIN:func.10">AutMt</a> (<span class="p5">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#D4" title="MATRIX_1:def.4">MATRIX_1:def 4</a></span>;<br/>
<a NAME="E3:60_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">A11</font></span>: 
<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#K10" title="MATRLIN:func.10">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 class="kw">by</span> <span class="lab"><a class="txt" href="#E5:60"><span class="lab"><font color="Green" title="E52">A7</font></span></a>, <a class="txt" href="#E2:60"><span class="lab"><font color="Green" title="E49">A2</font></span></a>, <a class="ref" href="matrix_1.html#D4" title="MATRIX_1:def.4">MATRIX_1:def 4</a></span>;<br/>
<a NAME="E4:60_3"/>
<span class="p1">(<span class="default"><a href="matrlin.html#K10" title="MATRLIN:func.10">AutMt</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</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="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrlin.html#K10" title="MATRLIN:func.10">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="60_3_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">consider </span><font color="Maroon" title="c10">p2</font> being   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c1">K</font><span class="kw"> such that </span><br/><a NAME="E2:60_3_1"/><span class="lab"><font color="Green" title="E57">A12</font></span>: 
<font color="Maroon" title="c10">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrlin.html#K10" title="MATRLIN:func.10">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>
 <span class="kw">and </span><br/><a NAME="E3:60_3_1"/><span class="lab"><font color="Green" title="E58">A13</font></span>: 
<span class="p1">(<span class="default"><a href="matrlin.html#K10" title="MATRLIN:func.10">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="c10">p2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:60_3"><span class="lab"><font color="Green" title="E56">A11</font></span></a>, <a class="ref" href="matrix_1.html#D5" title="MATRIX_1:def.5">MATRIX_1:def 5</a></span>;<br/>
<a NAME="E4:60_3_1"/><span class="lab"><font color="Green" title="E59">A14</font></span>: 
<font color="Maroon" title="c8">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="matrlin.html#K10" title="MATRLIN:func.10">AutMt</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</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 class="kw">by</span> <span class="lab"><a class="txt" href="#E2:60_3"><span class="lab"><font color="Green" title="E55">A10</font></span></a>, <a class="ref" href="zfmisc_1.html#T87" title="ZFMISC_1:th.87">ZFMISC_1:87</a></span>;<br/>
<a NAME="E5:60_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E60">A15</font></span>: 
<font color="Maroon" title="c8">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c6">b1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="matrlin.html#E48"><span class="lab"><font color="Green" title="E45">Lm3</font></span></a></span>;<br/>
<a NAME="E6:60_3_1"/><span class="kw">then </span>
<font color="Maroon" title="c8">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="matrlin.html#K10" title="MATRLIN:func.10">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 class="kw">by</span> <span class="lab"><a class="txt" href="matrlin.html#E48"><span class="lab"><font color="Green" title="E45">Lm3</font></span></a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E61">A16</font></span>: <font color="Maroon" title="c10">p2</font> = 
<span class="p1">(<span class="default"><a href="matrlin.html#K10" title="MATRLIN:func.10">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="pre_poly.html#K9" title="PRE_POLY:func.9">/.</a> <font color="Maroon" title="c8">i</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:60_3_1"><span class="lab"><font color="Green" title="E57">A12</font></span></a>, <a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>
<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="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c8">i</font></span>)</span></span>)</span> <a href="matrlin.html#K9" title="MATRLIN:func.9">|--</a> <font color="Maroon" title="c7">b2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:60_3_1"><span class="lab"><font color="Green" title="E60">A15</font></span></a>, <a class="ref" href="matrlin.html#D8" target="_self" title="MATRLIN:def.8">Def10</a></span>
;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c11">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> <span class="kw">by</span> <span class="lab"><a class="ref" href="matrlin.html#D2" target="_self" title="MATRLIN:def.2">Def4</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c12">p1</font> being   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c1">K</font><span class="kw"> such that </span><br/><a NAME="E10:60_3_1"/><span class="lab"><font color="Green" title="E62">A17</font></span>: 
<font color="Maroon" title="c12">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrlin.html#K10" title="MATRLIN:func.10">AutMt</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</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>
 <span class="kw">and </span><br/><a NAME="E11:60_3_1"/><span class="lab"><font color="Green" title="E63">A18</font></span>: 
<span class="p1">(<span class="default"><a href="matrlin.html#K10" title="MATRLIN:func.10">AutMt</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</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="c12">p1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:60_3"><span class="lab"><font color="Green" title="E54">A9</font></span></a>, <a class="ref" href="matrix_1.html#D5" title="MATRIX_1:def.5">MATRIX_1:def 5</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c13">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><span class="kw"> such that </span><br/><a NAME="E13:60_3_1"/><span class="lab"><font color="Green" title="E64">A19</font></span>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</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="partfun1.html#K7" title="PARTFUN1:func.7">/.</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">KL1</font> &amp;  <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c13">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> )
 <span class="kw">and </span><br/><a NAME="E14:60_3_1"/><span class="lab"><font color="Green" title="E65">A20</font></span>: 
 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#K4" title="MATRLIN:func.4">*</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="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c8">i</font></span>)</span></span>)</span> <a href="matrlin.html#K9" title="MATRLIN:func.9">|--</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#K4" title="MATRLIN:func.4">*</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="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c8">i</font></span>)</span></span>)</span> <a href="matrlin.html#K9" title="MATRLIN:func.9">|--</a> <font color="Maroon" title="c7">b2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">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="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">t</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="matrlin.html#D7" target="_self" title="MATRLIN:def.7">Def9</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c14">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><span class="kw"> such that </span><br/><a NAME="E16:60_3_1"/><span class="lab"><font color="Green" title="E66">A21</font></span>: 
( <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="partfun1.html#K7" title="PARTFUN1:func.7">/.</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="c14">KL2</font> &amp;  <a href="vectsp_6.html#K1" title="VECTSP_6:func.1">Carrier</a> <font color="Maroon" title="c14">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> )
 <span class="kw">and </span><br/><a NAME="E17:60_3_1"/><span class="lab"><font color="Green" title="E67">A22</font></span>: 
 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="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c8">i</font></span>)</span></span>)</span> <a href="matrlin.html#K9" title="MATRLIN:func.9">|--</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="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c8">i</font></span>)</span></span>)</span> <a href="matrlin.html#K9" title="MATRLIN:func.9">|--</a> <font color="Maroon" title="c7">b2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">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="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">t</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="matrlin.html#D7" target="_self" title="MATRLIN:def.7">Def9</a></span>;<br/>
<a NAME="E18:60_3_1"/>
( <font color="Maroon" title="c11">b4</font> is  <a href="vectsp_7.html#V1" title="VECTSP_7:attr.1">linearly-independent</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</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="partfun1.html#K7" title="PARTFUN1:func.7">/.</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#K4" title="VECTSP_1:func.4">*</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="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c8">i</font></span>)</span></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrlin.html#D4" target="_self" title="MATRLIN:def.4">Def6</a>, <a class="ref" href="vectsp_7.html#D3" title="VECTSP_7:def.3">VECTSP_7:def 3</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E68">A23</font></span>: <font color="Maroon" title="c13">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="partfun1.html#K7" title="PARTFUN1:func.7">/.</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="c14">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="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c9">j</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:60"><span class="lab"><font color="Green" title="E48">A1</font></span></a>, <a class="txt" href="#E13:60_3_1"><span class="lab"><font color="Green" title="E64">A19</font></span></a>, <a class="txt" href="#E16:60_3_1"><span class="lab"><font color="Green" title="E66">A21</font></span></a>, <a class="ref" href="matrlin.html#T7" target="_self" title="MATRLIN:th.7">Th11</a></span>
<br/>.= 
<font color="Maroon" title="c2">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">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="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c9">j</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_6.html#D9" title="VECTSP_6:def.9">VECTSP_6:def 9</a></span>
;<br/>
<a NAME="E20:60_3_1"/><span class="lab"><font color="Green" title="E69">A24</font></span>: 
<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#K10" title="MATRLIN:func.10">AutMt</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</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="kw">by</span> <span class="lab"><a class="txt" href="#E2:60_3"><span class="lab"><font color="Green" title="E55">A10</font></span></a>, <a class="ref" href="zfmisc_1.html#T87" title="ZFMISC_1:th.87">ZFMISC_1:87</a></span>;<br/>
<a NAME="E21:60_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E70">A25</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">j</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/>
<a NAME="E22:60_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#K10" title="MATRLIN:func.10">AutMt</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</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 class="kw">by</span> <span class="lab"><a class="ref" href="matrlin.html#D8" target="_self" title="MATRLIN:def.8">Def10</a></span>;<br/>
<a NAME="E23:60_3_1"/><span class="kw">then </span>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c6">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="matrlin.html#K10" title="MATRLIN:func.10">AutMt</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</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 class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T29" title="FINSEQ_3:th.29">FINSEQ_3:29</a></span>;<br/>
<a NAME="E24:60_3_1"/><span class="kw">then </span>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:60_3"><span class="lab"><font color="Green" title="E55">A10</font></span></a>, <a class="ref" href="zfmisc_1.html#T87" title="ZFMISC_1:th.87">ZFMISC_1:87</a></span>;<br/>
<a NAME="E25:60_3_1"/><span class="kw">then </span>
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="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> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E26:60_3_1"/><span class="kw">then </span>
 <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>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
;<br/>
<a NAME="E27:60_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E71">A26</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:60_3_1"><span class="lab"><font color="Green" title="E69">A24</font></span></a>, <a class="ref" href="matrlin.html#T39" target="_self" title="MATRLIN:th.39">Th44</a></span>;<br/>
<a NAME="E28:60_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E72">A27</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/>
<a NAME="E29:60_3_1"/><span class="kw">then </span>
<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="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c8">i</font></span>)</span></span>)</span> <a href="matrlin.html#K9" title="MATRLIN:func.9">|--</a> <font color="Maroon" title="c7">b2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrlin.html#D7" target="_self" title="MATRLIN:def.7">Def9</a></span>;<br/>
<a NAME="E30:60_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E73">A28</font></span>: 
<font color="Maroon" title="c10">p2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c9">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">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="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c9">j</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:60_3_1"><span class="lab"><font color="Green" title="E70">A25</font></span></a>, <a class="txt" href="#E7:60_3_1"><span class="lab"><font color="Green" title="E61">A16</font></span></a>, <a class="txt" href="#E17:60_3_1"><span class="lab"><font color="Green" title="E67">A22</font></span></a></span>;<br/>
<a NAME="E31:60_3_1"/><span class="lab"><font color="Green" title="E74">A29</font></span>: 
<font color="Maroon" title="c9">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c7">b2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:60_3_1"><span class="lab"><font color="Green" title="E71">A26</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E32:60_3_1"/><span class="kw">then </span>
<font color="Maroon" title="c9">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">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="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c8">i</font></span>)</span></span>)</span> <a href="matrlin.html#K9" title="MATRLIN:func.9">|--</a> <font color="Maroon" title="c7">b2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="matrlin.html#E41"><span class="lab"><font color="Green" title="E38">Lm1</font></span></a></span>;<br/>
<a NAME="E33:60_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E75">A30</font></span>: 
<span class="p1">(<span class="default"><a href="matrlin.html#K10" title="MATRLIN:func.10">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="c10">p2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c9">j</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:60_3_1"><span class="lab"><font color="Green" title="E58">A13</font></span></a>, <a class="txt" href="#E7:60_3_1"><span class="lab"><font color="Green" title="E61">A16</font></span></a>, <a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>;<br/>
<span class="lab"><font color="Green" title="E76">A31</font></span>: <font color="Maroon" title="c12">p1</font> = 
<span class="p1">(<span class="default"><a href="matrlin.html#K10" title="MATRLIN:func.10">AutMt</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</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="pre_poly.html#K9" title="PRE_POLY:func.9">/.</a> <font color="Maroon" title="c8">i</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:60_3_1"><span class="lab"><font color="Green" title="E62">A17</font></span></a>, <a class="txt" href="#E4:60_3_1"><span class="lab"><font color="Green" title="E59">A14</font></span></a>, <a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</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="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c8">i</font></span>)</span></span>)</span> <a href="matrlin.html#K9" title="MATRLIN:func.9">|--</a> <font color="Maroon" title="c7">b2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:60_3_1"><span class="lab"><font color="Green" title="E60">A15</font></span></a>, <a class="ref" href="matrlin.html#D8" target="_self" title="MATRLIN:def.8">Def10</a></span>
;<br/>
<a NAME="E35:60_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E77">A32</font></span>: 
<font color="Maroon" title="c9">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c12">p1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:60_3_1"><span class="lab"><font color="Green" title="E74">A29</font></span></a>, <a class="txt" href="matrlin.html#E41"><span class="lab"><font color="Green" title="E38">Lm1</font></span></a></span>;<br/>
<a NAME="E36:60_3_1"/>
<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#K4" title="MATRLIN:func.4">*</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="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c8">i</font></span>)</span></span>)</span> <a href="matrlin.html#K9" title="MATRLIN:func.9">|--</a> <font color="Maroon" title="c7">b2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:60_3_1"><span class="lab"><font color="Green" title="E72">A27</font></span></a>, <a class="ref" href="matrlin.html#D7" target="_self" title="MATRLIN:def.7">Def9</a></span>;<br/>
<a NAME="E37:60_3_1"/><span class="kw">then </span>
<font color="Maroon" title="c12">p1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c9">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">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="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c9">j</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:60_3_1"><span class="lab"><font color="Green" title="E70">A25</font></span></a>, <a class="txt" href="#E34:60_3_1"><span class="lab"><font color="Green" title="E76">A31</font></span></a>, <a class="txt" href="#E14:60_3_1"><span class="lab"><font color="Green" title="E65">A20</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E38:60_3_1"/>
<span class="p1">(<span class="default"><a href="matrlin.html#K10" title="MATRLIN:func.10">AutMt</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</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="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrlin.html#K10" title="MATRLIN:func.10">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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:60_3_1"><span class="lab"><font color="Green" title="E63">A18</font></span></a>, <a class="txt" href="#E35:60_3_1"><span class="lab"><font color="Green" title="E77">A32</font></span></a>, <a class="txt" href="#E33:60_3_1"><span class="lab"><font color="Green" title="E75">A30</font></span></a>, <a class="txt" href="#E30:60_3_1"><span class="lab"><font color="Green" title="E73">A28</font></span></a>, <a class="txt" href="#E19:60_3_1"><span class="lab"><font color="Green" title="E68">A23</font></span></a>, <a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E5:60_3"/>
<span class="p1">(<span class="default"><a href="matrlin.html#K10" title="MATRLIN:func.10">AutMt</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</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#K10" title="MATRLIN:func.10">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>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:60_3"><span class="lab"><font color="Green" title="E56">A11</font></span></a>, <a class="ref" href="matrix_3.html#D5" title="MATRIX_3:def.5">MATRIX_3:def 5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E7:60"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrlin.html#K10" title="MATRLIN:func.10">AutMt</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</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#K10" title="MATRLIN:func.10">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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:60"><span class="lab"><font color="Green" title="E51">A6</font></span></a>, <a class="ref" href="matrix_3.html#D5" title="MATRIX_3:def.5">MATRIX_3:def 5</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:60"/>
 <a href="matrlin.html#K10" title="MATRLIN:func.10">AutMt</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="matrlin.html#K4" title="MATRLIN:func.4">*</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#K10" title="MATRLIN:func.10">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 class="kw">by</span> <span class="lab"><a class="txt" href="#E3:60"><span class="lab"><font color="Green" title="E50">A5</font></span></a>, <a class="txt" href="#E6:60"><span class="lab"><font color="Green" title="E53">A8</font></span></a>, <a class="ref" href="matrix_1.html#T21" title="MATRIX_1:th.21">MATRIX_1:21</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
