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

<b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">V2</font>, <font color="Olive" title="b2">V1</font> being <a href="matrlin.html#V2" title="MATRLIN:attr.2">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="b2">V1</font>,<font color="Olive" title="b1">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="b2">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="b1">V2</font><br/> for <font color="Olive" title="b6">v1</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Olive" title="b2">V1</font><br/> for <font color="Olive" title="b7">M</font> being   <a href="matrlin.html#M1" title="MATRLIN:mode.1">OrdBasis</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b5">b2</font></span>)</span> <a href="matrix13.html#K13" title="MATRIX13:func.13">-VectSp_over</a> <font color="Maroon" title="c1">K</font>  st <font color="Olive" title="b7">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix13.html#K12" title="MATRIX13:func.12">MX2FinS</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> <font color="Maroon" title="c1">K</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b5">b2</font></span>)</span></span>)</span> holds <br/>for <font color="Olive" title="b8">A</font> being  <a href="matrix_1.html#NM6" title="MATRIX_1:NM.6">Matrix</a> of  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b4">b1</font>, <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b7">M</font>,<font color="Maroon" title="c1">K</font>  st <font color="Olive" title="b8">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrlin2.html#K1" title="MATRLIN2:func.1">AutMt</a> <font color="Olive" title="b3">f</font>,<font color="Olive" title="b4">b1</font>,<font color="Olive" title="b5">b2</font> &amp; <font color="Olive" title="b3">f</font> is <a href="mod_2.html#V1" title="MOD_2:attr.1">linear</a> holds <br/><span class="p1">(<span class="default"><a href="matrlin2.html#K5" title="MATRLIN2:func.5">Mx2Tran</a> <font color="Olive" title="b8">A</font>,<font color="Olive" title="b4">b1</font>,<font color="Olive" title="b7">M</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b6">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b6">v1</font></span>)</span> <a href="matrlin.html#K11" title="MATRLIN:func.11">|--</a> <font color="Olive" title="b5">b2</font></span><br/><b>let </b><font color="Maroon" title="c2">V2</font>, <font color="Maroon" title="c3">V1</font> be  <a href="matrlin.html#V2" title="MATRLIN:attr.2">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:()"><i><font color="Red">::  thesis: </font></i></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="c2">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="c2">V2</font><br/> for <font color="Olive" title="b4">v1</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">V1</font><br/> for <font color="Olive" title="b5">M</font> being   <a href="matrlin.html#M1" title="MATRLIN:mode.1">OrdBasis</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b3">b2</font></span>)</span> <a href="matrix13.html#K13" title="MATRIX13:func.13">-VectSp_over</a> <font color="Maroon" title="c1">K</font>  st <font color="Olive" title="b5">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix13.html#K12" title="MATRIX13:func.12">MX2FinS</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> <font color="Maroon" title="c1">K</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b3">b2</font></span>)</span></span>)</span> holds <br/>for <font color="Olive" title="b6">A</font> being  <a href="matrix_1.html#NM6" title="MATRIX_1:NM.6">Matrix</a> of  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">b1</font>, <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b5">M</font>,<font color="Maroon" title="c1">K</font>  st <font color="Olive" title="b6">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrlin2.html#K1" title="MATRLIN2:func.1">AutMt</a> <font color="Olive" title="b1">f</font>,<font color="Olive" title="b2">b1</font>,<font color="Olive" title="b3">b2</font> &amp; <font color="Olive" title="b1">f</font> is <a href="mod_2.html#V1" title="MOD_2:attr.1">linear</a> holds <br/><span class="p1">(<span class="default"><a href="matrlin2.html#K5" title="MATRLIN2:func.5">Mx2Tran</a> <font color="Olive" title="b6">A</font>,<font color="Olive" title="b2">b1</font>,<font color="Olive" title="b5">M</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b4">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b4">v1</font></span>)</span> <a href="matrlin.html#K11" title="MATRLIN:func.11">|--</a> <font color="Olive" title="b3">b2</font></span><br/><b>let </b><font color="Maroon" title="c4">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="c2">V2</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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="c2">V2</font><br/> for <font color="Olive" title="b3">v1</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">V1</font><br/> for <font color="Olive" title="b4">M</font> being   <a href="matrlin.html#M1" title="MATRLIN:mode.1">OrdBasis</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">b2</font></span>)</span> <a href="matrix13.html#K13" title="MATRIX13:func.13">-VectSp_over</a> <font color="Maroon" title="c1">K</font>  st <font color="Olive" title="b4">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix13.html#K12" title="MATRIX13:func.12">MX2FinS</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> <font color="Maroon" title="c1">K</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">b2</font></span>)</span></span>)</span> holds <br/>for <font color="Olive" title="b5">A</font> being  <a href="matrix_1.html#NM6" title="MATRIX_1:NM.6">Matrix</a> of  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">b1</font>, <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b4">M</font>,<font color="Maroon" title="c1">K</font>  st <font color="Olive" title="b5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrlin2.html#K1" title="MATRLIN2:func.1">AutMt</a> <font color="Maroon" title="c4">f</font>,<font color="Olive" title="b1">b1</font>,<font color="Olive" title="b2">b2</font> &amp; <font color="Maroon" title="c4">f</font> is <a href="mod_2.html#V1" title="MOD_2:attr.1">linear</a> holds <br/><span class="p1">(<span class="default"><a href="matrlin2.html#K5" title="MATRLIN2:func.5">Mx2Tran</a> <font color="Olive" title="b5">A</font>,<font color="Olive" title="b1">b1</font>,<font color="Olive" title="b4">M</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">v1</font></span>)</span> <a href="matrlin.html#K11" title="MATRLIN:func.11">|--</a> <font color="Olive" title="b2">b2</font></span><br/><b>let </b><font color="Maroon" title="c5">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:()"><i><font color="Red">::  thesis: </font></i></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="c2">V2</font><br/> for <font color="Olive" title="b2">v1</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">V1</font><br/> for <font color="Olive" title="b3">M</font> being   <a href="matrlin.html#M1" title="MATRLIN:mode.1">OrdBasis</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">b2</font></span>)</span> <a href="matrix13.html#K13" title="MATRIX13:func.13">-VectSp_over</a> <font color="Maroon" title="c1">K</font>  st <font color="Olive" title="b3">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix13.html#K12" title="MATRIX13:func.12">MX2FinS</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> <font color="Maroon" title="c1">K</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">b2</font></span>)</span></span>)</span> holds <br/>for <font color="Olive" title="b4">A</font> being  <a href="matrix_1.html#NM6" title="MATRIX_1:NM.6">Matrix</a> of  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">b1</font>, <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b3">M</font>,<font color="Maroon" title="c1">K</font>  st <font color="Olive" title="b4">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrlin2.html#K1" title="MATRLIN2:func.1">AutMt</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">b1</font>,<font color="Olive" title="b1">b2</font> &amp; <font color="Maroon" title="c4">f</font> is <a href="mod_2.html#V1" title="MOD_2:attr.1">linear</a> holds <br/><span class="p1">(<span class="default"><a href="matrlin2.html#K5" title="MATRLIN2:func.5">Mx2Tran</a> <font color="Olive" title="b4">A</font>,<font color="Maroon" title="c5">b1</font>,<font color="Olive" title="b3">M</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">v1</font></span>)</span> <a href="matrlin.html#K11" title="MATRLIN:func.11">|--</a> <font color="Olive" title="b1">b2</font></span><br/><b>let </b><font color="Maroon" title="c6">b2</font> be    <a href="matrlin.html#M1" title="MATRLIN:mode.1">OrdBasis</a> of <font color="Maroon" title="c2">V2</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">v1</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">V1</font><br/> for <font color="Olive" title="b2">M</font> being   <a href="matrlin.html#M1" title="MATRLIN:mode.1">OrdBasis</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">b2</font></span>)</span> <a href="matrix13.html#K13" title="MATRIX13:func.13">-VectSp_over</a> <font color="Maroon" title="c1">K</font>  st <font color="Olive" title="b2">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix13.html#K12" title="MATRIX13:func.12">MX2FinS</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> <font color="Maroon" title="c1">K</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">b2</font></span>)</span></span>)</span> holds <br/>for <font color="Olive" title="b3">A</font> being  <a href="matrix_1.html#NM6" title="MATRIX_1:NM.6">Matrix</a> of  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">b1</font>, <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">M</font>,<font color="Maroon" title="c1">K</font>  st <font color="Olive" title="b3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrlin2.html#K1" title="MATRLIN2:func.1">AutMt</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">b1</font>,<font color="Maroon" title="c6">b2</font> &amp; <font color="Maroon" title="c4">f</font> is <a href="mod_2.html#V1" title="MOD_2:attr.1">linear</a> holds <br/><span class="p1">(<span class="default"><a href="matrlin2.html#K5" title="MATRLIN2:func.5">Mx2Tran</a> <font color="Olive" title="b3">A</font>,<font color="Maroon" title="c5">b1</font>,<font color="Olive" title="b2">M</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">v1</font></span>)</span> <a href="matrlin.html#K11" title="MATRLIN:func.11">|--</a> <font color="Maroon" title="c6">b2</font></span><br/><b>let </b><font color="Maroon" title="c7">v1</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">V1</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">M</font> being   <a href="matrlin.html#M1" title="MATRLIN:mode.1">OrdBasis</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">b2</font></span>)</span> <a href="matrix13.html#K13" title="MATRIX13:func.13">-VectSp_over</a> <font color="Maroon" title="c1">K</font>  st <font color="Olive" title="b1">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix13.html#K12" title="MATRIX13:func.12">MX2FinS</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> <font color="Maroon" title="c1">K</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">b2</font></span>)</span></span>)</span> holds <br/>for <font color="Olive" title="b2">A</font> being  <a href="matrix_1.html#NM6" title="MATRIX_1:NM.6">Matrix</a> of  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">b1</font>, <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">M</font>,<font color="Maroon" title="c1">K</font>  st <font color="Olive" title="b2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrlin2.html#K1" title="MATRLIN2:func.1">AutMt</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">b1</font>,<font color="Maroon" title="c6">b2</font> &amp; <font color="Maroon" title="c4">f</font> is <a href="mod_2.html#V1" title="MOD_2:attr.1">linear</a> holds <br/><span class="p1">(<span class="default"><a href="matrlin2.html#K5" title="MATRLIN2:func.5">Mx2Tran</a> <font color="Olive" title="b2">A</font>,<font color="Maroon" title="c5">b1</font>,<font color="Olive" title="b1">M</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">v1</font></span>)</span> <a href="matrlin.html#K11" title="MATRLIN:func.11">|--</a> <font color="Maroon" title="c6">b2</font></span><br/><b>let </b><font color="Maroon" title="c8">M</font> be    <a href="matrlin.html#M1" title="MATRLIN:mode.1">OrdBasis</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">b2</font></span>)</span> <a href="matrix13.html#K13" title="MATRIX13:func.13">-VectSp_over</a> <font color="Maroon" title="c1">K</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c8">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix13.html#K12" title="MATRIX13:func.12">MX2FinS</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> <font color="Maroon" title="c1">K</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">b2</font></span>)</span></span>)</span> implies for <font color="Olive" title="b1">A</font> being  <a href="matrix_1.html#NM6" title="MATRIX_1:NM.6">Matrix</a> of  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">b1</font>, <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">M</font>,<font color="Maroon" title="c1">K</font>  st <font color="Olive" title="b1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrlin2.html#K1" title="MATRLIN2:func.1">AutMt</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">b1</font>,<font color="Maroon" title="c6">b2</font> &amp; <font color="Maroon" title="c4">f</font> is <a href="mod_2.html#V1" title="MOD_2:attr.1">linear</a> holds <br/><span class="p1">(<span class="default"><a href="matrlin2.html#K5" title="MATRLIN2:func.5">Mx2Tran</a> <font color="Olive" title="b1">A</font>,<font color="Maroon" title="c5">b1</font>,<font color="Maroon" title="c8">M</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">v1</font></span>)</span> <a href="matrlin.html#K11" title="MATRLIN:func.11">|--</a> <font color="Maroon" title="c6">b2</font> )</span><br/>















<b>assume </b><a NAME="E1:63"/><i><font color="Green" title="E37">A1</font></i>: 
<font color="Maroon" title="c8">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix13.html#K12" title="MATRIX13:func.12">MX2FinS</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> <font color="Maroon" title="c1">K</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">b2</font></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">A</font> being  <a href="matrix_1.html#NM6" title="MATRIX_1:NM.6">Matrix</a> of  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">b1</font>, <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">M</font>,<font color="Maroon" title="c1">K</font>  st <font color="Olive" title="b1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrlin2.html#K1" title="MATRLIN2:func.1">AutMt</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">b1</font>,<font color="Maroon" title="c6">b2</font> &amp; <font color="Maroon" title="c4">f</font> is <a href="mod_2.html#V1" title="MOD_2:attr.1">linear</a> holds <br/><span class="p1">(<span class="default"><a href="matrlin2.html#K5" title="MATRLIN2:func.5">Mx2Tran</a> <font color="Olive" title="b1">A</font>,<font color="Maroon" title="c5">b1</font>,<font color="Maroon" title="c8">M</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">v1</font></span>)</span> <a href="matrlin.html#K11" title="MATRLIN:func.11">|--</a> <font color="Maroon" title="c6">b2</font></span><br/>

<b>let </b><font color="Maroon" title="c9">A</font> be   <a href="matrix_1.html#NM6" title="MATRIX_1:NM.6">Matrix</a> of  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">b1</font>, <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">M</font>,<font color="Maroon" title="c1">K</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c9">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrlin2.html#K1" title="MATRLIN2:func.1">AutMt</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">b1</font>,<font color="Maroon" title="c6">b2</font> &amp; <font color="Maroon" title="c4">f</font> is <a href="mod_2.html#V1" title="MOD_2:attr.1">linear</a> implies <span class="p1">(<span class="default"><a href="matrlin2.html#K5" title="MATRLIN2:func.5">Mx2Tran</a> <font color="Maroon" title="c9">A</font>,<font color="Maroon" title="c5">b1</font>,<font color="Maroon" title="c8">M</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">v1</font></span>)</span> <a href="matrlin.html#K11" title="MATRLIN:func.11">|--</a> <font color="Maroon" title="c6">b2</font> )</span><br/>

<b>assume </b><a NAME="E2:63"/><i><font color="Green" title="E38">A2</font></i>: 
( <font color="Maroon" title="c9">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrlin2.html#K1" title="MATRLIN2:func.1">AutMt</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">b1</font>,<font color="Maroon" title="c6">b2</font> &amp; <font color="Maroon" title="c4">f</font> is <a href="mod_2.html#V1" title="MOD_2:attr.1">linear</a> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="matrlin2.html#K5" title="MATRLIN2:func.5">Mx2Tran</a> <font color="Maroon" title="c9">A</font>,<font color="Maroon" title="c5">b1</font>,<font color="Maroon" title="c8">M</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">v1</font></span>)</span> <a href="matrlin.html#K11" title="MATRLIN:func.11">|--</a> <font color="Maroon" title="c6">b2</font></span><br/>

<b>reconsider </b><font color="Maroon" title="c10">f'</font> = <font color="Maroon" title="c4">f</font> as   <a href="ranknull.html#NM1" title="RANKNULL:NM.1">linear-transformation</a> of <font color="Maroon" title="c3">V1</font>,<font color="Maroon" title="c2">V2</font> <b>by </b><i><a class="txt" href="#E2:63"><i><font color="Green" title="E38">A2</font></i></a></i>;<br/>
<b>set </b><font color="Maroon" title="c11">MM</font> =  <a href="matrlin2.html#K5" title="MATRLIN2:func.5">Mx2Tran</a> <font color="Maroon" title="c9">A</font>,<font color="Maroon" title="c5">b1</font>,<font color="Maroon" title="c8">M</font>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="63_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:63_1"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  or  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">b1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="63_1_1"><b>suppose </b></a><a NAME="E1:63_1_1"/><i><font color="Green" title="E39">A3</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="matrlin2.html#K5" title="MATRLIN2:func.5">Mx2Tran</a> <font color="Maroon" title="c9">A</font>,<font color="Maroon" title="c5">b1</font>,<font color="Maroon" title="c8">M</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">v1</font></span>)</span> <a href="matrlin.html#K11" title="MATRLIN:func.11">|--</a> <font color="Maroon" title="c6">b2</font></span><br/><div class="add"><a NAME="E2:63_1_1"/><b>then </b>
 <a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <font color="Maroon" title="c3">V1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> 
 <b>by </b><i><a class="ref" href="matrlin2.html#T21" target="_self" title="MATRLIN2:th.21">Th21</a></i>;<br/><a NAME="E3:63_1_1"/><b>then </b>
 <a href="vectsp_4.html#K2" title="VECTSP_4:func.2">(Omega).</a> <font color="Maroon" title="c3">V1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_4.html#K1" title="VECTSP_4:func.1">(0).</a> <font color="Maroon" title="c3">V1</font>
 <b>by </b><i><a class="ref" href="vectsp_9.html#T33" title="VECTSP_9:th.33">VECTSP_9:33</a></i>;<br/><a NAME="E4:63_1_1"/><b>then </b><i><font color="Green" title="E40">A4</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">V1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c3">V1</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="ref" href="vectsp_4.html#D3" title="VECTSP_4:def.3">VECTSP_4:def 3</a></i>;<br/><a NAME="E5:63_1_1"/>
<font color="Maroon" title="c7">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c3">V1</font>
 <b>by </b><i><a class="txt" href="#E4:63_1_1"><i><font color="Green" title="E40">A4</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E6:63_1_1"/><b>then </b>
<font color="Maroon" title="c7">v1</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="ranknull.html#K1" title="RANKNULL:func.1">ker</a> <font color="Maroon" title="c10">f'</font>
 <b>by </b><i><a class="ref" href="ranknull.html#T11" title="RANKNULL:th.11">RANKNULL:11</a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">v1</font></span>)</span> <a href="matrlin.html#K11" title="MATRLIN:func.11">|--</a> <font color="Maroon" title="c6">b2</font> = 
<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">V2</font></span>)</span> <a href="matrlin.html#K11" title="MATRLIN:func.11">|--</a> <font color="Maroon" title="c6">b2</font>
<b>by </b><i><a class="ref" href="ranknull.html#T10" title="RANKNULL:th.10">RANKNULL:10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">b2</font></span>)</span> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font></span>)</span>
<b>by </b><i><a class="ref" href="matrlin2.html#T20" target="_self" title="MATRLIN2:th.20">Th20</a></i>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">b2</font></span>)</span> <a href="matrix13.html#K13" title="MATRIX13:func.13">-VectSp_over</a> <font color="Maroon" title="c1">K</font></span>)</span>
<b>by </b><i><a class="ref" href="matrix13.html#T102" title="MATRIX13:th.102">MATRIX13:102</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="matrlin2.html#K5" title="MATRLIN2:func.5">Mx2Tran</a> <font color="Maroon" title="c9">A</font>,<font color="Maroon" title="c5">b1</font>,<font color="Maroon" title="c8">M</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">v1</font>
<b>by </b><i><a class="ref" href="matrlin2.html#T33" target="_self" title="MATRLIN2:th.33">Th33</a>, <a class="txt" href="#E1:63_1_1"><i><font color="Green" title="E39">A3</font></i></a></i>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="63_1_2"><b>suppose </b></a><a NAME="E1:63_1_2"/><i><font color="Green" title="E39">A5</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">b1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="matrlin2.html#K5" title="MATRLIN2:func.5">Mx2Tran</a> <font color="Maroon" title="c9">A</font>,<font color="Maroon" title="c5">b1</font>,<font color="Maroon" title="c8">M</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">v1</font></span>)</span> <a href="matrlin.html#K11" title="MATRLIN:func.11">|--</a> <font color="Maroon" title="c6">b2</font></span><br/><div class="add"><b>then </b> <a href="matrix15.html#K2" title="MATRIX15:func.2">LineVec2Mx</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrlin2.html#K5" title="MATRLIN2:func.5">Mx2Tran</a> <font color="Maroon" title="c9">A</font>,<font color="Maroon" title="c5">b1</font>,<font color="Maroon" title="c8">M</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">v1</font></span>)</span> <a href="matrlin.html#K11" title="MATRLIN:func.11">|--</a> <font color="Maroon" title="c8">M</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="matrix15.html#K2" title="MATRIX15:func.2">LineVec2Mx</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">v1</font> <a href="matrlin.html#K11" title="MATRLIN:func.11">|--</a> <font color="Maroon" title="c5">b1</font></span>)</span></span>)</span> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <font color="Maroon" title="c9">A</font>
<b>by </b><i><a class="ref" href="matrlin2.html#T32" target="_self" title="MATRLIN2:th.32">Th32</a></i>
<br/>.= 
 <a href="matrix15.html#K2" title="MATRIX15:func.2">LineVec2Mx</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">v1</font></span>)</span> <a href="matrlin.html#K11" title="MATRLIN:func.11">|--</a> <font color="Maroon" title="c6">b2</font></span>)</span>
<b>by </b><i><a class="txt" href="#E2:63"><i><font color="Green" title="E38">A2</font></i></a>, <a class="txt" href="#E1:63_1_2"><i><font color="Green" title="E39">A5</font></i></a>, <a class="ref" href="matrlin2.html#T31" target="_self" title="MATRLIN2:th.31">Th31</a></i>
;<br/><b>hence </b><span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">v1</font></span>)</span> <a href="matrlin.html#K11" title="MATRLIN:func.11">|--</a> <font color="Maroon" title="c6">b2</font> = 
 <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p1">(<span class="default"><a href="matrix15.html#K2" title="MATRIX15:func.2">LineVec2Mx</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="matrlin2.html#K5" title="MATRLIN2:func.5">Mx2Tran</a> <font color="Maroon" title="c9">A</font>,<font color="Maroon" title="c5">b1</font>,<font color="Maroon" title="c8">M</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">v1</font></span>)</span> <a href="matrlin.html#K11" title="MATRLIN:func.11">|--</a> <font color="Maroon" title="c8">M</font></span>)</span></span>)</span>,1
<b>by </b><i><a class="ref" href="matrix15.html#T25" title="MATRIX15:th.25">MATRIX15:25</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrlin2.html#K5" title="MATRLIN2:func.5">Mx2Tran</a> <font color="Maroon" title="c9">A</font>,<font color="Maroon" title="c5">b1</font>,<font color="Maroon" title="c8">M</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">v1</font></span>)</span> <a href="matrlin.html#K11" title="MATRLIN:func.11">|--</a> <font color="Maroon" title="c8">M</font>
<b>by </b><i><a class="ref" href="matrix15.html#T25" title="MATRIX15:th.25">MATRIX15:25</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="matrlin2.html#K5" title="MATRLIN2:func.5">Mx2Tran</a> <font color="Maroon" title="c9">A</font>,<font color="Maroon" title="c5">b1</font>,<font color="Maroon" title="c8">M</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">v1</font>
<b>by </b><i><a class="txt" href="#E1:63"><i><font color="Green" title="E37">A1</font></i></a>, <a class="ref" href="matrlin2.html#T46" target="_self" title="MATRLIN2:th.46">Th46</a></i>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
