<?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">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="b4">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="b5">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="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="b3">b1</font>, <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b4">b2</font>,<font color="Maroon" title="c1">K</font>  st  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b3">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>  &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b4">b2</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>  holds <br/>( <font color="Olive" title="b5">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> <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="b3">b1</font>,<font color="Olive" title="b4">b2</font></span>)</span> iff <font color="Olive" title="b5">v1</font> <a href="matrlin.html#K11" title="MATRLIN:func.11">|--</a> <font color="Olive" title="b3">b1</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="matrix15.html#K7" title="MATRIX15:func.7">Space_of_Solutions_of</a> <span class="p1">(<span class="default"><font color="Olive" title="b6">A</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span> )</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">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">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="b2">b2</font>,<font color="Maroon" title="c1">K</font>  st  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">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>  &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">b2</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>  holds <br/>( <font color="Olive" title="b3">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> <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="Olive" title="b1">b1</font>,<font color="Olive" title="b2">b2</font></span>)</span> iff <font color="Olive" title="b3">v1</font> <a href="matrlin.html#K11" title="MATRLIN:func.11">|--</a> <font color="Olive" title="b1">b1</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="matrix15.html#K7" title="MATRIX15:func.7">Space_of_Solutions_of</a> <span class="p1">(<span class="default"><font color="Olive" title="b4">A</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span> )</span><br/><b>let </b><font color="Maroon" title="c4">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">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="c4">b1</font>, <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">b2</font>,<font color="Maroon" title="c1">K</font>  st  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">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>  &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">b2</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>  holds <br/>( <font color="Olive" title="b2">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> <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="c4">b1</font>,<font color="Olive" title="b1">b2</font></span>)</span> iff <font color="Olive" title="b2">v1</font> <a href="matrlin.html#K11" title="MATRLIN:func.11">|--</a> <font color="Maroon" title="c4">b1</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="matrix15.html#K7" title="MATRIX15:func.7">Space_of_Solutions_of</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">A</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span> )</span><br/><b>let </b><font color="Maroon" title="c5">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">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="c4">b1</font>, <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">b2</font>,<font color="Maroon" title="c1">K</font>  st  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">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>  &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">b2</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>  holds <br/>( <font color="Olive" title="b1">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> <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="c4">b1</font>,<font color="Maroon" title="c5">b2</font></span>)</span> iff <font color="Olive" title="b1">v1</font> <a href="matrlin.html#K11" title="MATRLIN:func.11">|--</a> <font color="Maroon" title="c4">b1</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="matrix15.html#K7" title="MATRIX15:func.7">Space_of_Solutions_of</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">A</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span> )</span><br/><b>let </b><font color="Maroon" title="c6">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">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="c4">b1</font>, <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">b2</font>,<font color="Maroon" title="c1">K</font>  st  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">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>  &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">b2</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>  holds <br/>( <font color="Maroon" title="c6">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> <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="c4">b1</font>,<font color="Maroon" title="c5">b2</font></span>)</span> iff <font color="Maroon" title="c6">v1</font> <a href="matrlin.html#K11" title="MATRLIN:func.11">|--</a> <font color="Maroon" title="c4">b1</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="matrix15.html#K7" title="MATRIX15:func.7">Space_of_Solutions_of</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">A</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span> )</span><br/><b>let </b><font color="Maroon" title="c7">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="c4">b1</font>, <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">b2</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"> (  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">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>  &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">b2</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>  implies ( <font color="Maroon" title="c6">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> <span class="p1">(<span class="default"><a href="matrlin2.html#K5" title="MATRLIN2:func.5">Mx2Tran</a> <font color="Maroon" title="c7">A</font>,<font color="Maroon" title="c4">b1</font>,<font color="Maroon" title="c5">b2</font></span>)</span> iff <font color="Maroon" title="c6">v1</font> <a href="matrlin.html#K11" title="MATRLIN:func.11">|--</a> <font color="Maroon" title="c4">b1</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="matrix15.html#K7" title="MATRIX15:func.7">Space_of_Solutions_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span> ) )</span><br/>













<b>assume </b><a NAME="E1:52"/><i><font color="Green" title="E30">A1</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">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>  &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">b2</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"> ( <font color="Maroon" title="c6">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> <span class="p1">(<span class="default"><a href="matrlin2.html#K5" title="MATRLIN2:func.5">Mx2Tran</a> <font color="Maroon" title="c7">A</font>,<font color="Maroon" title="c4">b1</font>,<font color="Maroon" title="c5">b2</font></span>)</span> iff <font color="Maroon" title="c6">v1</font> <a href="matrlin.html#K11" title="MATRLIN:func.11">|--</a> <font color="Maroon" title="c4">b1</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="matrix15.html#K7" title="MATRIX15:func.7">Space_of_Solutions_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span> )</span><br/>

<b>set </b><font color="Maroon" title="c8">AT</font> = <font color="Maroon" title="c7">A</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> ;<br/>
<b>set </b><font color="Maroon" title="c9">SA</font> =  <a href="matrix15.html#K7" title="MATRIX15:func.7">Space_of_Solutions_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c10">M</font> =  <a href="matrlin2.html#K5" title="MATRLIN2:func.5">Mx2Tran</a> <font color="Maroon" title="c7">A</font>,<font color="Maroon" title="c4">b1</font>,<font color="Maroon" title="c5">b2</font>;<br/>
<a NAME="E2:52"/><i><font color="Green" title="E31">A2</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">b1</font> &amp;  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c7">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">b2</font> )
 
<b>by </b><i><a class="txt" href="matrlin2.html#E1:52"><i><font color="Green" title="E30">A1</font></i></a>, <a class="ref" href="matrix_1.html#T24" title="MATRIX_1:th.24">MATRIX_1:24</a></i>;<br/>
<a NAME="E3:52"/><b>then </b><i><font color="Green" title="E32">A3</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">b2</font> &amp;  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">b1</font> &amp; (  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  implies  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span> <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="txt" href="matrlin2.html#E1:52"><i><font color="Green" title="E30">A1</font></i></a>, <a class="ref" href="matrix_2.html#T12" title="MATRIX_2:th.12">MATRIX_2:12</a></i>;<br/>
<b>set </b><font color="Maroon" title="c11">L</font> =  <a href="matrix15.html#K2" title="MATRIX15:func.2">LineVec2Mx</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">v1</font> <a href="matrlin.html#K11" title="MATRLIN:func.11">|--</a> <font color="Maroon" title="c4">b1</font></span>)</span>;<br/>
<a NAME="E4:52"/><i><font color="Green" title="E33">A4</font></i>: 
(  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <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="c6">v1</font> <a href="matrlin.html#K11" title="MATRLIN:func.11">|--</a> <font color="Maroon" title="c4">b1</font></span>)</span></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="c6">v1</font> <a href="matrlin.html#K11" title="MATRLIN:func.11">|--</a> <font color="Maroon" title="c4">b1</font></span>)</span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">v1</font> <a href="matrlin.html#K11" title="MATRLIN:func.11">|--</a> <font color="Maroon" title="c4">b1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">b1</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">b2</font></span>)</span> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <span class="p2">(<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></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">b2</font> )
 
<b>by </b><i><a class="ref" href="matrix_1.html#T24" title="MATRIX_1:th.24">MATRIX_1:24</a>, <a class="ref" href="matrlin.html#D9" title="MATRLIN:def.9">MATRLIN:def 9</a>, <a class="ref" href="finseq_2.html#T69" title="FINSEQ_2:th.69">FINSEQ_2:69</a></i>;<br/>
<a NAME="E5:52"/><b>then </b><i><font color="Green" title="E34">A5</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix15.html#K3" title="MATRIX15:func.3">ColVec2Mx</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">v1</font> <a href="matrlin.html#K11" title="MATRLIN:func.11">|--</a> <font color="Maroon" title="c4">b1</font></span>)</span></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="c6">v1</font> <a href="matrlin.html#K11" title="MATRLIN:func.11">|--</a> <font color="Maroon" title="c4">b1</font></span>)</span> &amp;  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="matrix15.html#K3" title="MATRIX15:func.3">ColVec2Mx</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">v1</font> <a href="matrlin.html#K11" title="MATRLIN:func.11">|--</a> <font color="Maroon" title="c4">b1</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp;  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="matrix15.html#K3" title="MATRIX15:func.3">ColVec2Mx</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">b2</font></span>)</span> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <span class="p3">(<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></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp;  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix15.html#K2" title="MATRIX15:func.2">LineVec2Mx</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">v1</font> <a href="matrlin.html#K11" title="MATRLIN:func.11">|--</a> <font color="Maroon" title="c4">b1</font></span>)</span></span>)</span> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <font color="Maroon" title="c7">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c7">A</font> &amp;  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</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"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">b2</font></span>)</span> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <span class="p3">(<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></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">b2</font> )
 
<b>by </b><i><a class="ref" href="matrix_1.html#T24" title="MATRIX_1:th.24">MATRIX_1:24</a>, <a class="ref" href="matrix_3.html#D4" title="MATRIX_3:def.4">MATRIX_3:def 4</a>, <a class="txt" href="matrlin2.html#E1:52"><i><font color="Green" title="E30">A1</font></i></a>, <a class="txt" href="matrlin2.html#E2:52"><i><font color="Green" title="E31">A2</font></i></a></i>;<br/>
<b>thus </b><a NAME="E6:52"/>
( <font color="Maroon" title="c6">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> <span class="p1">(<span class="default"><a href="matrlin2.html#K5" title="MATRLIN2:func.5">Mx2Tran</a> <font color="Maroon" title="c7">A</font>,<font color="Maroon" title="c4">b1</font>,<font color="Maroon" title="c5">b2</font></span>)</span> implies <font color="Maroon" title="c6">v1</font> <a href="matrlin.html#K11" title="MATRLIN:func.11">|--</a> <font color="Maroon" title="c4">b1</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="matrix15.html#K7" title="MATRIX15:func.7">Space_of_Solutions_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c6">v1</font> <a href="matrlin.html#K11" title="MATRLIN:func.11">|--</a> <font color="Maroon" title="c4">b1</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="matrix15.html#K7" title="MATRIX15:func.7">Space_of_Solutions_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span> implies <font color="Maroon" title="c6">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> <span class="p1">(<span class="default"><a href="matrlin2.html#K5" title="MATRLIN2:func.5">Mx2Tran</a> <font color="Maroon" title="c7">A</font>,<font color="Maroon" title="c4">b1</font>,<font color="Maroon" title="c5">b2</font></span>)</span> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:52_1"/><i><font color="Green" title="E35">A6</font></i>: 
<font color="Maroon" title="c6">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> <span class="p1">(<span class="default"><a href="matrlin2.html#K5" title="MATRLIN2:func.5">Mx2Tran</a> <font color="Maroon" title="c7">A</font>,<font color="Maroon" title="c4">b1</font>,<font color="Maroon" title="c5">b2</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c6">v1</font> <a href="matrlin.html#K11" title="MATRLIN:func.11">|--</a> <font color="Maroon" title="c4">b1</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="matrix15.html#K7" title="MATRIX15:func.7">Space_of_Solutions_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span></span><br/>

<a NAME="E2:52_1"/>
<span class="p1">(<span class="default"><a href="matrlin2.html#K5" title="MATRLIN2:func.5">Mx2Tran</a> <font color="Maroon" title="c7">A</font>,<font color="Maroon" title="c4">b1</font>,<font color="Maroon" title="c5">b2</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">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="c2">V2</font>
 
<b>by </b><i><a class="txt" href="matrlin2.html#E1:52_1"><i><font color="Green" title="E35">A6</font></i></a>, <a class="ref" href="ranknull.html#T10" title="RANKNULL:th.10">RANKNULL:10</a></i>;<br/>
<b>then </b><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="c6">v1</font> <a href="matrlin.html#K11" title="MATRLIN:func.11">|--</a> <font color="Maroon" title="c4">b1</font></span>)</span></span>)</span> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <font color="Maroon" title="c7">A</font> = 
 <a href="matrix15.html#K2" title="MATRIX15:func.2">LineVec2Mx</a> <span class="p1">(<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="c2">V2</font></span>)</span> <a href="matrlin.html#K11" title="MATRLIN:func.11">|--</a> <font color="Maroon" title="c5">b2</font></span>)</span>
<b>by </b><i><a class="ref" href="matrlin2.html#T32" target="_self" title="MATRLIN2:th.32">Th32</a>, <a class="txt" href="matrlin2.html#E1:52"><i><font color="Green" title="E30">A1</font></i></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"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">b2</font></span>)</span> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <span class="p2">(<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></span>)</span>
<b>by </b><i><a class="ref" href="matrlin2.html#T20" target="_self" title="MATRLIN2:th.20">Th20</a></i>
;<br/>
<a NAME="E4:52_1"/><b>then </b>
 <a href="matrix15.html#K3" title="MATRIX15:func.3">ColVec2Mx</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="c5">b2</font></span>)</span> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <span class="p2">(<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></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <span class="p1">(<span class="default"><a href="matrix15.html#K3" title="MATRIX15:func.3">ColVec2Mx</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">v1</font> <a href="matrlin.html#K11" title="MATRLIN:func.11">|--</a> <font color="Maroon" title="c4">b1</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="matrlin2.html#E4:52"><i><font color="Green" title="E33">A4</font></i></a>, <a class="txt" href="matrlin2.html#E1:52"><i><font color="Green" title="E30">A1</font></i></a>, <a class="txt" href="matrlin2.html#E2:52"><i><font color="Green" title="E31">A2</font></i></a>, <a class="ref" href="matrix_3.html#T24" title="MATRIX_3:th.24">MATRIX_3:24</a></i>;<br/>
<a NAME="E5:52_1"/><b>then </b>
 <a href="matrix15.html#K3" title="MATRIX15:func.3">ColVec2Mx</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">v1</font> <a href="matrlin.html#K11" title="MATRLIN:func.11">|--</a> <font color="Maroon" title="c4">b1</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix15.html#K4" title="MATRIX15:func.4">Solutions_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span>,<span class="p1">(<span class="default"><a href="matrix15.html#K3" title="MATRIX15:func.3">ColVec2Mx</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">b2</font></span>)</span> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <span class="p3">(<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></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="matrlin2.html#E3:52"><i><font color="Green" title="E32">A3</font></i></a>, <a class="txt" href="matrlin2.html#E4:52"><i><font color="Green" title="E33">A4</font></i></a>, <a class="txt" href="matrlin2.html#E5:52"><i><font color="Green" title="E34">A5</font></i></a></i>;<br/>
<a NAME="E6:52_1"/><b>then </b>
<font color="Maroon" title="c6">v1</font> <a href="matrlin.html#K11" title="MATRLIN:func.11">|--</a> <font color="Maroon" title="c4">b1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix15.html#K6" title="MATRIX15:func.6">Solutions_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span>,<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="c5">b2</font></span>)</span> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <span class="p2">(<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></span>)</span>
 
;<br/>
<a NAME="E7:52_1"/><b>then </b>
<font color="Maroon" title="c6">v1</font> <a href="matrlin.html#K11" title="MATRLIN:func.11">|--</a> <font color="Maroon" title="c4">b1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="matrix15.html#K7" title="MATRIX15:func.7">Space_of_Solutions_of</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="matrlin2.html#E3:52"><i><font color="Green" title="E32">A3</font></i></a>, <a class="ref" href="matrix15.html#D5" title="MATRIX15:def.5">MATRIX15:def 5</a></i>;<br/>
<b>hence </b><a NAME="E8:52_1"/>
<font color="Maroon" title="c6">v1</font> <a href="matrlin.html#K11" title="MATRLIN:func.11">|--</a> <font color="Maroon" title="c4">b1</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="matrix15.html#K7" title="MATRIX15:func.7">Space_of_Solutions_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span>
 <b>by </b><i><a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></i>; <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>

<b>assume </b><a NAME="E7:52"/>
<font color="Maroon" title="c6">v1</font> <a href="matrlin.html#K11" title="MATRLIN:func.11">|--</a> <font color="Maroon" title="c4">b1</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="matrix15.html#K7" title="MATRIX15:func.7">Space_of_Solutions_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c6">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> <span class="p1">(<span class="default"><a href="matrlin2.html#K5" title="MATRLIN2:func.5">Mx2Tran</a> <font color="Maroon" title="c7">A</font>,<font color="Maroon" title="c4">b1</font>,<font color="Maroon" title="c5">b2</font></span>)</span></span><br/>

<a NAME="E8:52"/><b>then </b>
<font color="Maroon" title="c6">v1</font> <a href="matrlin.html#K11" title="MATRLIN:func.11">|--</a> <font color="Maroon" title="c4">b1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="matrix15.html#K7" title="MATRIX15:func.7">Space_of_Solutions_of</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></i>;<br/>
<a NAME="E9:52"/><b>then </b>
<font color="Maroon" title="c6">v1</font> <a href="matrlin.html#K11" title="MATRLIN:func.11">|--</a> <font color="Maroon" title="c4">b1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix15.html#K6" title="MATRIX15:func.6">Solutions_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span>,<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="c5">b2</font></span>)</span> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <span class="p2">(<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></span>)</span>
 
<b>by </b><i><a class="txt" href="matrlin2.html#E3:52"><i><font color="Green" title="E32">A3</font></i></a>, <a class="ref" href="matrix15.html#D5" title="MATRIX15:def.5">MATRIX15:def 5</a></i>;<br/>
<a NAME="E10:52"/><b>then </b>
 ex <font color="Olive" title="b1">f</font> being  <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c1">K</font> st <br/>( <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">v1</font> <a href="matrlin.html#K11" title="MATRLIN:func.11">|--</a> <font color="Maroon" title="c4">b1</font> &amp;  <a href="matrix15.html#K3" title="MATRIX15:func.3">ColVec2Mx</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix15.html#K4" title="MATRIX15:func.4">Solutions_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span>,<span class="p1">(<span class="default"><a href="matrix15.html#K3" title="MATRIX15:func.3">ColVec2Mx</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">b2</font></span>)</span> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <span class="p3">(<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></span>)</span></span>)</span> )
 
;<br/>
<a NAME="E11:52"/><b>then </b>
 ex <font color="Olive" title="b1">X</font> being  <a href="matrix_1.html#NM4" title="MATRIX_1:NM.4">Matrix</a> of <font color="Maroon" title="c1">K</font> st <br/>( <font color="Olive" title="b1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix15.html#K3" title="MATRIX15:func.3">ColVec2Mx</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">v1</font> <a href="matrlin.html#K11" title="MATRLIN:func.11">|--</a> <font color="Maroon" title="c4">b1</font></span>)</span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">X</font> <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="c7">A</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span> &amp;  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Olive" title="b1">X</font> <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="matrix15.html#K3" title="MATRIX15:func.3">ColVec2Mx</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">b2</font></span>)</span> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <span class="p3">(<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></span>)</span></span>)</span> &amp;  <a href="matrix15.html#K3" title="MATRIX15:func.3">ColVec2Mx</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="c5">b2</font></span>)</span> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <span class="p2">(<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></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">A</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <font color="Olive" title="b1">X</font> )
 
;<br/>
<a NAME="E12:52"/><b>then </b>
 <a href="matrix15.html#K3" title="MATRIX15:func.3">ColVec2Mx</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="c5">b2</font></span>)</span> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <span class="p2">(<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></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix15.html#K2" title="MATRIX15:func.2">LineVec2Mx</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">v1</font> <a href="matrlin.html#K11" title="MATRLIN:func.11">|--</a> <font color="Maroon" title="c4">b1</font></span>)</span></span>)</span> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <font color="Maroon" title="c7">A</font></span>)</span> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> 
 
<b>by </b><i><a class="txt" href="matrlin2.html#E4:52"><i><font color="Green" title="E33">A4</font></i></a>, <a class="txt" href="matrlin2.html#E1:52"><i><font color="Green" title="E30">A1</font></i></a>, <a class="txt" href="matrlin2.html#E2:52"><i><font color="Green" title="E31">A2</font></i></a>, <a class="ref" href="matrix_3.html#T24" title="MATRIX_3:th.24">MATRIX_3:24</a></i>;<br/>
<b>then </b><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="c6">v1</font> <a href="matrlin.html#K11" title="MATRLIN:func.11">|--</a> <font color="Maroon" title="c4">b1</font></span>)</span></span>)</span> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <font color="Maroon" title="c7">A</font> = 
 <a href="matrix15.html#K2" title="MATRIX15:func.2">LineVec2Mx</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="c5">b2</font></span>)</span> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <span class="p2">(<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></span>)</span>
<b>by </b><i><a class="txt" href="matrlin2.html#E1:52"><i><font color="Green" title="E30">A1</font></i></a>, <a class="txt" href="matrlin2.html#E2:52"><i><font color="Green" title="E31">A2</font></i></a>, <a class="txt" href="matrlin2.html#E5:52"><i><font color="Green" title="E34">A5</font></i></a>, <a class="ref" href="matrix_2.html#T14" title="MATRIX_2:th.14">MATRIX_2:14</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"><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="c5">b2</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 NAME="E14:52"/><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"><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="c5">b2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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="c7">A</font>,<font color="Maroon" title="c4">b1</font>,<font color="Maroon" title="c5">b2</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">v1</font></span>)</span> <a href="matrlin.html#K11" title="MATRLIN:func.11">|--</a> <font color="Maroon" title="c5">b2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="matrlin2.html#T32" target="_self" title="MATRLIN2:th.32">Th32</a>, <a class="txt" href="matrlin2.html#E1:52"><i><font color="Green" title="E30">A1</font></i></a></i>;<br/>
<b>then </b><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="c5">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="c7">A</font>,<font color="Maroon" title="c4">b1</font>,<font color="Maroon" title="c5">b2</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">v1</font></span>)</span> <a href="matrlin.html#K11" title="MATRLIN:func.11">|--</a> <font color="Maroon" title="c5">b2</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="c7">A</font>,<font color="Maroon" title="c4">b1</font>,<font color="Maroon" title="c5">b2</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">v1</font></span>)</span> <a href="matrlin.html#K11" title="MATRLIN:func.11">|--</a> <font color="Maroon" title="c5">b2</font>
<b>by </b><i><a class="ref" href="matrix15.html#T25" title="MATRIX15:th.25">MATRIX15:25</a></i>
;<br/>
<a NAME="E16:52"/><b>then </b>
<span class="p1">(<span class="default"><a href="matrlin2.html#K5" title="MATRLIN2:func.5">Mx2Tran</a> <font color="Maroon" title="c7">A</font>,<font color="Maroon" title="c4">b1</font>,<font color="Maroon" title="c5">b2</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">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="c2">V2</font>
 
<b>by </b><i><a class="ref" href="matrlin.html#T39" title="MATRLIN:th.39">MATRLIN:39</a></i>;<br/>
<b>hence </b><a NAME="E17:52"/>
<font color="Maroon" title="c6">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> <span class="p1">(<span class="default"><a href="matrlin2.html#K5" title="MATRLIN2:func.5">Mx2Tran</a> <font color="Maroon" title="c7">A</font>,<font color="Maroon" title="c4">b1</font>,<font color="Maroon" title="c5">b2</font></span>)</span>
 <b>by </b><i><a class="ref" href="ranknull.html#T10" title="RANKNULL:th.10">RANKNULL:10</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
