<?xml version="1.0"?>
<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="Red">::  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">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="b4">b2</font> being   <a href="matrlin.html#M1" title="MATRLIN:mode.1">OrdBasis</a> of <font color="Olive" title="b2">V2</font><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="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/> <a href="ranknull.html#K11" title="RANKNULL:func.11">nullity</a> <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="b3">b1</font>,<font color="Olive" title="b4">b2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b3">b1</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <font color="Olive" title="b5">A</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c2">V1</font>, <font color="Maroon" title="c3">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="Red">::  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="c2">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="c3">V2</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="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/> <a href="ranknull.html#K11" title="RANKNULL:func.11">nullity</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="Olive" title="b1">b1</font>,<font color="Olive" title="b2">b2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">b1</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <font color="Olive" title="b3">A</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c4">b1</font> be    <a href="matrlin.html#M1" title="MATRLIN:mode.1">OrdBasis</a> of <font color="Maroon" title="c2">V1</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="c3">V2</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="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/> <a href="ranknull.html#K11" title="RANKNULL:func.11">nullity</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="Olive" title="b1">b2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">b1</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <font color="Olive" title="b2">A</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c5">b2</font> be    <a href="matrlin.html#M1" title="MATRLIN:mode.1">OrdBasis</a> of <font color="Maroon" title="c3">V2</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/> <a href="ranknull.html#K11" title="RANKNULL:func.11">nullity</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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">b1</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <font color="Olive" title="b1">A</font></span>)</span></span><br/>









<span class="kw">set </span><font color="Maroon" title="c6">I</font> =  <a href="matrlin2.html#K7" title="MATRLIN2:func.7">id</a> <font color="Maroon" title="c2">V1</font>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c7">BB</font> =  <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="c4">b1</font></span>)</span></span>)</span> as    <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="c4">b1</font></span>)</span> <a href="matrix13.html#K13" title="MATRIX13:func.13">-VectSp_over</a> <font color="Maroon" title="c1">K</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="matrlin2.html#T45" target="_self" title="MATRLIN2:th.45">Th45</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c8">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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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  <a href="ranknull.html#K11" title="RANKNULL:func.11">nullity</a> <span class="p1">(<span class="default"><a href="matrlin2.html#K5" title="MATRLIN2:func.5">Mx2Tran</a> <font color="Maroon" title="c8">A</font>,<font color="Maroon" title="c4">b1</font>,<font color="Maroon" title="c5">b2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">b1</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <font color="Maroon" title="c8">A</font></span>)</span> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E2:66"/><span class="lab"><font color="Green" title="E38">A1</font></span>: 
 <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> 
 <span class="kw">and </span><br/><a NAME="E3:66"/><span class="lab"><font color="Green" title="E39">A2</font></span>: 
 <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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="ranknull.html#K11" title="RANKNULL:func.11">nullity</a> <span class="p1">(<span class="default"><a href="matrlin2.html#K5" title="MATRLIN2:func.5">Mx2Tran</a> <font color="Maroon" title="c8">A</font>,<font color="Maroon" title="c4">b1</font>,<font color="Maroon" title="c5">b2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">b1</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <font color="Maroon" title="c8">A</font></span>)</span></span><br/>

 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">BB</font> = 
 <a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</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="c4">b1</font></span>)</span> <a href="matrix13.html#K13" title="MATRIX13:func.13">-VectSp_over</a> <font color="Maroon" title="c1">K</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrlin2.html#T21" target="_self" title="MATRLIN2:th.21">Th21</a></span>
<br/>.= 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">b1</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix13.html#T112" title="MATRIX13:th.112">MATRIX13:112</a></span>
;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c9">AI</font> =  <a href="matrlin2.html#K1" title="MATRLIN2:func.1">AutMt</a> <span class="p1">(<span class="default"><a href="matrlin2.html#K7" title="MATRLIN2:func.7">id</a> <font color="Maroon" title="c2">V1</font></span>)</span>,<font color="Maroon" title="c4">b1</font>,<font color="Maroon" title="c4">b1</font> as   <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="c7">BB</font>,<font color="Maroon" title="c1">K</font> ;<br/>
<a NAME="E6:66"/><span class="lab"><font color="Green" title="E40">A3</font></span>: 
(  <a href="matrlin2.html#K1" title="MATRLIN2:func.1">AutMt</a> <span class="p1">(<span class="default"><a href="matrlin2.html#K7" title="MATRLIN2:func.7">id</a> <font color="Maroon" title="c2">V1</font></span>)</span>,<font color="Maroon" title="c4">b1</font>,<font color="Maroon" title="c4">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> <font color="Maroon" title="c1">K</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">b1</font></span>)</span> &amp;  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">K</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrlin2.html#T28" target="_self" title="MATRLIN2:th.28">Th28</a></span>;<br/>
<a NAME="E7:66"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">b1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  <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 class="kw">by</span> <span class="lab"><a class="txt" href="#E2:66"><span class="lab"><font color="Green" title="E38">A1</font></span></a></span>;<br/>
<a NAME="E8:66"/><span class="kw">then </span>
 <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#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T19" title="NAT_1:th.19">NAT_1:19</a></span>;<br/>
<a NAME="E9:66"/><span class="kw">then </span>
 <a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</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="c4">b1</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">K</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_7.html#T16" title="MATRIX_7:th.16">MATRIX_7:16</a></span>;<br/>
<a NAME="E10:66"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A4</font></span>: 
 <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <font color="Maroon" title="c9">AI</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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:66"><span class="lab"><font color="Green" title="E40">A3</font></span></a>, <a class="ref" href="matrix13.html#T83" title="MATRIX13:th.83">MATRIX13:83</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c10">S</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="c8">A</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c11">KER</font> =  <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="c8">A</font>,<font color="Maroon" title="c4">b1</font>,<font color="Maroon" title="c5">b2</font></span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c12">MAI</font> =  <a href="matrlin2.html#K5" title="MATRLIN2:func.5">Mx2Tran</a> <font color="Maroon" title="c9">AI</font>,<font color="Maroon" title="c4">b1</font>,<font color="Maroon" title="c7">BB</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c13">MK</font> = <span class="p1">(<span class="default"><a href="matrlin2.html#K5" title="MATRLIN2:func.5">Mx2Tran</a> <font color="Maroon" title="c9">AI</font>,<font color="Maroon" title="c4">b1</font>,<font color="Maroon" title="c7">BB</font></span>)</span> <a href="matrlin2.html#K12" title="MATRLIN2:func.12">|</a> <span class="p1">(<span class="default"><a href="ranknull.html#K1" title="RANKNULL:func.1">ker</a> <span class="p2">(<span class="default"><a href="matrlin2.html#K5" title="MATRLIN2:func.5">Mx2Tran</a> <font color="Maroon" title="c8">A</font>,<font color="Maroon" title="c4">b1</font>,<font color="Maroon" title="c5">b2</font></span>)</span></span>)</span>;<br/>
<a NAME="E11:66"/><span class="lab"><font color="Green" title="E42">A5</font></span>: 
 <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"><a href="matrlin2.html#K5" title="MATRLIN2:func.5">Mx2Tran</a> <font color="Maroon" title="c9">AI</font>,<font color="Maroon" title="c4">b1</font>,<font color="Maroon" title="c7">BB</font></span>)</span> <a href="matrlin2.html#K12" title="MATRLIN2:func.12">|</a> <span class="p2">(<span class="default"><a href="ranknull.html#K1" title="RANKNULL:func.1">ker</a> <span class="p3">(<span class="default"><a href="matrlin2.html#K5" title="MATRLIN2:func.5">Mx2Tran</a> <font color="Maroon" title="c8">A</font>,<font color="Maroon" title="c4">b1</font>,<font color="Maroon" title="c5">b2</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="ranknull.html#K1" title="RANKNULL:func.1">ker</a> <span class="p2">(<span class="default"><a href="matrlin2.html#K5" title="MATRLIN2:func.5">Mx2Tran</a> <font color="Maroon" title="c8">A</font>,<font color="Maroon" title="c4">b1</font>,<font color="Maroon" title="c5">b2</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E12:66"/><span class="lab"><font color="Green" title="E43">A6</font></span>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="ranknull.html#K3" title="RANKNULL:func.3">im</a> <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">AI</font>,<font color="Maroon" title="c4">b1</font>,<font color="Maroon" title="c7">BB</font></span>)</span> <a href="matrlin2.html#K12" title="MATRLIN2:func.12">|</a> <span class="p3">(<span class="default"><a href="ranknull.html#K1" title="RANKNULL:func.1">ker</a> <span class="p4">(<span class="default"><a href="matrlin2.html#K5" title="MATRLIN2:func.5">Mx2Tran</a> <font color="Maroon" title="c8">A</font>,<font color="Maroon" title="c4">b1</font>,<font color="Maroon" title="c5">b2</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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="c8">A</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c14">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c14">x</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="ranknull.html#K3" title="RANKNULL:func.3">im</a> <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">AI</font>,<font color="Maroon" title="c4">b1</font>,<font color="Maroon" title="c7">BB</font></span>)</span> <a href="matrlin2.html#K12" title="MATRLIN2:func.12">|</a> <span class="p3">(<span class="default"><a href="ranknull.html#K1" title="RANKNULL:func.1">ker</a> <span class="p4">(<span class="default"><a href="matrlin2.html#K5" title="MATRLIN2:func.5">Mx2Tran</a> <font color="Maroon" title="c8">A</font>,<font color="Maroon" title="c4">b1</font>,<font color="Maroon" title="c5">b2</font></span>)</span></span>)</span></span>)</span></span>)</span> or <font color="Maroon" title="c14">x</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="c8">A</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:66_2"/><span class="lab"><font color="Green" title="E44">A7</font></span>: 
<font color="Maroon" title="c14">x</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="ranknull.html#K3" title="RANKNULL:func.3">im</a> <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">AI</font>,<font color="Maroon" title="c4">b1</font>,<font color="Maroon" title="c7">BB</font></span>)</span> <a href="matrlin2.html#K12" title="MATRLIN2:func.12">|</a> <span class="p3">(<span class="default"><a href="ranknull.html#K1" title="RANKNULL:func.1">ker</a> <span class="p4">(<span class="default"><a href="matrlin2.html#K5" title="MATRLIN2:func.5">Mx2Tran</a> <font color="Maroon" title="c8">A</font>,<font color="Maroon" title="c4">b1</font>,<font color="Maroon" title="c5">b2</font></span>)</span></span>)</span></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c14">x</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="c8">A</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span></span>)</span></span><br/>

<a NAME="E2:66_2"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="ranknull.html#K3" title="RANKNULL:func.3">im</a> <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">AI</font>,<font color="Maroon" title="c4">b1</font>,<font color="Maroon" title="c7">BB</font></span>)</span> <a href="matrlin2.html#K12" title="MATRLIN2:func.12">|</a> <span class="p3">(<span class="default"><a href="ranknull.html#K1" title="RANKNULL:func.1">ker</a> <span class="p4">(<span class="default"><a href="matrlin2.html#K5" title="MATRLIN2:func.5">Mx2Tran</a> <font color="Maroon" title="c8">A</font>,<font color="Maroon" title="c4">b1</font>,<font color="Maroon" title="c5">b2</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <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="c4">b1</font></span>)</span> <a href="matrix13.html#K13" title="MATRIX13:func.13">-VectSp_over</a> <font color="Maroon" title="c1">K</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_4.html#D2" title="VECTSP_4:def.2">VECTSP_4:def 2</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c15">v</font> = <font color="Maroon" title="c14">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <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="c4">b1</font></span>)</span> <a href="matrix13.html#K13" title="MATRIX13:func.13">-VectSp_over</a> <font color="Maroon" title="c1">K</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:66_2"><span class="lab"><font color="Green" title="E44">A7</font></span></a></span>;<br/>
<a NAME="E4:66_2"/>
<font color="Maroon" title="c15">v</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="ranknull.html#K3" title="RANKNULL:func.3">im</a> <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">AI</font>,<font color="Maroon" title="c4">b1</font>,<font color="Maroon" title="c7">BB</font></span>)</span> <a href="matrlin2.html#K12" title="MATRLIN2:func.12">|</a> <span class="p2">(<span class="default"><a href="ranknull.html#K1" title="RANKNULL:func.1">ker</a> <span class="p3">(<span class="default"><a href="matrlin2.html#K5" title="MATRLIN2:func.5">Mx2Tran</a> <font color="Maroon" title="c8">A</font>,<font color="Maroon" title="c4">b1</font>,<font color="Maroon" title="c5">b2</font></span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:66_2"><span class="lab"><font color="Green" title="E44">A7</font></span></a>, <a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c16">w</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="ranknull.html#K1" title="RANKNULL:func.1">ker</a> <span class="p2">(<span class="default"><a href="matrlin2.html#K5" title="MATRLIN2:func.5">Mx2Tran</a> <font color="Maroon" title="c8">A</font>,<font color="Maroon" title="c4">b1</font>,<font color="Maroon" title="c5">b2</font></span>)</span></span>)</span><span class="kw"> such that </span><br/><a NAME="E6:66_2"/><span class="lab"><font color="Green" title="E45">A8</font></span>: 
<font color="Maroon" title="c15">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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">AI</font>,<font color="Maroon" title="c4">b1</font>,<font color="Maroon" title="c7">BB</font></span>)</span> <a href="matrlin2.html#K12" title="MATRLIN2:func.12">|</a> <span class="p2">(<span class="default"><a href="ranknull.html#K1" title="RANKNULL:func.1">ker</a> <span class="p3">(<span class="default"><a href="matrlin2.html#K5" title="MATRLIN2:func.5">Mx2Tran</a> <font color="Maroon" title="c8">A</font>,<font color="Maroon" title="c4">b1</font>,<font color="Maroon" title="c5">b2</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">w</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ranknull.html#T13" title="RANKNULL:th.13">RANKNULL:13</a></span>;<br/>
<a NAME="E7:66_2"/><span class="lab"><font color="Green" title="E46">A9</font></span>: 
<font color="Maroon" title="c16">w</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="c8">A</font>,<font color="Maroon" title="c4">b1</font>,<font color="Maroon" title="c5">b2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></span>;<br/>
<a NAME="E8:66_2"/><span class="kw">then </span>
<font color="Maroon" title="c16">w</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c2">V1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_4.html#T17" title="VECTSP_4:th.17">VECTSP_4:17</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c17">W</font> = <font color="Maroon" title="c16">w</font> as   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c2">V1</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></span>;<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">AI</font>,<font color="Maroon" title="c4">b1</font>,<font color="Maroon" title="c7">BB</font></span>)</span> <a href="matrlin2.html#K12" title="MATRLIN2:func.12">|</a> <span class="p2">(<span class="default"><a href="ranknull.html#K1" title="RANKNULL:func.1">ker</a> <span class="p3">(<span class="default"><a href="matrlin2.html#K5" title="MATRLIN2:func.5">Mx2Tran</a> <font color="Maroon" title="c8">A</font>,<font color="Maroon" title="c4">b1</font>,<font color="Maroon" title="c5">b2</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">w</font> = 
<span class="p1">(<span class="default"><a href="matrlin2.html#K5" title="MATRLIN2:func.5">Mx2Tran</a> <font color="Maroon" title="c9">AI</font>,<font color="Maroon" title="c4">b1</font>,<font color="Maroon" title="c7">BB</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">w</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:66"><span class="lab"><font color="Green" title="E42">A5</font></span></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrlin2.html#K7" title="MATRLIN2:func.7">id</a> <font color="Maroon" title="c2">V1</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">W</font></span>)</span> <a href="matrlin.html#K9" title="MATRLIN:func.9">|--</a> <font color="Maroon" title="c4">b1</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrlin2.html#T47" target="_self" title="MATRLIN2:th.47">Th47</a></span>
<br/>.= 
<font color="Maroon" title="c17">W</font> <a href="matrlin.html#K9" title="MATRLIN:func.9">|--</a> <font color="Maroon" title="c4">b1</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T34" title="FUNCT_1:th.34">FUNCT_1:34</a></span>
;<br/>
<a NAME="E11:66_2"/><span class="kw">then </span>
<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">AI</font>,<font color="Maroon" title="c4">b1</font>,<font color="Maroon" title="c7">BB</font></span>)</span> <a href="matrlin2.html#K12" title="MATRLIN2:func.12">|</a> <span class="p2">(<span class="default"><a href="ranknull.html#K1" title="RANKNULL:func.1">ker</a> <span class="p3">(<span class="default"><a href="matrlin2.html#K5" title="MATRLIN2:func.5">Mx2Tran</a> <font color="Maroon" title="c8">A</font>,<font color="Maroon" title="c4">b1</font>,<font color="Maroon" title="c5">b2</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">w</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="c8">A</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:66"><span class="lab"><font color="Green" title="E38">A1</font></span></a>, <a class="txt" href="#E3:66"><span class="lab"><font color="Green" title="E39">A2</font></span></a>, <a class="txt" href="#E7:66_2"><span class="lab"><font color="Green" title="E46">A9</font></span></a>, <a class="ref" href="matrlin2.html#T41" target="_self" title="MATRLIN2:th.41">Th41</a></span>;<br/>
<span class="kw">hence </span><a NAME="E12:66_2"/>
<font color="Maroon" title="c14">x</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="c8">A</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:66_2"><span class="lab"><font color="Green" title="E45">A8</font></span></a>, <a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E13:66"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">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="c8">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> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:66"><span class="lab"><font color="Green" title="E38">A1</font></span></a>, <a class="ref" href="matrix_1.html#T24" title="MATRIX_1:th.24">MATRIX_1:24</a></span>;<br/>
<a NAME="E14:66"/><span class="kw">then </span><span class="lab"><font color="Green" title="E44">A10</font></span>: 
 <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:66"><span class="lab"><font color="Green" title="E39">A2</font></span></a>, <a class="ref" href="matrix_2.html#T12" title="MATRIX_2:th.12">MATRIX_2:12</a></span>;<br/>
 <a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</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="c4">b1</font></span>)</span> <a href="matrix13.html#K13" title="MATRIX13:func.13">-VectSp_over</a> <font color="Maroon" title="c1">K</font></span>)</span> = 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">b1</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix13.html#T112" title="MATRIX13:th.112">MATRIX13:112</a></span>
<br/>.= 
 <a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <font color="Maroon" title="c2">V1</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrlin2.html#T21" target="_self" title="MATRLIN2:th.21">Th21</a></span>
<br/>.= 
 <a href="ranknull.html#K10" title="RANKNULL:func.10">rank</a> <span class="p1">(<span class="default"><a href="matrlin2.html#K5" title="MATRLIN2:func.5">Mx2Tran</a> <font color="Maroon" title="c9">AI</font>,<font color="Maroon" title="c4">b1</font>,<font color="Maroon" title="c7">BB</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:66"><span class="lab"><font color="Green" title="E41">A4</font></span></a>, <a class="ref" href="matrlin2.html#T44" target="_self" title="MATRLIN2:th.44">Th44</a>, <a class="ref" href="ranknull.html#T45" title="RANKNULL:th.45">RANKNULL:45</a></span>
<br/>.= 
 <a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <span class="p1">(<span class="default"><a href="ranknull.html#K3" title="RANKNULL:func.3">im</a> <span class="p2">(<span class="default"><a href="matrlin2.html#K5" title="MATRLIN2:func.5">Mx2Tran</a> <font color="Maroon" title="c9">AI</font>,<font color="Maroon" title="c4">b1</font>,<font color="Maroon" title="c7">BB</font></span>)</span></span>)</span>

;<br/>
<a NAME="E16:66"/><span class="kw">then </span><span class="lab"><font color="Green" title="E45">A11</font></span>: 
 <a href="vectsp_4.html#K2" title="VECTSP_4:func.2">(Omega).</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="c4">b1</font></span>)</span> <a href="matrix13.html#K13" title="MATRIX13:func.13">-VectSp_over</a> <font color="Maroon" title="c1">K</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_4.html#K2" title="VECTSP_4:func.2">(Omega).</a> <span class="p1">(<span class="default"><a href="ranknull.html#K3" title="RANKNULL:func.3">im</a> <span class="p2">(<span class="default"><a href="matrlin2.html#K5" title="MATRLIN2:func.5">Mx2Tran</a> <font color="Maroon" title="c9">AI</font>,<font color="Maroon" title="c4">b1</font>,<font color="Maroon" title="c7">BB</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_9.html#T32" title="VECTSP_9:th.32">VECTSP_9:32</a></span>;<br/>
<a NAME="E17:66"/>
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="c8">A</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="ranknull.html#K3" title="RANKNULL:func.3">im</a> <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">AI</font>,<font color="Maroon" title="c4">b1</font>,<font color="Maroon" title="c7">BB</font></span>)</span> <a href="matrlin2.html#K12" title="MATRLIN2:func.12">|</a> <span class="p3">(<span class="default"><a href="ranknull.html#K1" title="RANKNULL:func.1">ker</a> <span class="p4">(<span class="default"><a href="matrlin2.html#K5" title="MATRLIN2:func.5">Mx2Tran</a> <font color="Maroon" title="c8">A</font>,<font color="Maroon" title="c4">b1</font>,<font color="Maroon" title="c5">b2</font></span>)</span></span>)</span></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c14">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c14">x</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="c8">A</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span></span>)</span> or <font color="Maroon" title="c14">x</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="ranknull.html#K3" title="RANKNULL:func.3">im</a> <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">AI</font>,<font color="Maroon" title="c4">b1</font>,<font color="Maroon" title="c7">BB</font></span>)</span> <a href="matrlin2.html#K12" title="MATRLIN2:func.12">|</a> <span class="p3">(<span class="default"><a href="ranknull.html#K1" title="RANKNULL:func.1">ker</a> <span class="p4">(<span class="default"><a href="matrlin2.html#K5" title="MATRLIN2:func.5">Mx2Tran</a> <font color="Maroon" title="c8">A</font>,<font color="Maroon" title="c4">b1</font>,<font color="Maroon" title="c5">b2</font></span>)</span></span>)</span></span>)</span></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:66_4"/><span class="lab"><font color="Green" title="E46">A12</font></span>: 
<font color="Maroon" title="c14">x</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="c8">A</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c14">x</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="ranknull.html#K3" title="RANKNULL:func.3">im</a> <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">AI</font>,<font color="Maroon" title="c4">b1</font>,<font color="Maroon" title="c7">BB</font></span>)</span> <a href="matrlin2.html#K12" title="MATRLIN2:func.12">|</a> <span class="p3">(<span class="default"><a href="ranknull.html#K1" title="RANKNULL:func.1">ker</a> <span class="p4">(<span class="default"><a href="matrlin2.html#K5" title="MATRLIN2:func.5">Mx2Tran</a> <font color="Maroon" title="c8">A</font>,<font color="Maroon" title="c4">b1</font>,<font color="Maroon" title="c5">b2</font></span>)</span></span>)</span></span>)</span></span>)</span></span><br/>

<a NAME="E2:66_4"/>
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="c8">A</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <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="c4">b1</font></span>)</span> <a href="matrix13.html#K13" title="MATRIX13:func.13">-VectSp_over</a> <font color="Maroon" title="c1">K</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:66"><span class="lab"><font color="Green" title="E44">A10</font></span></a>, <a class="ref" href="vectsp_4.html#D2" title="VECTSP_4:def.2">VECTSP_4:def 2</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c15">v</font> = <font color="Maroon" title="c14">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <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="c4">b1</font></span>)</span> <a href="matrix13.html#K13" title="MATRIX13:func.13">-VectSp_over</a> <font color="Maroon" title="c1">K</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:66_4"><span class="lab"><font color="Green" title="E46">A12</font></span></a></span>;<br/>
<a NAME="E4:66_4"/><span class="lab"><font color="Green" title="E47">A13</font></span>: 
<font color="Maroon" title="c15">v</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="c8">A</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:66_4"><span class="lab"><font color="Green" title="E46">A12</font></span></a>, <a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></span>;<br/>
<a NAME="E5:66_4"/>
<font color="Maroon" title="c15">v</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="ranknull.html#K3" title="RANKNULL:func.3">im</a> <span class="p1">(<span class="default"><a href="matrlin2.html#K5" title="MATRLIN2:func.5">Mx2Tran</a> <font color="Maroon" title="c9">AI</font>,<font color="Maroon" title="c4">b1</font>,<font color="Maroon" title="c7">BB</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:66"><span class="lab"><font color="Green" title="E45">A11</font></span></a>, <a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c16">w</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">V1</font><span class="kw"> such that </span><br/><a NAME="E7:66_4"/><span class="lab"><font color="Green" title="E48">A14</font></span>: 
<font color="Maroon" title="c15">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrlin2.html#K5" title="MATRLIN2:func.5">Mx2Tran</a> <font color="Maroon" title="c9">AI</font>,<font color="Maroon" title="c4">b1</font>,<font color="Maroon" title="c7">BB</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">w</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ranknull.html#T13" title="RANKNULL:th.13">RANKNULL:13</a></span>;<br/>
<span class="p1">(<span class="default"><a href="matrlin2.html#K5" title="MATRLIN2:func.5">Mx2Tran</a> <font color="Maroon" title="c9">AI</font>,<font color="Maroon" title="c4">b1</font>,<font color="Maroon" title="c7">BB</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">w</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrlin2.html#K7" title="MATRLIN2:func.7">id</a> <font color="Maroon" title="c2">V1</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">w</font></span>)</span> <a href="matrlin.html#K9" title="MATRLIN:func.9">|--</a> <font color="Maroon" title="c4">b1</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrlin2.html#T47" target="_self" title="MATRLIN2:th.47">Th47</a></span>
<br/>.= 
<font color="Maroon" title="c16">w</font> <a href="matrlin.html#K9" title="MATRLIN:func.9">|--</a> <font color="Maroon" title="c4">b1</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T34" title="FUNCT_1:th.34">FUNCT_1:34</a></span>
;<br/>
<a NAME="E9:66_4"/><span class="kw">then </span>
<font color="Maroon" title="c16">w</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="c8">A</font>,<font color="Maroon" title="c4">b1</font>,<font color="Maroon" title="c5">b2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:66"><span class="lab"><font color="Green" title="E38">A1</font></span></a>, <a class="txt" href="#E3:66"><span class="lab"><font color="Green" title="E39">A2</font></span></a>, <a class="txt" href="#E7:66_4"><span class="lab"><font color="Green" title="E48">A14</font></span></a>, <a class="txt" href="#E4:66_4"><span class="lab"><font color="Green" title="E47">A13</font></span></a>, <a class="ref" href="matrlin2.html#T41" target="_self" title="MATRLIN2:th.41">Th41</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c17">W</font> = <font color="Maroon" title="c16">w</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="ranknull.html#K1" title="RANKNULL:func.1">ker</a> <span class="p2">(<span class="default"><a href="matrlin2.html#K5" title="MATRLIN2:func.5">Mx2Tran</a> <font color="Maroon" title="c8">A</font>,<font color="Maroon" title="c4">b1</font>,<font color="Maroon" title="c5">b2</font></span>)</span></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></span>;<br/>
<a NAME="E11:66_4"/>
<font color="Maroon" title="c15">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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">AI</font>,<font color="Maroon" title="c4">b1</font>,<font color="Maroon" title="c7">BB</font></span>)</span> <a href="matrlin2.html#K12" title="MATRLIN2:func.12">|</a> <span class="p2">(<span class="default"><a href="ranknull.html#K1" title="RANKNULL:func.1">ker</a> <span class="p3">(<span class="default"><a href="matrlin2.html#K5" title="MATRLIN2:func.5">Mx2Tran</a> <font color="Maroon" title="c8">A</font>,<font color="Maroon" title="c4">b1</font>,<font color="Maroon" title="c5">b2</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">W</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:66"><span class="lab"><font color="Green" title="E42">A5</font></span></a>, <a class="txt" href="#E7:66_4"><span class="lab"><font color="Green" title="E48">A14</font></span></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></span>;<br/>
<a NAME="E12:66_4"/><span class="kw">then </span>
<font color="Maroon" title="c15">v</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="ranknull.html#K3" title="RANKNULL:func.3">im</a> <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">AI</font>,<font color="Maroon" title="c4">b1</font>,<font color="Maroon" title="c7">BB</font></span>)</span> <a href="matrlin2.html#K12" title="MATRLIN2:func.12">|</a> <span class="p2">(<span class="default"><a href="ranknull.html#K1" title="RANKNULL:func.1">ker</a> <span class="p3">(<span class="default"><a href="matrlin2.html#K5" title="MATRLIN2:func.5">Mx2Tran</a> <font color="Maroon" title="c8">A</font>,<font color="Maroon" title="c4">b1</font>,<font color="Maroon" title="c5">b2</font></span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ranknull.html#T13" title="RANKNULL:th.13">RANKNULL:13</a></span>;<br/>
<span class="kw">hence </span><a NAME="E13:66_4"/>
<font color="Maroon" title="c14">x</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="ranknull.html#K3" title="RANKNULL:func.3">im</a> <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">AI</font>,<font color="Maroon" title="c4">b1</font>,<font color="Maroon" title="c7">BB</font></span>)</span> <a href="matrlin2.html#K12" title="MATRLIN2:func.12">|</a> <span class="p3">(<span class="default"><a href="ranknull.html#K1" title="RANKNULL:func.1">ker</a> <span class="p4">(<span class="default"><a href="matrlin2.html#K5" title="MATRLIN2:func.5">Mx2Tran</a> <font color="Maroon" title="c8">A</font>,<font color="Maroon" title="c4">b1</font>,<font color="Maroon" title="c5">b2</font></span>)</span></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E18:66"/><span class="kw">then </span>
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="c8">A</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="ranknull.html#K3" title="RANKNULL:func.3">im</a> <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">AI</font>,<font color="Maroon" title="c4">b1</font>,<font color="Maroon" title="c7">BB</font></span>)</span> <a href="matrlin2.html#K12" title="MATRLIN2:func.12">|</a> <span class="p3">(<span class="default"><a href="ranknull.html#K1" title="RANKNULL:func.1">ker</a> <span class="p4">(<span class="default"><a href="matrlin2.html#K5" title="MATRLIN2:func.5">Mx2Tran</a> <font color="Maroon" title="c8">A</font>,<font color="Maroon" title="c4">b1</font>,<font color="Maroon" title="c5">b2</font></span>)</span></span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:66"><span class="lab"><font color="Green" title="E43">A6</font></span></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></span>;<br/>
<a NAME="E19:66"/><span class="kw">then </span><span class="lab"><font color="Green" title="E46">A15</font></span>: 
 <a href="ranknull.html#K3" title="RANKNULL:func.3">im</a> <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">AI</font>,<font color="Maroon" title="c4">b1</font>,<font color="Maroon" title="c7">BB</font></span>)</span> <a href="matrlin2.html#K12" title="MATRLIN2:func.12">|</a> <span class="p2">(<span class="default"><a href="ranknull.html#K1" title="RANKNULL:func.1">ker</a> <span class="p3">(<span class="default"><a href="matrlin2.html#K5" title="MATRLIN2:func.5">Mx2Tran</a> <font color="Maroon" title="c8">A</font>,<font color="Maroon" title="c4">b1</font>,<font color="Maroon" title="c5">b2</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c8">A</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:66"><span class="lab"><font color="Green" title="E44">A10</font></span></a>, <a class="ref" href="vectsp_4.html#T37" title="VECTSP_4:th.37">VECTSP_4:37</a></span>;<br/>
<a NAME="E20:66"/>
 <a href="matrlin2.html#K5" title="MATRLIN2:func.5">Mx2Tran</a> <font color="Maroon" title="c9">AI</font>,<font color="Maroon" title="c4">b1</font>,<font color="Maroon" title="c7">BB</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:66"><span class="lab"><font color="Green" title="E41">A4</font></span></a>, <a class="ref" href="matrlin2.html#T44" target="_self" title="MATRLIN2:th.44">Th44</a></span>;<br/>
<a NAME="E21:66"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="matrlin2.html#K5" title="MATRLIN2:func.5">Mx2Tran</a> <font color="Maroon" title="c9">AI</font>,<font color="Maroon" title="c4">b1</font>,<font color="Maroon" title="c7">BB</font></span>)</span> <a href="matrlin2.html#K12" title="MATRLIN2:func.12">|</a> <span class="p1">(<span class="default"><a href="ranknull.html#K1" title="RANKNULL:func.1">ker</a> <span class="p2">(<span class="default"><a href="matrlin2.html#K5" title="MATRLIN2:func.5">Mx2Tran</a> <font color="Maroon" title="c8">A</font>,<font color="Maroon" title="c4">b1</font>,<font color="Maroon" title="c5">b2</font></span>)</span></span>)</span> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T84" title="FUNCT_1:th.84">FUNCT_1:84</a></span>;<br/>
<span class="kw">hence </span> <a href="ranknull.html#K11" title="RANKNULL:func.11">nullity</a> <span class="p1">(<span class="default"><a href="matrlin2.html#K5" title="MATRLIN2:func.5">Mx2Tran</a> <font color="Maroon" title="c8">A</font>,<font color="Maroon" title="c4">b1</font>,<font color="Maroon" title="c5">b2</font></span>)</span> = 
 <a href="ranknull.html#K10" title="RANKNULL:func.10">rank</a> <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">AI</font>,<font color="Maroon" title="c4">b1</font>,<font color="Maroon" title="c7">BB</font></span>)</span> <a href="matrlin2.html#K12" title="MATRLIN2:func.12">|</a> <span class="p2">(<span class="default"><a href="ranknull.html#K1" title="RANKNULL:func.1">ker</a> <span class="p3">(<span class="default"><a href="matrlin2.html#K5" title="MATRLIN2:func.5">Mx2Tran</a> <font color="Maroon" title="c8">A</font>,<font color="Maroon" title="c4">b1</font>,<font color="Maroon" title="c5">b2</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="ranknull.html#T45" title="RANKNULL:th.45">RANKNULL:45</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">b1</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">A</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:66"><span class="lab"><font color="Green" title="E38">A1</font></span></a>, <a class="txt" href="#E14:66"><span class="lab"><font color="Green" title="E44">A10</font></span></a>, <a class="txt" href="#E19:66"><span class="lab"><font color="Green" title="E46">A15</font></span></a>, <a class="ref" href="matrix15.html#T68" title="MATRIX15:th.68">MATRIX15:68</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">b1</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <font color="Maroon" title="c8">A</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix13.html#T84" title="MATRIX13:th.84">MATRIX13:84</a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>

<span class="kw">set </span><font color="Maroon" title="c14">M</font> =  <a href="matrlin2.html#K5" title="MATRLIN2:func.5">Mx2Tran</a> <font color="Maroon" title="c8">A</font>,<font color="Maroon" title="c4">b1</font>,<font color="Maroon" title="c5">b2</font>;<br/>

</div>
