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

<span class="kw">set </span><font color="Maroon" title="c4">V</font> = <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="prvect_1.html#K7" title="PRVECT_1:func.7">-VectSp_over</a> <font color="Maroon" title="c1">K</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c5">k0</font> = <font color="Maroon" title="c3">k</font> <a href="prvect_1.html#K4" title="PRVECT_1:func.4">|-&gt;</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font></span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c6">S</font> =  <a href="matrix15.html#K6" title="MATRIX15:func.6">Solutions_of</a> (<font color="Maroon" title="c2">A</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="prvect_1.html#K4" title="PRVECT_1: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="E1:73_1_1"/><span class="lab"><font color="Green" title="E57">A1</font></span>: 
 <a href="matrix15.html#K3" title="MATRIX15:func.3">ColVec2Mx</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="prvect_1.html#K4" title="PRVECT_1: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="matrix_3.html#K1" title="MATRIX_3:func.1">0.</a> (<font color="Maroon" title="c1">K</font>,<font color="Maroon" title="c3">k</font>,1)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix15.html#T32" target="_self" title="MATRIX15:th.32">Th32</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E58">A2</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="73_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c7">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">K</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">v</font> being   <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="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="prvect_1.html#K7" title="PRVECT_1:func.7">-VectSp_over</a> <font color="Maroon" title="c1">K</font></span>)</span>  st <font color="Olive" title="b1">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix15.html#K6" title="MATRIX15:func.6">Solutions_of</a> (<font color="Maroon" title="c2">A</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="prvect_1.html#K4" title="PRVECT_1: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>) holds <br/><font color="Maroon" title="c7">a</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Olive" title="b1">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix15.html#K6" title="MATRIX15:func.6">Solutions_of</a> (<font color="Maroon" title="c2">A</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="prvect_1.html#K4" title="PRVECT_1: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>)</span><br/><span class="kw">let </span><font color="Maroon" title="c8">v</font> be   <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="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="prvect_1.html#K7" title="PRVECT_1:func.7">-VectSp_over</a> <font color="Maroon" title="c1">K</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c8">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix15.html#K6" title="MATRIX15:func.6">Solutions_of</a> (<font color="Maroon" title="c2">A</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="prvect_1.html#K4" title="PRVECT_1: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>) implies <font color="Maroon" title="c7">a</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c8">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix15.html#K6" title="MATRIX15:func.6">Solutions_of</a> (<font color="Maroon" title="c2">A</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="prvect_1.html#K4" title="PRVECT_1: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>) )</span><br/><span class="kw">assume </span><a NAME="E1:73_1_1_1"/>
<font color="Maroon" title="c8">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix15.html#K6" title="MATRIX15:func.6">Solutions_of</a> (<font color="Maroon" title="c2">A</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="prvect_1.html#K4" title="PRVECT_1: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 class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">a</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c8">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix15.html#K6" title="MATRIX15:func.6">Solutions_of</a> (<font color="Maroon" title="c2">A</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="prvect_1.html#K4" title="PRVECT_1: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>)</span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">f</font> being   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c1">K</font><span class="kw"> such that </span><br/><a NAME="E3:73_1_1_1"/><span class="lab"><font color="Green" title="E59">A3</font></span>: 
<font color="Maroon" title="c8">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">f</font>
 <span class="kw">and </span><br/><a NAME="E4:73_1_1_1"/><span class="lab"><font color="Green" title="E60">A4</font></span>: 
 <a href="matrix15.html#K3" title="MATRIX15:func.3">ColVec2Mx</a> <font color="Maroon" title="c9">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> (<font color="Maroon" title="c2">A</font>,<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="c3">k</font> <a href="prvect_1.html#K4" title="PRVECT_1: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/><span class="kw">reconsider </span><font color="Maroon" title="c10">f</font> = <font color="Maroon" title="c9">f</font> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">K</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:73_1_1_1"><span class="lab"><font color="Green" title="E59">A3</font></span></a>, <a class="ref" href="matrix13.html#T102" title="MATRIX13:th.102">MATRIX13:102</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_1_1_1_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="73_1_1_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:73_1_1_1_1_1"/>
( <font color="Maroon" title="c3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  or <font color="Maroon" title="c3">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="73_1_1_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:73_1_1_1_1_1_1"/>
<font color="Maroon" title="c3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="matrix_3.html#K1" title="MATRIX_3:func.1">0.</a> (<font color="Maroon" title="c1">K</font>,<font color="Maroon" title="c3">k</font>,1) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">a</font> <a href="matrix13.html#K2" title="MATRIX13:func.2">*</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K1" title="MATRIX_3:func.1">0.</a> (<font color="Maroon" title="c1">K</font>,<font color="Maroon" title="c3">k</font>,1)</span>)</span></span><br/><div class="add"><a NAME="E2:73_1_1_1_1_1_1"/><span class="kw">then </span>
(  <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="matrix13.html#K2" title="MATRIX13:func.2">*</a> <span class="p2">(<span class="default"><a href="matrix_3.html#K1" title="MATRIX_3:func.1">0.</a> (<font color="Maroon" title="c1">K</font>,<font color="Maroon" title="c3">k</font>,1)</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"><a href="matrix_3.html#K1" title="MATRIX_3:func.1">0.</a> (<font color="Maroon" title="c1">K</font>,<font color="Maroon" title="c3">k</font>,1)</span>)</span> &amp;  <a href="matrix_3.html#K1" title="MATRIX_3:func.1">0.</a> (<font color="Maroon" title="c1">K</font>,<font color="Maroon" title="c3">k</font>,1) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_3.html#D5" title="MATRIX_3:def.5">MATRIX_3:def 5</a></span>;<br/><span class="kw">hence </span><a NAME="E3:73_1_1_1_1_1_1"/>
 <a href="matrix_3.html#K1" title="MATRIX_3:func.1">0.</a> (<font color="Maroon" title="c1">K</font>,<font color="Maroon" title="c3">k</font>,1) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">a</font> <a href="matrix13.html#K2" title="MATRIX13:func.2">*</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K1" title="MATRIX_3:func.1">0.</a> (<font color="Maroon" title="c1">K</font>,<font color="Maroon" title="c3">k</font>,1)</span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="73_1_1_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:73_1_1_1_1_1_2"/><span class="lab"><font color="Green" title="E61">A5</font></span>: 
<font color="Maroon" title="c3">k</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="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">a</font> <a href="matrix13.html#K2" title="MATRIX13:func.2">*</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K1" title="MATRIX_3:func.1">0.</a> (<font color="Maroon" title="c1">K</font>,<font color="Maroon" title="c3">k</font>,1)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_3.html#K1" title="MATRIX_3:func.1">0.</a> (<font color="Maroon" title="c1">K</font>,<font color="Maroon" title="c3">k</font>,1)</span><br/><div class="add"><a NAME="E2:73_1_1_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E62">A6</font></span>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K1" title="MATRIX_3:func.1">0.</a> (<font color="Maroon" title="c1">K</font>,<font color="Maroon" title="c3">k</font>,1)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">k</font> &amp;  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K1" title="MATRIX_3:func.1">0.</a> (<font color="Maroon" title="c1">K</font>,<font color="Maroon" title="c3">k</font>,1)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#T23" title="MATRIX_1:th.23">MATRIX_1:23</a></span>;<br/><span class="kw">hence </span><font color="Maroon" title="c7">a</font> <a href="matrix13.html#K2" title="MATRIX13:func.2">*</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K1" title="MATRIX_3:func.1">0.</a> (<font color="Maroon" title="c1">K</font>,<font color="Maroon" title="c3">k</font>,1)</span>)</span> = 
<font color="Maroon" title="c7">a</font> <a href="matrix13.html#K2" title="MATRIX13:func.2">*</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="c1">K</font></span>)</span> <a href="matrix13.html#K2" title="MATRIX13:func.2">*</a> <span class="p2">(<span class="default"><a href="matrix_3.html#K1" title="MATRIX_3:func.1">0.</a> (<font color="Maroon" title="c1">K</font>,<font color="Maroon" title="c3">k</font>,1)</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:73_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E61">A5</font></span></a>, <a class="ref" href="matrix_5.html#T24" title="MATRIX_5:th.24">MATRIX_5:24</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</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="matrix13.html#K2" title="MATRIX13:func.2">*</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K1" title="MATRIX_3:func.1">0.</a> (<font color="Maroon" title="c1">K</font>,<font color="Maroon" title="c3">k</font>,1)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_5.html#T11" title="MATRIX_5:th.11">MATRIX_5:11</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font></span>)</span> <a href="matrix13.html#K2" title="MATRIX13:func.2">*</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K1" title="MATRIX_3:func.1">0.</a> (<font color="Maroon" title="c1">K</font>,<font color="Maroon" title="c3">k</font>,1)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_1.html#T6" title="VECTSP_1:th.6">VECTSP_1:6</a></span>
<br/>.= 
 <a href="matrix_3.html#K1" title="MATRIX_3:func.1">0.</a> (<font color="Maroon" title="c1">K</font>,<font color="Maroon" title="c3">k</font>,1)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:73_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E61">A5</font></span></a>, <a class="txt" href="#E2:73_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E62">A6</font></span></a>, <a class="ref" href="matrix_5.html#T24" title="MATRIX_5:th.24">MATRIX_5:24</a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><a NAME="E7:73_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">a</font> <a href="matrix13.html#K2" title="MATRIX13:func.2">*</a> <span class="p1">(<span class="default"><a href="matrix15.html#K3" title="MATRIX15:func.3">ColVec2Mx</a> <font color="Maroon" title="c10">f</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> (<font color="Maroon" title="c2">A</font>,<span class="p1">(<span class="default"><a href="matrix_3.html#K1" title="MATRIX_3:func.1">0.</a> (<font color="Maroon" title="c1">K</font>,<font color="Maroon" title="c3">k</font>,1)</span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:73_1_1"><span class="lab"><font color="Green" title="E57">A1</font></span></a>, <a class="txt" href="#E4:73_1_1_1"><span class="lab"><font color="Green" title="E60">A4</font></span></a>, <a class="ref" href="matrix15.html#T35" target="_self" title="MATRIX15:th.35">Th35</a></span>;<br/><a NAME="E8:73_1_1_1"/><span class="kw">then </span>
 <a href="matrix15.html#K3" title="MATRIX15:func.3">ColVec2Mx</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <font color="Maroon" title="c10">f</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> (<font color="Maroon" title="c2">A</font>,<span class="p1">(<span class="default"><a href="matrix_3.html#K1" title="MATRIX_3:func.1">0.</a> (<font color="Maroon" title="c1">K</font>,<font color="Maroon" title="c3">k</font>,1)</span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="matrix15.html#T30" target="_self" title="MATRIX15:th.30">Th30</a></span>;<br/><a NAME="E9:73_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <font color="Maroon" title="c10">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix15.html#K6" title="MATRIX15:func.6">Solutions_of</a> (<font color="Maroon" title="c2">A</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="prvect_1.html#K4" title="PRVECT_1: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>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:73_1_1"><span class="lab"><font color="Green" title="E57">A1</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E10:73_1_1_1"/>
<font color="Maroon" title="c7">a</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c8">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix15.html#K6" title="MATRIX15:func.6">Solutions_of</a> (<font color="Maroon" title="c2">A</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="prvect_1.html#K4" title="PRVECT_1: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>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:73_1_1_1"><span class="lab"><font color="Green" title="E59">A3</font></span></a>, <a class="ref" href="matrix13.html#T102" title="MATRIX13:th.102">MATRIX13:102</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_1_1_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c7">v</font>, <font color="Maroon" title="c8">u</font> be   <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="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="prvect_1.html#K7" title="PRVECT_1:func.7">-VectSp_over</a> <font color="Maroon" title="c1">K</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c7">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix15.html#K6" title="MATRIX15:func.6">Solutions_of</a> (<font color="Maroon" title="c2">A</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="prvect_1.html#K4" title="PRVECT_1: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>) &amp; <font color="Maroon" title="c8">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix15.html#K6" title="MATRIX15:func.6">Solutions_of</a> (<font color="Maroon" title="c2">A</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="prvect_1.html#K4" title="PRVECT_1: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>) implies <font color="Maroon" title="c7">v</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c8">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix15.html#K6" title="MATRIX15:func.6">Solutions_of</a> (<font color="Maroon" title="c2">A</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="prvect_1.html#K4" title="PRVECT_1: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>) )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:73_1_1_2"/><span class="lab"><font color="Green" title="E59">A7</font></span>: 
<font color="Maroon" title="c7">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix15.html#K6" title="MATRIX15:func.6">Solutions_of</a> (<font color="Maroon" title="c2">A</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="prvect_1.html#K4" title="PRVECT_1: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>)
 <span class="kw">and </span><br/><a NAME="E2:73_1_1_2"/><span class="lab"><font color="Green" title="E60">A8</font></span>: 
<font color="Maroon" title="c8">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix15.html#K6" title="MATRIX15:func.6">Solutions_of</a> (<font color="Maroon" title="c2">A</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="prvect_1.html#K4" title="PRVECT_1: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 class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">v</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c8">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix15.html#K6" title="MATRIX15:func.6">Solutions_of</a> (<font color="Maroon" title="c2">A</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="prvect_1.html#K4" title="PRVECT_1: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>)</span><br/><span class="kw">consider </span><font color="Maroon" title="c9">f</font> being   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c1">K</font><span class="kw"> such that </span><br/><a NAME="E4:73_1_1_2"/><span class="lab"><font color="Green" title="E61">A9</font></span>: 
<font color="Maroon" title="c7">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">f</font>
 <span class="kw">and </span><br/><a NAME="E5:73_1_1_2"/><span class="lab"><font color="Green" title="E62">A10</font></span>: 
 <a href="matrix15.html#K3" title="MATRIX15:func.3">ColVec2Mx</a> <font color="Maroon" title="c9">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> (<font color="Maroon" title="c2">A</font>,<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="c3">k</font> <a href="prvect_1.html#K4" title="PRVECT_1: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>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:73_1_1_2"><span class="lab"><font color="Green" title="E59">A7</font></span></a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c10">g</font> being   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c1">K</font><span class="kw"> such that </span><br/><a NAME="E7:73_1_1_2"/><span class="lab"><font color="Green" title="E63">A11</font></span>: 
<font color="Maroon" title="c8">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">g</font>
 <span class="kw">and </span><br/><a NAME="E8:73_1_1_2"/><span class="lab"><font color="Green" title="E64">A12</font></span>: 
 <a href="matrix15.html#K3" title="MATRIX15:func.3">ColVec2Mx</a> <font color="Maroon" title="c10">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix15.html#K4" title="MATRIX15:func.4">Solutions_of</a> (<font color="Maroon" title="c2">A</font>,<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="c3">k</font> <a href="prvect_1.html#K4" title="PRVECT_1: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>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:73_1_1_2"><span class="lab"><font color="Green" title="E60">A8</font></span></a></span>;<br/><a NAME="E9:73_1_1_2"/><span class="lab"><font color="Green" title="E65">A13</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">g</font> <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="c2">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:73_1_1_2"><span class="lab"><font color="Green" title="E64">A12</font></span></a>, <a class="ref" href="matrix15.html#T59" target="_self" title="MATRIX15:th.59">Th59</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c11">f</font> = <font color="Maroon" title="c9">f</font>, <font color="Maroon" title="c12">g</font> = <font color="Maroon" title="c10">g</font> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">K</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:73_1_1_2"><span class="lab"><font color="Green" title="E61">A9</font></span></a>, <a class="txt" href="#E7:73_1_1_2"><span class="lab"><font color="Green" title="E63">A11</font></span></a>, <a class="ref" href="matrix13.html#T102" title="MATRIX13:th.102">MATRIX13:102</a></span>;<br/><a NAME="E11:73_1_1_2"/>
<span class="p1">(<span class="default"><a href="matrix15.html#K3" title="MATRIX15:func.3">ColVec2Mx</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="matrix_3.html#K3" title="MATRIX_3:func.3">+</a> <span class="p1">(<span class="default"><a href="matrix15.html#K3" title="MATRIX15:func.3">ColVec2Mx</a> <font color="Maroon" title="c12">g</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> (<font color="Maroon" title="c2">A</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_3.html#K1" title="MATRIX_3:func.1">0.</a> (<font color="Maroon" title="c1">K</font>,<font color="Maroon" title="c3">k</font>,1)</span>)</span> <a href="matrix_3.html#K3" title="MATRIX_3:func.3">+</a> <span class="p2">(<span class="default"><a href="matrix_3.html#K1" title="MATRIX_3:func.1">0.</a> (<font color="Maroon" title="c1">K</font>,<font color="Maroon" title="c3">k</font>,1)</span>)</span></span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:73_1_1"><span class="lab"><font color="Green" title="E57">A1</font></span></a>, <a class="txt" href="#E5:73_1_1_2"><span class="lab"><font color="Green" title="E62">A10</font></span></a>, <a class="txt" href="#E8:73_1_1_2"><span class="lab"><font color="Green" title="E64">A12</font></span></a>, <a class="ref" href="matrix15.html#T37" target="_self" title="MATRIX15:th.37">Th37</a></span>;<br/><a NAME="E12:73_1_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="matrix15.html#K3" title="MATRIX15:func.3">ColVec2Mx</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="matrix_3.html#K3" title="MATRIX_3:func.3">+</a> <span class="p1">(<span class="default"><a href="matrix15.html#K3" title="MATRIX15:func.3">ColVec2Mx</a> <font color="Maroon" title="c12">g</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> (<font color="Maroon" title="c2">A</font>,<span class="p1">(<span class="default"><a href="matrix_3.html#K1" title="MATRIX_3:func.1">0.</a> (<font color="Maroon" title="c1">K</font>,<font color="Maroon" title="c3">k</font>,1)</span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_3.html#T4" title="MATRIX_3:th.4">MATRIX_3:4</a></span>;<br/><a NAME="E13:73_1_1_2"/><span class="kw">then </span>
 <a href="matrix15.html#K3" title="MATRIX15:func.3">ColVec2Mx</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">f</font> <a href="fvsum_1.html#K4" title="FVSUM_1:func.4">+</a> <font color="Maroon" title="c12">g</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> (<font color="Maroon" title="c2">A</font>,<span class="p1">(<span class="default"><a href="matrix_3.html#K1" title="MATRIX_3:func.1">0.</a> (<font color="Maroon" title="c1">K</font>,<font color="Maroon" title="c3">k</font>,1)</span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:73_1_1_2"><span class="lab"><font color="Green" title="E62">A10</font></span></a>, <a class="txt" href="#E9:73_1_1_2"><span class="lab"><font color="Green" title="E65">A13</font></span></a>, <a class="ref" href="matrix15.html#T28" target="_self" title="MATRIX15:th.28">Th28</a>, <a class="ref" href="matrix15.html#T59" target="_self" title="MATRIX15:th.59">Th59</a></span>;<br/><a NAME="E14:73_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c11">f</font> <a href="fvsum_1.html#K4" title="FVSUM_1:func.4">+</a> <font color="Maroon" title="c12">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix15.html#K6" title="MATRIX15:func.6">Solutions_of</a> (<font color="Maroon" title="c2">A</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="prvect_1.html#K4" title="PRVECT_1: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>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:73_1_1"><span class="lab"><font color="Green" title="E57">A1</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E15:73_1_1_2"/>
<font color="Maroon" title="c7">v</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c8">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix15.html#K6" title="MATRIX15:func.6">Solutions_of</a> (<font color="Maroon" title="c2">A</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="prvect_1.html#K4" title="PRVECT_1: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>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:73_1_1_2"><span class="lab"><font color="Green" title="E61">A9</font></span></a>, <a class="txt" href="#E7:73_1_1_2"><span class="lab"><font color="Green" title="E63">A11</font></span></a>, <a class="ref" href="matrix13.html#T102" title="MATRIX13:th.102">MATRIX13:102</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E4:73_1_1"/>
 for <font color="Olive">b<sub>1</sub></font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="prvect_1.html#K7" title="PRVECT_1:func.7">-VectSp_over</a> <font color="Maroon" title="c1">K</font></span>)</span>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix15.html#K6" title="MATRIX15:func.6">Solutions_of</a> (<font color="Maroon" title="c2">A</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="prvect_1.html#K4" title="PRVECT_1: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>) holds <br/><font color="Olive">b<sub>1</sub></font> is  <a href="vectsp_4.html#V1" title="VECTSP_4:attr.1">linearly-closed</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:73_1_1"><span class="lab"><font color="Green" title="E58">A2</font></span></a>, <a class="ref" href="vectsp_4.html#D1" title="VECTSP_4:def.1">VECTSP_4:def 1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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