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<b>set </b><font color="Maroon" title="c4">k0</font> = <font color="Maroon" title="c3">k</font> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c5">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="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font></span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c6">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="matrix13.html#K13" title="MATRIX13:func.13">-VectSp_over</a> <font color="Maroon" title="c1">K</font>;<br/>
<a NAME="E1:73_1_1"/><i><font color="Green" title="E57">A1</font></i>: 
 <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="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="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
 
<b>by </b><i><a class="ref" href="matrix15.html#T32" target="_self" title="MATRIX15:th.32">Th32</a></i>;<br/>
<div><i><font color="Green" title="E58">A2</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="73_1_1_1"><b>now </b></a><div class="add"><b>let </b><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="matrix13.html#K13" title="MATRIX13:func.13">-VectSp_over</a> <font color="Maroon" title="c1">K</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="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="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font></span>)</span></span>)</span> &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="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font></span>)</span></span>)</span> 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="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font></span>)</span></span>)</span> )</span><br/><b>assume </b><a NAME="E1:73_1_1_1"/><i><font color="Green" title="E59">A3</font></i>: 
( <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="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font></span>)</span></span>)</span> &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="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font></span>)</span></span>)</span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font></span>)</span></span>)</span></span><br/><b>consider </b><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><b> such that </b><br/><a NAME="E3:73_1_1_1"/><i><font color="Green" title="E60">A4</font></i>: 
( <font color="Maroon" title="c7">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">f</font> &amp;  <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="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <span class="p3">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font></span>)</span></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="matrix15.html#E1:73_1_1_1"><i><font color="Green" title="E59">A3</font></i></a></i>;<br/><b>consider </b><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><b> such that </b><br/><a NAME="E5:73_1_1_1"/><i><font color="Green" title="E61">A5</font></i>: 
( <font color="Maroon" title="c8">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">g</font> &amp;  <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="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <span class="p3">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font></span>)</span></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="matrix15.html#E1:73_1_1_1"><i><font color="Green" title="E59">A3</font></i></a></i>;<br/><a NAME="E6:73_1_1_1"/><i><font color="Green" title="E62">A6</font></i>: 
(  <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> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">f</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> )
 <b>by </b><i><a class="txt" href="matrix15.html#E3:73_1_1_1"><i><font color="Green" title="E60">A4</font></i></a>, <a class="txt" href="matrix15.html#E5:73_1_1_1"><i><font color="Green" title="E61">A5</font></i></a>, <a class="ref" href="matrix15.html#T59" target="_self" title="MATRIX15:th.59">Th59</a></i>;<br/><b>reconsider </b><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> <b>by </b><i><a class="txt" href="matrix15.html#E3:73_1_1_1"><i><font color="Green" title="E60">A4</font></i></a>, <a class="txt" href="matrix15.html#E5:73_1_1_1"><i><font color="Green" title="E61">A5</font></i></a>, <a class="ref" href="matrix13.html#T102" title="MATRIX13:th.102">MATRIX13:102</a></i>;<br/><a NAME="E8:73_1_1_1"/>
<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>
 <b>by </b><i><a class="txt" href="matrix15.html#E3:73_1_1_1"><i><font color="Green" title="E60">A4</font></i></a>, <a class="txt" href="matrix15.html#E5:73_1_1_1"><i><font color="Green" title="E61">A5</font></i></a>, <a class="ref" href="matrix15.html#T37" target="_self" title="MATRIX15:th.37">Th37</a>, <a class="txt" href="matrix15.html#E1:73_1_1"><i><font color="Green" title="E57">A1</font></i></a></i>;<br/><a NAME="E9:73_1_1_1"/><b>then </b>
<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>
 <b>by </b><i><a class="ref" href="matrix_3.html#T6" title="MATRIX_3:th.6">MATRIX_3:6</a></i>;<br/><a NAME="E10:73_1_1_1"/><b>then </b>
 <a href="matrix15.html#K3" title="MATRIX15:func.3">ColVec2Mx</a> <span class="p1">(<span class="default"><font color="Maroon" title="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>
 <b>by </b><i><a class="txt" href="matrix15.html#E6:73_1_1_1"><i><font color="Green" title="E62">A6</font></i></a>, <a class="ref" href="matrix15.html#T28" target="_self" title="MATRIX15:th.28">Th28</a></i>;<br/><a NAME="E11:73_1_1_1"/><b>then </b>
<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="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="matrix15.html#E1:73_1_1"><i><font color="Green" title="E57">A1</font></i></a></i>;<br/><b>hence </b><a NAME="E12:73_1_1_1"/>
<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="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="matrix15.html#E3:73_1_1_1"><i><font color="Green" title="E60">A4</font></i></a>, <a class="txt" href="matrix15.html#E5:73_1_1_1"><i><font color="Green" title="E61">A5</font></i></a>, <a class="ref" href="matrix13.html#T102" title="MATRIX13:th.102">MATRIX13:102</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_1_1_2"><b>now </b></a><div class="add"><b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></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="matrix13.html#K13" title="MATRIX13:func.13">-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="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font></span>)</span></span>)</span> 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="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font></span>)</span></span>)</span></span><br/><b>let </b><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="matrix13.html#K13" title="MATRIX13:func.13">-VectSp_over</a> <font color="Maroon" title="c1">K</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="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="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font></span>)</span></span>)</span> 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="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font></span>)</span></span>)</span> )</span><br/><b>assume </b><a NAME="E1:73_1_1_2"/><i><font color="Green" title="E59">A7</font></i>: 
<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="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font></span>)</span></span>)</span></span><br/><b>consider </b><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><b> such that </b><br/><a NAME="E3:73_1_1_2"/><i><font color="Green" title="E60">A8</font></i>: 
( <font color="Maroon" title="c8">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">f</font> &amp;  <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="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <span class="p3">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font></span>)</span></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="matrix15.html#E1:73_1_1_2"><i><font color="Green" title="E59">A7</font></i></a></i>;<br/><b>reconsider </b><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> <b>by </b><i><a class="txt" href="matrix15.html#E3:73_1_1_2"><i><font color="Green" title="E60">A8</font></i></a>, <a class="ref" href="matrix13.html#T102" title="MATRIX13:th.102">MATRIX13:102</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="73_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:73_1_1_2_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_2_1_1_1"><b>suppose </b></a><a NAME="E1:73_1_1_2_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:()"><i><font color="Red">::  thesis: </font></i></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_2_1_1_1"/><b>then </b>
(  <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>  <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> <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> )
 <b>by </b><i><a class="ref" href="matrix_1.html#T23" title="MATRIX_1:th.23">MATRIX_1:23</a>, <a class="ref" href="matrix_3.html#D5" title="MATRIX_3:def.5">MATRIX_3:def 5</a></i>;<br/><a NAME="E3:73_1_1_2_1_1_1"/><b>then </b>
(  <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>  &amp; <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="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 <b>by </b><i><a class="ref" href="card_2.html#T59" title="CARD_2:th.59">CARD_2:59</a></i>;<br/><b>hence </b><a NAME="E4:73_1_1_2_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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="73_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:73_1_1_2_1_1_2"/><i><font color="Green" title="E61">A9</font></i>: 
<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:()"><i><font color="Red">::  thesis: </font></i></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_2_1_1_2"/><b>then </b><i><font color="Green" title="E62">A10</font></i>: 
(  <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 )
 <b>by </b><i><a class="ref" href="matrix_1.html#T24" title="MATRIX_1:th.24">MATRIX_1:24</a></i>;<br/><b>hence </b><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>
<b>by </b><i><a class="txt" href="matrix15.html#E1:73_1_1_2_1_1_2"><i><font color="Green" title="E61">A9</font></i></a>, <a class="ref" href="matrix_5.html#T40" title="MATRIX_5:th.40">MATRIX_5:40</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">a</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</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>
<b>by </b><i><a class="ref" href="matrix_5.html#T12" title="MATRIX_5:th.12">MATRIX_5:12</a></i>
<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>
<b>by </b><i><a class="ref" href="vectsp_1.html#T36" title="VECTSP_1:th.36">VECTSP_1:36</a></i>
<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
<b>by </b><i><a class="txt" href="matrix15.html#E1:73_1_1_2_1_1_2"><i><font color="Green" title="E61">A9</font></i></a>, <a class="txt" href="matrix15.html#E2:73_1_1_2_1_1_2"><i><font color="Green" title="E62">A10</font></i></a>, <a class="ref" href="matrix_5.html#T40" title="MATRIX_5:th.40">MATRIX_5:40</a></i>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><a NAME="E6:73_1_1_2"/><b>then </b>
<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>
 <b>by </b><i><a class="txt" href="matrix15.html#E3:73_1_1_2"><i><font color="Green" title="E60">A8</font></i></a>, <a class="txt" href="matrix15.html#E1:73_1_1"><i><font color="Green" title="E57">A1</font></i></a>, <a class="ref" href="matrix15.html#T35" target="_self" title="MATRIX15:th.35">Th35</a></i>;<br/><a NAME="E7:73_1_1_2"/><b>then </b>
 <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>
 <b>by </b><i><a class="ref" href="matrix15.html#T30" target="_self" title="MATRIX15:th.30">Th30</a></i>;<br/><a NAME="E8:73_1_1_2"/><b>then </b>
<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="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="matrix15.html#E1:73_1_1"><i><font color="Green" title="E57">A1</font></i></a></i>;<br/><b>hence </b><a NAME="E9:73_1_1_2"/>
<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="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="matrix15.html#E3:73_1_1_2"><i><font color="Green" title="E60">A8</font></i></a>, <a class="ref" href="matrix13.html#T102" title="MATRIX13:th.102">MATRIX13:102</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:73_1_1"/>
 <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="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font></span>)</span></span>)</span> is  <a href="vectsp_4.html#V1" title="VECTSP_4:attr.1">linearly-closed</a> <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="matrix13.html#K13" title="MATRIX13:func.13">-VectSp_over</a> <font color="Maroon" title="c1">K</font></span>)</span>
 <b>by </b><i><a class="txt" href="matrix15.html#E2:73_1_1"><i><font color="Green" title="E58">A2</font></i></a>, <a class="ref" href="vectsp_4.html#D1" title="VECTSP_4:def.1">VECTSP_4:def 1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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