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<div class="add">

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">n</font> <a href="matrix_1.html#K10" title="MATRIX_1:func.10">-Matrices_over</a> <font color="Maroon" title="c1">F</font>,   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">n</font> <a href="matrix_1.html#K10" title="MATRIX_1:func.10">-Matrices_over</a> <font color="Maroon" title="c1">F</font>,   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">n</font> <a href="matrix_1.html#K10" title="MATRIX_1:func.10">-Matrices_over</a> <font color="Maroon" title="c1">F</font>] <b>means </b> ex <font color="Olive" title="b1">A</font>, <font color="Olive" title="b2">B</font> being  <a href="matrix_1.html#NM5" title="MATRIX_1:NM.5">Matrix</a> of <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">F</font> st <br/>( $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">A</font> &amp; $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">B</font> &amp; $3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">A</font> <a href="matrix_1.html#K14" title="MATRIX_1:func.14">+</a> <font color="Olive" title="b2">B</font> );<br/>
<a NAME="E1:59_1_1"/><i><font color="Green" title="E37">A1</font></i>: 
for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">n</font> <a href="matrix_1.html#K10" title="MATRIX_1:func.10">-Matrices_over</a> <font color="Maroon" title="c1">F</font>  ex <font color="Olive" title="b3">c</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">n</font> <a href="matrix_1.html#K10" title="MATRIX_1:func.10">-Matrices_over</a> <font color="Maroon" title="c1">F</font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>,<font color="Olive" title="b3">c</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">a</font>, <font color="Maroon" title="c4">b</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">n</font> <a href="matrix_1.html#K10" title="MATRIX_1:func.10">-Matrices_over</a> <font color="Maroon" title="c1">F</font>;<br/>

<b>reconsider </b><font color="Maroon" title="c5">A</font> = <font color="Maroon" title="c3">a</font> as   <a href="matrix_1.html#NM5" title="MATRIX_1:NM.5">Matrix</a> of <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">F</font> <b>by </b><i><a class="ref" href="matrix_1.html#T31" target="_self" title="MATRIX_1:th.31">Th31</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c6">B</font> = <font color="Maroon" title="c4">b</font> as   <a href="matrix_1.html#NM5" title="MATRIX_1:NM.5">Matrix</a> of <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">F</font> <b>by </b><i><a class="ref" href="matrix_1.html#T31" target="_self" title="MATRIX_1:th.31">Th31</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">c</font> = <font color="Maroon" title="c5">A</font> <a href="matrix_1.html#K14" title="MATRIX_1:func.14">+</a> <font color="Maroon" title="c6">B</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">n</font> <a href="matrix_1.html#K10" title="MATRIX_1:func.10">-Matrices_over</a> <font color="Maroon" title="c1">F</font> <b>by </b><i><a class="ref" href="matrix_1.html#T31" target="_self" title="MATRIX_1:th.31">Th31</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c7">c</font>
;<br/>

<b>thus </b><a NAME="E4:59_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c7">c</font>]
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c3">h</font> being   <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <font color="Maroon" title="c2">n</font> <a href="matrix_1.html#K10" title="MATRIX_1:func.10">-Matrices_over</a> <font color="Maroon" title="c1">F</font><b> such that </b><br/><a NAME="E3:59_1_1"/><i><font color="Green" title="E38">A2</font></i>: 
for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">n</font> <a href="matrix_1.html#K10" title="MATRIX_1:func.10">-Matrices_over</a> <font color="Maroon" title="c1">F</font> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>,<font color="Maroon" title="c3">h</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>]
 <b>from </b><i><a class="ref" href="binop_1.html#S3" title="BINOP_1:sch.3">BINOP_1:sch 3</a>(<a class="txt" href="matrix_1.html#E1:59_1_1"><i><font color="Green" title="E37">A1</font></i></a>);<br/></i>
<a NAME="E4:59_1_1"/><i><font color="Green" title="E39">A3</font></i>: 
for <font color="Olive" title="b1">A</font>, <font color="Olive" title="b2">B</font> being  <a href="matrix_1.html#NM5" title="MATRIX_1:NM.5">Matrix</a> of <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">F</font> holds  <font color="Maroon" title="c3">h</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive" title="b1">A</font>,<font color="Olive" title="b2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">A</font> <a href="matrix_1.html#K14" title="MATRIX_1:func.14">+</a> <font color="Olive" title="b2">B</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">A</font>, <font color="Maroon" title="c5">B</font> be   <a href="matrix_1.html#NM5" title="MATRIX_1:NM.5">Matrix</a> of <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">F</font>;<br/>



<a NAME="E1:59_1_1_2"/>
( <font color="Maroon" title="c4">A</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">n</font> <a href="matrix_1.html#K10" title="MATRIX_1:func.10">-Matrices_over</a> <font color="Maroon" title="c1">F</font> &amp; <font color="Maroon" title="c5">B</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">n</font> <a href="matrix_1.html#K10" title="MATRIX_1:func.10">-Matrices_over</a> <font color="Maroon" title="c1">F</font> )
 
<b>by </b><i><a class="ref" href="matrix_1.html#T31" target="_self" title="MATRIX_1:th.31">Th31</a></i>;<br/>
<a NAME="E2:59_1_1_2"/><b>then </b>
 ex <font color="Olive" title="b1">A'</font>, <font color="Olive" title="b2">B'</font> being  <a href="matrix_1.html#NM5" title="MATRIX_1:NM.5">Matrix</a> of <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">F</font> st <br/>( <font color="Maroon" title="c4">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">A'</font> &amp; <font color="Maroon" title="c5">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">B'</font> &amp; <font color="Maroon" title="c3">h</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c5">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">A'</font> <a href="matrix_1.html#K14" title="MATRIX_1:func.14">+</a> <font color="Olive" title="b2">B'</font> )
 
<b>by </b><i><a class="txt" href="matrix_1.html#E3:59_1_1"><i><font color="Green" title="E38">A2</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:59_1_1_2"/>
<font color="Maroon" title="c3">h</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c5">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">A</font> <a href="matrix_1.html#K14" title="MATRIX_1:func.14">+</a> <font color="Maroon" title="c5">B</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E5:59_1_1"/><i><font color="Green" title="E40">A4</font></i>: 
for <font color="Olive" title="b1">a</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">n</font> <a href="matrix_1.html#K10" title="MATRIX_1:func.10">-Matrices_over</a> <font color="Maroon" title="c1">F</font>  ex <font color="Olive" title="b2">b</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">n</font> <a href="matrix_1.html#K10" title="MATRIX_1:func.10">-Matrices_over</a> <font color="Maroon" title="c1">F</font> ex <font color="Olive" title="b3">A</font> being  <a href="matrix_1.html#NM5" title="MATRIX_1:NM.5">Matrix</a> of <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">F</font> st <br/>( <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">A</font> &amp; <font color="Olive" title="b2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K13" title="MATRIX_1:func.13">-</a> <font color="Olive" title="b3">A</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">a</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">n</font> <a href="matrix_1.html#K10" title="MATRIX_1:func.10">-Matrices_over</a> <font color="Maroon" title="c1">F</font>;<br/>

<b>reconsider </b><font color="Maroon" title="c5">A</font> = <font color="Maroon" title="c4">a</font> as   <a href="matrix_1.html#NM5" title="MATRIX_1:NM.5">Matrix</a> of <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">F</font> <b>by </b><i><a class="ref" href="matrix_1.html#T31" target="_self" title="MATRIX_1:th.31">Th31</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c6">b</font> =  <a href="matrix_1.html#K13" title="MATRIX_1:func.13">-</a> <font color="Maroon" title="c5">A</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">n</font> <a href="matrix_1.html#K10" title="MATRIX_1:func.10">-Matrices_over</a> <font color="Maroon" title="c1">F</font> <b>by </b><i><a class="ref" href="matrix_1.html#T31" target="_self" title="MATRIX_1:th.31">Th31</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c6">b</font>
;<br/>

<b>thus </b><a NAME="E3:59_1_1_3"/>
 ex <font color="Olive" title="b1">A</font> being  <a href="matrix_1.html#NM5" title="MATRIX_1:NM.5">Matrix</a> of <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">F</font> st <br/>( <font color="Maroon" title="c4">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">A</font> &amp; <font color="Maroon" title="c6">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K13" title="MATRIX_1:func.13">-</a> <font color="Olive" title="b1">A</font> )
 ;<br/>


</div><b>end;</b></div>
<b>reconsider </b><font color="Maroon" title="c4">A0</font> =  <a href="matrix_1.html#K11" title="MATRIX_1:func.11">0.</a> <font color="Maroon" title="c1">F</font>,<font color="Maroon" title="c2">n</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">n</font> <a href="matrix_1.html#K10" title="MATRIX_1:func.10">-Matrices_over</a> <font color="Maroon" title="c1">F</font> <b>by </b><i><a class="ref" href="matrix_1.html#T31" target="_self" title="MATRIX_1:th.31">Th31</a></i>;<br/>
<b>set </b><font color="Maroon" title="c5">G</font> =  <a href="rlvect_1.html#G1" title="RLVECT_1:aggr.1">LoopStr</a>(# <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="matrix_1.html#K10" title="MATRIX_1:func.10">-Matrices_over</a> <font color="Maroon" title="c1">F</font></span>)</span>,<font color="Maroon" title="c3">h</font>,<font color="Maroon" title="c4">A0</font> #);<br/>
<a NAME="E7:59_1_1"/>
(  <a href="rlvect_1.html#G1" title="RLVECT_1:aggr.1">LoopStr</a>(# <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="matrix_1.html#K10" title="MATRIX_1:func.10">-Matrices_over</a> <font color="Maroon" title="c1">F</font></span>)</span>,<font color="Maroon" title="c3">h</font>,<font color="Maroon" title="c4">A0</font> #) is <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">Abelian</a> &amp;  <a href="rlvect_1.html#G1" title="RLVECT_1:aggr.1">LoopStr</a>(# <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="matrix_1.html#K10" title="MATRIX_1:func.10">-Matrices_over</a> <font color="Maroon" title="c1">F</font></span>)</span>,<font color="Maroon" title="c3">h</font>,<font color="Maroon" title="c4">A0</font> #) is <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">add-associative</a> &amp;  <a href="rlvect_1.html#G1" title="RLVECT_1:aggr.1">LoopStr</a>(# <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="matrix_1.html#K10" title="MATRIX_1:func.10">-Matrices_over</a> <font color="Maroon" title="c1">F</font></span>)</span>,<font color="Maroon" title="c3">h</font>,<font color="Maroon" title="c4">A0</font> #) is <a href="rlvect_1.html#V5" title="RLVECT_1:attr.5">right_zeroed</a> &amp;  <a href="rlvect_1.html#G1" title="RLVECT_1:aggr.1">LoopStr</a>(# <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="matrix_1.html#K10" title="MATRIX_1:func.10">-Matrices_over</a> <font color="Maroon" title="c1">F</font></span>)</span>,<font color="Maroon" title="c3">h</font>,<font color="Maroon" title="c4">A0</font> #) is <a href="rlvect_1.html#V6" title="RLVECT_1:attr.6">right_complementable</a> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1_1_4"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:59_1_1_4"/>
 <a href="rlvect_1.html#G1" title="RLVECT_1:aggr.1">LoopStr</a>(# <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="matrix_1.html#K10" title="MATRIX_1:func.10">-Matrices_over</a> <font color="Maroon" title="c1">F</font></span>)</span>,<font color="Maroon" title="c3">h</font>,<font color="Maroon" title="c4">A0</font> #) is <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">Abelian</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1_1_4_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">a</font>, <font color="Maroon" title="c7">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="rlvect_1.html#G1" title="RLVECT_1:aggr.1">LoopStr</a>(# <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="matrix_1.html#K10" title="MATRIX_1:func.10">-Matrices_over</a> <font color="Maroon" title="c1">F</font></span>)</span>,<font color="Maroon" title="c3">h</font>,<font color="Maroon" title="c4">A0</font> #); <i><font color="Red">:: according to </font></i><a class="ref" href="rlvect_1.html#D5" title="RLVECT_1:def.5">RLVECT_1:def 5</a><br/>

<b>reconsider </b><font color="Maroon" title="c8">A</font> = <font color="Maroon" title="c6">a</font>, <font color="Maroon" title="c9">B</font> = <font color="Maroon" title="c7">b</font> as   <a href="matrix_1.html#NM5" title="MATRIX_1:NM.5">Matrix</a> of <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">F</font> <b>by </b><i><a class="ref" href="matrix_1.html#T31" target="_self" title="MATRIX_1:th.31">Th31</a></i>;<br/>
<b>thus </b><font color="Maroon" title="c6">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Maroon" title="c7">b</font> = 
<font color="Maroon" title="c8">A</font> <a href="matrix_1.html#K14" title="MATRIX_1:func.14">+</a> <font color="Maroon" title="c9">B</font>
<b>by </b><i><a class="txt" href="matrix_1.html#E4:59_1_1"><i><font color="Green" title="E39">A3</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c9">B</font> <a href="matrix_1.html#K14" title="MATRIX_1:func.14">+</a> <font color="Maroon" title="c8">A</font>
<b>by </b><i><a class="ref" href="matrix_1.html#T32" target="_self" title="MATRIX_1:th.32">Th32</a></i>
<br/>.= 
<font color="Maroon" title="c7">b</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Maroon" title="c6">a</font>
<b>by </b><i><a class="txt" href="matrix_1.html#E4:59_1_1"><i><font color="Green" title="E39">A3</font></i></a></i>
;<br/>


</div><b>end;</b></div>

<div><b>hereby </b> <i><font color="Red">:: according to </font></i><a class="ref" href="rlvect_1.html#D6" title="RLVECT_1:def.6">RLVECT_1:def 6</a>
<div class="add"><b>let </b><font color="Maroon" title="c6">a</font>, <font color="Maroon" title="c7">b</font>, <font color="Maroon" title="c8">c</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="rlvect_1.html#G1" title="RLVECT_1:aggr.1">LoopStr</a>(# <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="matrix_1.html#K10" title="MATRIX_1:func.10">-Matrices_over</a> <font color="Maroon" title="c1">F</font></span>)</span>,<font color="Maroon" title="c3">h</font>,<font color="Maroon" title="c4">A0</font> #);<br/><b>reconsider </b><font color="Maroon" title="c9">A</font> = <font color="Maroon" title="c6">a</font>, <font color="Maroon" title="c10">B</font> = <font color="Maroon" title="c7">b</font>, <font color="Maroon" title="c11">C</font> = <font color="Maroon" title="c8">c</font> as   <a href="matrix_1.html#NM5" title="MATRIX_1:NM.5">Matrix</a> of <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">F</font> <b>by </b><i><a class="ref" href="matrix_1.html#T31" target="_self" title="MATRIX_1:th.31">Th31</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c6">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Maroon" title="c7">b</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Maroon" title="c8">c</font> = 
<font color="Maroon" title="c3">h</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">A</font> <a href="matrix_1.html#K14" title="MATRIX_1:func.14">+</a> <font color="Maroon" title="c10">B</font></span>)</span>,<font color="Maroon" title="c11">C</font>
<b>by </b><i><a class="txt" href="matrix_1.html#E4:59_1_1"><i><font color="Green" title="E39">A3</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">A</font> <a href="matrix_1.html#K14" title="MATRIX_1:func.14">+</a> <font color="Maroon" title="c10">B</font></span>)</span> <a href="matrix_1.html#K14" title="MATRIX_1:func.14">+</a> <font color="Maroon" title="c11">C</font>
<b>by </b><i><a class="txt" href="matrix_1.html#E4:59_1_1"><i><font color="Green" title="E39">A3</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c9">A</font> <a href="matrix_1.html#K14" title="MATRIX_1:func.14">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">B</font> <a href="matrix_1.html#K14" title="MATRIX_1:func.14">+</a> <font color="Maroon" title="c11">C</font></span>)</span>
<b>by </b><i><a class="ref" href="matrix_1.html#T33" target="_self" title="MATRIX_1:th.33">Th33</a></i>
<br/>.= 
<font color="Maroon" title="c3">h</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c9">A</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">B</font> <a href="matrix_1.html#K14" title="MATRIX_1:func.14">+</a> <font color="Maroon" title="c11">C</font></span>)</span>
<b>by </b><i><a class="txt" href="matrix_1.html#E4:59_1_1"><i><font color="Green" title="E39">A3</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c6">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">b</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Maroon" title="c8">c</font></span>)</span>
<b>by </b><i><a class="txt" href="matrix_1.html#E4:59_1_1"><i><font color="Green" title="E39">A3</font></i></a></i>
;<br/></div>
<b>end;</b></div>

<div><b>hereby </b> <i><font color="Red">:: according to </font></i><a class="ref" href="rlvect_1.html#D7" title="RLVECT_1:def.7">RLVECT_1:def 7</a>
<div class="add"><b>let </b><font color="Maroon" title="c6">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="rlvect_1.html#G1" title="RLVECT_1:aggr.1">LoopStr</a>(# <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="matrix_1.html#K10" title="MATRIX_1:func.10">-Matrices_over</a> <font color="Maroon" title="c1">F</font></span>)</span>,<font color="Maroon" title="c3">h</font>,<font color="Maroon" title="c4">A0</font> #);<br/><b>reconsider </b><font color="Maroon" title="c7">A</font> = <font color="Maroon" title="c6">a</font> as   <a href="matrix_1.html#NM5" title="MATRIX_1:NM.5">Matrix</a> of <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">F</font> <b>by </b><i><a class="ref" href="matrix_1.html#T31" target="_self" title="MATRIX_1:th.31">Th31</a></i>;<br/><b>thus </b><font color="Maroon" title="c6">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="rlvect_1.html#G1" title="RLVECT_1:aggr.1">LoopStr</a>(# <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="matrix_1.html#K10" title="MATRIX_1:func.10">-Matrices_over</a> <font color="Maroon" title="c1">F</font></span>)</span>,<font color="Maroon" title="c3">h</font>,<font color="Maroon" title="c4">A0</font> #)</span>)</span> = 
<font color="Maroon" title="c7">A</font> <a href="matrix_1.html#K14" title="MATRIX_1:func.14">+</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K11" title="MATRIX_1:func.11">0.</a> <font color="Maroon" title="c1">F</font>,<font color="Maroon" title="c2">n</font></span>)</span>
<b>by </b><i><a class="txt" href="matrix_1.html#E4:59_1_1"><i><font color="Green" title="E39">A3</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c6">a</font>
<b>by </b><i><a class="ref" href="matrix_1.html#T34" target="_self" title="MATRIX_1:th.34">Th34</a></i>
;<br/></div>
<b>end;</b></div>

<b>let </b><font color="Maroon" title="c6">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="rlvect_1.html#G1" title="RLVECT_1:aggr.1">LoopStr</a>(# <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="matrix_1.html#K10" title="MATRIX_1:func.10">-Matrices_over</a> <font color="Maroon" title="c1">F</font></span>)</span>,<font color="Maroon" title="c3">h</font>,<font color="Maroon" title="c4">A0</font> #); <i><font color="Red">:: according to </font></i><a class="ref" href="rlvect_1.html#D8" title="RLVECT_1:def.8">RLVECT_1:def 8</a><br/>

<b>consider </b><font color="Maroon" title="c7">b</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">n</font> <a href="matrix_1.html#K10" title="MATRIX_1:func.10">-Matrices_over</a> <font color="Maroon" title="c1">F</font><b> such that </b><br/><a NAME="E5:59_1_1_4"/><i><font color="Green" title="E41">A5</font></i>: 
 ex <font color="Olive" title="b1">A</font> being  <a href="matrix_1.html#NM5" title="MATRIX_1:NM.5">Matrix</a> of <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">F</font> st <br/>( <font color="Maroon" title="c6">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">A</font> &amp; <font color="Maroon" title="c7">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K13" title="MATRIX_1:func.13">-</a> <font color="Olive" title="b1">A</font> )
 <b>by </b><i><a class="txt" href="matrix_1.html#E5:59_1_1"><i><font color="Green" title="E40">A4</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c8">A</font> being   <a href="matrix_1.html#NM5" title="MATRIX_1:NM.5">Matrix</a> of <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">F</font><b> such that </b><br/><a NAME="E7:59_1_1_4"/><i><font color="Green" title="E42">A6</font></i>: 
( <font color="Maroon" title="c6">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">A</font> &amp; <font color="Maroon" title="c7">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K13" title="MATRIX_1:func.13">-</a> <font color="Maroon" title="c8">A</font> )
 <b>by </b><i><a class="txt" href="matrix_1.html#E5:59_1_1_4"><i><font color="Green" title="E41">A5</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c9">b</font> =  <a href="matrix_1.html#K13" title="MATRIX_1:func.13">-</a> <font color="Maroon" title="c8">A</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="rlvect_1.html#G1" title="RLVECT_1:aggr.1">LoopStr</a>(# <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="matrix_1.html#K10" title="MATRIX_1:func.10">-Matrices_over</a> <font color="Maroon" title="c1">F</font></span>)</span>,<font color="Maroon" title="c3">h</font>,<font color="Maroon" title="c4">A0</font> #) <b>by </b><i><a class="txt" href="matrix_1.html#E7:59_1_1_4"><i><font color="Green" title="E42">A6</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c9">b</font>
;<br/>

<b>thus </b><font color="Maroon" title="c6">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Maroon" title="c9">b</font> = 
<font color="Maroon" title="c8">A</font> <a href="matrix_1.html#K14" title="MATRIX_1:func.14">+</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K13" title="MATRIX_1:func.13">-</a> <font color="Maroon" title="c8">A</font></span>)</span>
<b>by </b><i><a class="txt" href="matrix_1.html#E4:59_1_1"><i><font color="Green" title="E39">A3</font></i></a>, <a class="txt" href="matrix_1.html#E7:59_1_1_4"><i><font color="Green" title="E42">A6</font></i></a></i>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="rlvect_1.html#G1" title="RLVECT_1:aggr.1">LoopStr</a>(# <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="matrix_1.html#K10" title="MATRIX_1:func.10">-Matrices_over</a> <font color="Maroon" title="c1">F</font></span>)</span>,<font color="Maroon" title="c3">h</font>,<font color="Maroon" title="c4">A0</font> #)
<b>by </b><i><a class="ref" href="matrix_1.html#T35" target="_self" title="MATRIX_1:th.35">Th35</a></i>
;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c6">G</font> =  <a href="rlvect_1.html#G1" title="RLVECT_1:aggr.1">LoopStr</a>(# <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="matrix_1.html#K10" title="MATRIX_1:func.10">-Matrices_over</a> <font color="Maroon" title="c1">F</font></span>)</span>,<font color="Maroon" title="c3">h</font>,<font color="Maroon" title="c4">A0</font> #) as  <a href="rlvect_1.html#V1" title="RLVECT_1:attr.1">strict</a> <a href="vectsp_1.html#NM2" title="VECTSP_1:NM.2">AbGroup</a> ;<br/>
<b>take </b>
<font color="Maroon" title="c6">G</font>
;<br/>

<b>thus </b><a NAME="E9:59_1_1"/>
( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c6">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">n</font> <a href="matrix_1.html#K10" title="MATRIX_1:func.10">-Matrices_over</a> <font color="Maroon" title="c1">F</font> &amp; ( for <font color="Olive" title="b1">A</font>, <font color="Olive" title="b2">B</font> being  <a href="matrix_1.html#NM5" title="MATRIX_1:NM.5">Matrix</a> of <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">F</font> holds  the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">add</a> of <font color="Maroon" title="c6">G</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive" title="b1">A</font>,<font color="Olive" title="b2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">A</font> <a href="matrix_1.html#K14" title="MATRIX_1:func.14">+</a> <font color="Olive" title="b2">B</font> ) &amp;  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c6">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K11" title="MATRIX_1:func.11">0.</a> <font color="Maroon" title="c1">F</font>,<font color="Maroon" title="c2">n</font> )
 <b>by </b><i><a class="txt" href="matrix_1.html#E4:59_1_1"><i><font color="Green" title="E39">A3</font></i></a></i>;<br/>


</div>
