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<div class="add">

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>2</sub></font> <a href="matrix_1.html#K10" target="_self">-Matrices_over</a> <font color="Maroon">c<sub>1</sub></font>,   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>2</sub></font> <a href="matrix_1.html#K10" target="_self">-Matrices_over</a> <font color="Maroon">c<sub>1</sub></font>,   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>2</sub></font> <a href="matrix_1.html#K10" target="_self">-Matrices_over</a> <font color="Maroon">c<sub>1</sub></font>] means ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="matrix_1.html#NM5">Matrix</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Maroon">a<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="matrix_1.html#K14" target="_self">+</a> <font color="Olive">b<sub>2</sub></font> );<br/>
<a NAME="E1:59_1_1"/><i><font color="Green">E37</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>2</sub></font> <a href="matrix_1.html#K10" target="_self">-Matrices_over</a> <font color="Maroon">c<sub>1</sub></font>ex <font color="Olive">b<sub>3</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>2</sub></font> <a href="matrix_1.html#K10" target="_self">-Matrices_over</a> <font color="Maroon">c<sub>1</sub></font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></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">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>2</sub></font> <a href="matrix_1.html#K10" target="_self">-Matrices_over</a> <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>3</sub></font> as   <a href="matrix_1.html#NM5">Matrix</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="matrix_1.html#T31" target="_self">Th31</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>4</sub></font> as   <a href="matrix_1.html#NM5">Matrix</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="matrix_1.html#T31" target="_self">Th31</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>5</sub></font> <a href="matrix_1.html#K14" target="_self">+</a> <font color="Maroon">c<sub>6</sub></font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>2</sub></font> <a href="matrix_1.html#K10" target="_self">-Matrices_over</a> <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="matrix_1.html#T31" target="_self">Th31</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>7</sub></font>
;<br/>

<b>thus </b><a NAME="E4:59_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>7</sub></font>]
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">c<sub>3</sub></font> being   <a href="binop_1.html#NM2">BinOp</a> of <font color="Maroon">c<sub>2</sub></font> <a href="matrix_1.html#K10" target="_self">-Matrices_over</a> <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:59_1_1"/><i><font color="Green">E38</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>2</sub></font> <a href="matrix_1.html#K10" target="_self">-Matrices_over</a> <font color="Maroon">c<sub>1</sub></font> holds <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="binop_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>]
 <b>from </b><i><a class="ref" href="binop_1.html#S3">BINOP_1:sch 3</a>(<a class="txt" href="matrix_1.html#E1:59_1_1"><i><font color="Green">E37</font></i></a>);<br/></i>
<a NAME="E4:59_1_1"/><i><font color="Green">E39</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="matrix_1.html#NM5">Matrix</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font> holds <font color="Maroon">c<sub>3</sub></font> <a href="binop_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="matrix_1.html#K14" target="_self">+</a> <font color="Olive">b<sub>2</sub></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">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font> be   <a href="matrix_1.html#NM5">Matrix</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font>;<br/>



<a NAME="E1:59_1_1_2"/>
( <font color="Maroon">c<sub>4</sub></font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>2</sub></font> <a href="matrix_1.html#K10" target="_self">-Matrices_over</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>2</sub></font> <a href="matrix_1.html#K10" target="_self">-Matrices_over</a> <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="ref" href="matrix_1.html#T31" target="_self">Th31</a></i>;<br/>
<a NAME="E2:59_1_1_2"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="matrix_1.html#NM5">Matrix</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="matrix_1.html#K14" target="_self">+</a> <font color="Olive">b<sub>2</sub></font> )
 
<b>by </b><i><a class="txt" href="matrix_1.html#E3:59_1_1"><i><font color="Green">E38</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:59_1_1_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="matrix_1.html#K14" target="_self">+</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E5:59_1_1"/><i><font color="Green">E40</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>2</sub></font> <a href="matrix_1.html#K10" target="_self">-Matrices_over</a> <font color="Maroon">c<sub>1</sub></font>ex <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>2</sub></font> <a href="matrix_1.html#K10" target="_self">-Matrices_over</a> <font color="Maroon">c<sub>1</sub></font>ex <font color="Olive">b<sub>3</sub></font> being  <a href="matrix_1.html#NM5">Matrix</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>3</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K13" target="_self">-</a> <font color="Olive">b<sub>3</sub></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">c<sub>4</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>2</sub></font> <a href="matrix_1.html#K10" target="_self">-Matrices_over</a> <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</sub></font> as   <a href="matrix_1.html#NM5">Matrix</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="matrix_1.html#T31" target="_self">Th31</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>6</sub></font> =  <a href="matrix_1.html#K13" target="_self">-</a> <font color="Maroon">c<sub>5</sub></font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>2</sub></font> <a href="matrix_1.html#K10" target="_self">-Matrices_over</a> <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="matrix_1.html#T31" target="_self">Th31</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>6</sub></font>
;<br/>

<b>thus </b><a NAME="E3:59_1_1_3"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="matrix_1.html#NM5">Matrix</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K13" target="_self">-</a> <font color="Olive">b<sub>1</sub></font> )
 ;<br/>


</div><b>end;</b></div>
<b>reconsider </b><font color="Maroon">c<sub>4</sub></font> =  <a href="matrix_1.html#K11" target="_self">0.</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>2</sub></font> <a href="matrix_1.html#K10" target="_self">-Matrices_over</a> <font color="Maroon">c<sub>1</sub></font> ;<br/>
<b>set </b><font color="Maroon">c<sub>5</sub></font> =  <a href="rlvect_1.html#G1">LoopStr</a>(# <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="matrix_1.html#K10" target="_self">-Matrices_over</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> #);<br/>
<a NAME="E7:59_1_1"/>
(  <a href="rlvect_1.html#G1">LoopStr</a>(# <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="matrix_1.html#K10" target="_self">-Matrices_over</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> #) is <a href="rlvect_1.html#V3">Abelian</a> &amp;  <a href="rlvect_1.html#G1">LoopStr</a>(# <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="matrix_1.html#K10" target="_self">-Matrices_over</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> #) is <a href="rlvect_1.html#V4">add-associative</a> &amp;  <a href="rlvect_1.html#G1">LoopStr</a>(# <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="matrix_1.html#K10" target="_self">-Matrices_over</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> #) is <a href="rlvect_1.html#V5">right_zeroed</a> &amp;  <a href="rlvect_1.html#G1">LoopStr</a>(# <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="matrix_1.html#K10" target="_self">-Matrices_over</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> #) is <a href="rlvect_1.html#V6">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">LoopStr</a>(# <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="matrix_1.html#K10" target="_self">-Matrices_over</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> #) is <a href="rlvect_1.html#V3">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">c<sub>6</sub></font>, <font color="Maroon">c<sub>7</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <a href="rlvect_1.html#G1">LoopStr</a>(# <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="matrix_1.html#K10" target="_self">-Matrices_over</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> #); <i><font color="Red">:: according to </font></i><a class="ref" href="rlvect_1.html#D5">RLVECT_1:def 5</a><br/>

<b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>7</sub></font> as   <a href="matrix_1.html#NM5">Matrix</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="matrix_1.html#T31" target="_self">Th31</a></i>;<br/>
<b>thus </b><font color="Maroon">c<sub>6</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>7</sub></font> = 
<font color="Maroon">c<sub>3</sub></font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T5">RLVECT_1:5</a></i>
<br/>.= 
<font color="Maroon">c<sub>8</sub></font> <a href="matrix_1.html#K14" target="_self">+</a> <font color="Maroon">c<sub>9</sub></font>
<b>by </b><i><a class="txt" href="matrix_1.html#E4:59_1_1"><i><font color="Green">E39</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>9</sub></font> <a href="matrix_1.html#K14" target="_self">+</a> <font color="Maroon">c<sub>8</sub></font>
<b>by </b><i><a class="ref" href="matrix_1.html#T32" target="_self">Th32</a></i>
<br/>.= 
<font color="Maroon">c<sub>3</sub></font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>8</sub></font>
<b>by </b><i><a class="txt" href="matrix_1.html#E4:59_1_1"><i><font color="Green">E39</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>7</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>6</sub></font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T5">RLVECT_1:5</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">RLVECT_1:def 6</a>
<div class="add"><b>let </b><font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>8</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <a href="rlvect_1.html#G1">LoopStr</a>(# <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="matrix_1.html#K10" target="_self">-Matrices_over</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> #);<br/><b>reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>8</sub></font> as   <a href="matrix_1.html#NM5">Matrix</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="matrix_1.html#T31" target="_self">Th31</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>8</sub></font> = 
<font color="Maroon">c<sub>3</sub></font> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>8</sub></font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T5">RLVECT_1:5</a></i>
<br/>.= 
<font color="Maroon">c<sub>3</sub></font> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>8</sub></font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T5">RLVECT_1:5</a></i>
<br/>.= 
<font color="Maroon">c<sub>3</sub></font> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="matrix_1.html#K14" target="_self">+</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>,<font color="Maroon">c<sub>11</sub></font>
<b>by </b><i><a class="txt" href="matrix_1.html#E4:59_1_1"><i><font color="Green">E39</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="matrix_1.html#K14" target="_self">+</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="matrix_1.html#K14" target="_self">+</a> <font color="Maroon">c<sub>11</sub></font>
<b>by </b><i><a class="txt" href="matrix_1.html#E4:59_1_1"><i><font color="Green">E39</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>9</sub></font> <a href="matrix_1.html#K14" target="_self">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="matrix_1.html#K14" target="_self">+</a> <font color="Maroon">c<sub>11</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="matrix_1.html#T33" target="_self">Th33</a></i>
<br/>.= 
<font color="Maroon">c<sub>3</sub></font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="matrix_1.html#K14" target="_self">+</a> <font color="Maroon">c<sub>11</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="matrix_1.html#E4:59_1_1"><i><font color="Green">E39</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>3</sub></font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="matrix_1.html#E4:59_1_1"><i><font color="Green">E39</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>3</sub></font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T5">RLVECT_1:5</a></i>
<br/>.= 
<font color="Maroon">c<sub>6</sub></font> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T5">RLVECT_1:5</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">RLVECT_1:def 7</a>
<div class="add"><b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <a href="rlvect_1.html#G1">LoopStr</a>(# <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="matrix_1.html#K10" target="_self">-Matrices_over</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> #);<br/><b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>6</sub></font> as   <a href="matrix_1.html#NM5">Matrix</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="matrix_1.html#T31" target="_self">Th31</a></i>;<br/><b>thus </b><font color="Maroon">c<sub>6</sub></font> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <a href="rlvect_1.html#G1">LoopStr</a>(# <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="matrix_1.html#K10" target="_self">-Matrices_over</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> #)</span>)</span> = 
<font color="Maroon">c<sub>3</sub></font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font>,<span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <a href="rlvect_1.html#G1">LoopStr</a>(# <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="matrix_1.html#K10" target="_self">-Matrices_over</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> #)</span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T5">RLVECT_1:5</a></i>
<br/>.= 
<font color="Maroon">c<sub>3</sub></font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>4</sub></font>

<br/>.= 
<font color="Maroon">c<sub>7</sub></font> <a href="matrix_1.html#K14" target="_self">+</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K11" target="_self">0.</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="matrix_1.html#E4:59_1_1"><i><font color="Green">E39</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>6</sub></font>
<b>by </b><i><a class="ref" href="matrix_1.html#T34" target="_self">Th34</a></i>
;<br/></div>
<b>end;</b></div>

<b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <a href="rlvect_1.html#G1">LoopStr</a>(# <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="matrix_1.html#K10" target="_self">-Matrices_over</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> #); <i><font color="Red">:: according to </font></i><a class="ref" href="rlvect_1.html#D8">RLVECT_1:def 8</a><br/>

<b>consider </b><font color="Maroon">c<sub>7</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>2</sub></font> <a href="matrix_1.html#K10" target="_self">-Matrices_over</a> <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E5:59_1_1_4"/><i><font color="Green">E41</font></i>: 
ex <font color="Olive">b<sub>1</sub></font> being  <a href="matrix_1.html#NM5">Matrix</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K13" target="_self">-</a> <font color="Olive">b<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="matrix_1.html#E5:59_1_1"><i><font color="Green">E40</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>8</sub></font> being   <a href="matrix_1.html#NM5">Matrix</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E7:59_1_1_4"/><i><font color="Green">E42</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K13" target="_self">-</a> <font color="Maroon">c<sub>8</sub></font> )
 <b>by </b><i><a class="txt" href="matrix_1.html#E5:59_1_1_4"><i><font color="Green">E41</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>9</sub></font> =  <a href="matrix_1.html#K13" target="_self">-</a> <font color="Maroon">c<sub>8</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <a href="rlvect_1.html#G1">LoopStr</a>(# <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="matrix_1.html#K10" target="_self">-Matrices_over</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> #) <b>by </b><i><a class="txt" href="matrix_1.html#E7:59_1_1_4"><i><font color="Green">E42</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>9</sub></font>
;<br/>

<b>thus </b><font color="Maroon">c<sub>6</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>9</sub></font> = 
<font color="Maroon">c<sub>3</sub></font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font>,<span class="p1">(<span class="default"><a href="matrix_1.html#K13" target="_self">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="matrix_1.html#E7:59_1_1_4"><i><font color="Green">E42</font></i></a>, <a class="ref" href="rlvect_1.html#T5">RLVECT_1:5</a></i>
<br/>.= 
<font color="Maroon">c<sub>8</sub></font> <a href="matrix_1.html#K14" target="_self">+</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K13" target="_self">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="matrix_1.html#E4:59_1_1"><i><font color="Green">E39</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="ref" href="matrix_1.html#T35" target="_self">Th35</a></i>
<br/>.= 
 <a href="rlvect_1.html#K1">0.</a> <a href="rlvect_1.html#G1">LoopStr</a>(# <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="matrix_1.html#K10" target="_self">-Matrices_over</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> #)

;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>6</sub></font> =  <a href="rlvect_1.html#G1">LoopStr</a>(# <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="matrix_1.html#K10" target="_self">-Matrices_over</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> #) as  <a href="rlvect_1.html#V1">strict</a> <a href="vectsp_1.html#NM1">AbGroup</a> ;<br/>
<b>take </b>
<font color="Maroon">c<sub>6</sub></font>
;<br/>

<b>thus </b><a NAME="E9:59_1_1"/>
( the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="matrix_1.html#K10" target="_self">-Matrices_over</a> <font color="Maroon">c<sub>1</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="matrix_1.html#NM5">Matrix</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font> holds the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">c<sub>6</sub></font> <a href="binop_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="matrix_1.html#K14" target="_self">+</a> <font color="Olive">b<sub>2</sub></font> ) &amp; the <a href="struct_0.html#U2">Zero</a> of <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K11" target="_self">0.</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="matrix_1.html#E4:59_1_1"><i><font color="Green">E39</font></i></a></i>;<br/>


</div>
