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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="funcsdom.html#NM1">Ring</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="vectsp_2.html#NM6">LeftMod</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<b>set </b><font color="Maroon">c<sub>3</sub></font> =  <a href="mod_4.html#K2" target="_self">opp</a> <font color="Maroon">c<sub>1</sub></font>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>4</sub></font> =  <a href="mod_4.html#K3" target="_self">opp</a> <font color="Maroon">c<sub>2</sub></font> as  non <a href="struct_0.html#V3">empty</a>  <a href="vectsp_2.html#L1">RightModStr</a> of  <a href="mod_4.html#K2" target="_self">opp</a> <font color="Maroon">c<sub>1</sub></font> ;<br/>
<a NAME="E2:37"/><i><font color="Green">E28</font></i>: 
 <a href="rlvect_1.html#G1">LoopStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="mod_4.html#K3" target="_self">opp</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,the <a href="rlvect_1.html#U1">add</a> of <span class="p1">(<span class="default"><a href="mod_4.html#K3" target="_self">opp</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,the <a href="struct_0.html#U2">Zero</a> of <span class="p1">(<span class="default"><a href="mod_4.html#K3" target="_self">opp</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> #) <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#G1">LoopStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>,the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">c<sub>2</sub></font>,the <a href="struct_0.html#U2">Zero</a> of <font color="Maroon">c<sub>2</sub></font> #)
 
<b>by </b><i><a class="ref" href="mod_4.html#T9" target="_self">Th9</a></i>;<br/>
<a NAME="E3:37"/><i><font color="Green">E29</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="mod_4.html#K3" target="_self">opp</a> <font color="Maroon">c<sub>2</sub></font></span>)</span><br/>for <font color="Olive">b<sub>3</sub></font>, <font color="Olive">b<sub>4</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font> st <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>4</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>3</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Olive">b<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="mod_4.html#E2:37"><i><font color="Green">E28</font></i></a></i>;<br/>
<i><font color="Green">E30</font></i>:  <a href="rlvect_1.html#K1">0.</a> <span class="p1">(<span class="default"><a href="mod_4.html#K3" target="_self">opp</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> = 
the <a href="struct_0.html#U2">Zero</a> of <span class="p1">(<span class="default"><a href="mod_4.html#K3" target="_self">opp</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>

<br/>.= 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="txt" href="mod_4.html#E2:37"><i><font color="Green">E28</font></i></a></i>
;<br/>
<a NAME="E5:37"/><i><font color="Green">E31</font></i>: 
(  <a href="mod_4.html#K3" target="_self">opp</a> <font color="Maroon">c<sub>2</sub></font> is <a href="rlvect_1.html#V3">Abelian</a> &amp;  <a href="mod_4.html#K3" target="_self">opp</a> <font color="Maroon">c<sub>2</sub></font> is <a href="rlvect_1.html#V4">add-associative</a> &amp;  <a href="mod_4.html#K3" target="_self">opp</a> <font color="Maroon">c<sub>2</sub></font> is <a href="rlvect_1.html#V5">right_zeroed</a> &amp;  <a href="mod_4.html#K3" target="_self">opp</a> <font color="Maroon">c<sub>2</sub></font> is <a href="rlvect_1.html#V6">right_complementable</a> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:37_2"/>
 <a href="mod_4.html#K3" target="_self">opp</a> <font color="Maroon">c<sub>2</sub></font> is <a href="rlvect_1.html#V3">Abelian</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="mod_4.html#K3" target="_self">opp</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>; <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>7</sub></font> = <font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>6</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="ref" href="mod_4.html#T9" target="_self">Th9</a></i>;<br/>
<b>thus </b><font color="Maroon">c<sub>5</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>6</sub></font> = 
<font color="Maroon">c<sub>8</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>7</sub></font>
<b>by </b><i><a class="txt" href="mod_4.html#E3:37"><i><font color="Green">E29</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>6</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="txt" href="mod_4.html#E3:37"><i><font color="Green">E29</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">RLVECT_1:def 6</a>
<div class="add"><b>let </b><font color="Maroon">c<sub>5</sub></font>, <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 <span class="p1">(<span class="default"><a href="mod_4.html#K3" target="_self">opp</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>;<br/><b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>5</sub></font>, <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> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="ref" href="mod_4.html#T9" target="_self">Th9</a></i>;<br/><a NAME="E2:37_2_2"/><i><font color="Green">E32</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>10</sub></font> )
 <b>by </b><i><a class="txt" href="mod_4.html#E3:37"><i><font color="Green">E29</font></i></a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>7</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>10</sub></font>
<b>by </b><i><a class="txt" href="mod_4.html#E3:37"><i><font color="Green">E29</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>8</sub></font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D6">RLVECT_1:def 6</a></i>
<br/>.= 
<font color="Maroon">c<sub>5</sub></font> <a href="rlvect_1.html#K2">+</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>
<b>by </b><i><a class="txt" href="mod_4.html#E3:37"><i><font color="Green">E29</font></i></a>, <a class="txt" href="mod_4.html#E2:37_2_2"><i><font color="Green">E32</font></i></a></i>
;<br/><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>5</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="mod_4.html#K3" target="_self">opp</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>;<br/><b>reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>5</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="ref" href="mod_4.html#T9" target="_self">Th9</a></i>;<br/><b>thus </b><font color="Maroon">c<sub>5</sub></font> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <span class="p2">(<span class="default"><a href="mod_4.html#K3" target="_self">opp</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> = 
<font color="Maroon">c<sub>6</sub></font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="mod_4.html#E3:37"><i><font color="Green">E29</font></i></a>, <a class="txt" href="mod_4.html#E4:37"><i><font color="Green">E30</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T10">RLVECT_1:10</a></i>
;<br/></div>
<b>end;</b></div>

<b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="mod_4.html#K3" target="_self">opp</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="rlvect_1.html#D8">RLVECT_1:def 8</a><br/>

<b>reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>5</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="ref" href="mod_4.html#T9" target="_self">Th9</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E6:37_2"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="rlvect_1.html#D8">RLVECT_1:def 8</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>7</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="mod_4.html#K3" target="_self">opp</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <b>by </b><i><a class="ref" href="mod_4.html#T9" target="_self">Th9</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>8</sub></font>
;<br/>

<b>thus </b><a NAME="E8:37_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <span class="p1">(<span class="default"><a href="mod_4.html#K3" target="_self">opp</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="mod_4.html#E3:37"><i><font color="Green">E29</font></i></a>, <a class="txt" href="mod_4.html#E4:37"><i><font color="Green">E30</font></i></a>, <a class="txt" href="mod_4.html#E6:37_2"><i><font color="Green">E32</font></i></a></i>;<br/>


</div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font> be   <a href="vectsp_1.html#NM3">Scalar</a> of <span class="p1">(<span class="default"><a href="mod_4.html#K2" target="_self">opp</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/><b>let </b><font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>8</sub></font> be   <a href="vectsp_2.html#NM5">Vector</a> of <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>5</sub></font>, <font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>6</sub></font> as   <a href="vectsp_1.html#NM3">Scalar</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/><b>reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>8</sub></font> as   <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="ref" href="mod_4.html#T9" target="_self">Th9</a></i>;<br/><a NAME="E3:37_3"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="ref" href="mod_4.html#T18" target="_self">Th18</a></i>;<br/><a NAME="E4:37_3"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>10</sub></font> &amp;  <a href="group_1.html#K2">1.</a> <span class="p1">(<span class="default"><a href="mod_4.html#K2" target="_self">opp</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>10</sub></font> )
 <b>by </b><i><a class="txt" href="mod_4.html#E7"><i><font color="Green">Lemma6</font></i></a>, <a class="txt" href="mod_4.html#E5"><i><font color="Green">Lemma4</font></i></a></i>;<br/><a NAME="E5:37_3"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">c<sub>10</sub></font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="vectsp_2.html#K6">*</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="vectsp_2.html#K6">*</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="vectsp_2.html#K6">*</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="vectsp_2.html#K6">*</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="ref" href="mod_4.html#T17" target="_self">Th17</a></i>;<br/><b>thus </b><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> <a href="vectsp_2.html#K6">*</a> <font color="Maroon">c<sub>5</sub></font> = 
<font color="Maroon">c<sub>9</sub></font> <a href="vectsp_1.html#K6">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>12</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="mod_4.html#E3:37_3"><i><font color="Green">E32</font></i></a>, <a class="ref" href="mod_4.html#T17" target="_self">Th17</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">c<sub>12</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_1.html#D26">VECTSP_1:def 26</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="vectsp_2.html#K6">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="vectsp_2.html#K6">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="mod_4.html#E5:37_3"><i><font color="Green">E34</font></i></a>, <a class="ref" href="mod_4.html#T18" target="_self">Th18</a></i>
;<br/><b>thus </b><font color="Maroon">c<sub>7</sub></font> <a href="vectsp_2.html#K6">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">c<sub>11</sub></font>
<b>by </b><i><a class="ref" href="mod_4.html#T17" target="_self">Th17</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">c<sub>11</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_1.html#D26">VECTSP_1:def 26</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="vectsp_2.html#K6">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="vectsp_2.html#K6">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="mod_4.html#E5:37_3"><i><font color="Green">E34</font></i></a>, <a class="ref" href="mod_4.html#T18" target="_self">Th18</a></i>
;<br/><b>thus </b><font color="Maroon">c<sub>7</sub></font> <a href="vectsp_2.html#K6">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">c<sub>11</sub></font>
<b>by </b><i><a class="txt" href="mod_4.html#E4:37_3"><i><font color="Green">E33</font></i></a>, <a class="ref" href="mod_4.html#T17" target="_self">Th17</a></i>
<br/>.= 
<font color="Maroon">c<sub>9</sub></font> <a href="vectsp_1.html#K6">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">c<sub>11</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_1.html#D26">VECTSP_1:def 26</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="vectsp_2.html#K6">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="vectsp_2.html#K6">*</a> <font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="txt" href="mod_4.html#E5:37_3"><i><font color="Green">E34</font></i></a>, <a class="ref" href="mod_4.html#T17" target="_self">Th17</a></i>
;<br/><b>thus </b><font color="Maroon">c<sub>7</sub></font> <a href="vectsp_2.html#K6">*</a> <span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <span class="p2">(<span class="default"><a href="mod_4.html#K2" target="_self">opp</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">c<sub>11</sub></font>
<b>by </b><i><a class="txt" href="mod_4.html#E4:37_3"><i><font color="Green">E33</font></i></a>, <a class="ref" href="mod_4.html#T17" target="_self">Th17</a></i>
<br/>.= 
<font color="Maroon">c<sub>7</sub></font>
<b>by </b><i><a class="ref" href="vectsp_1.html#D26">VECTSP_1:def 26</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:37"/>
 <a href="mod_4.html#K3" target="_self">opp</a> <font color="Maroon">c<sub>2</sub></font> is  <a href="vectsp_2.html#V3">strict</a> <a href="vectsp_2.html#NM7">RightMod</a> of  <a href="mod_4.html#K2" target="_self">opp</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="mod_4.html#E5:37"><i><font color="Green">E31</font></i></a>, <a class="ref" href="vectsp_2.html#D23">VECTSP_2:def 23</a></i>;<br/>


</div>
