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

<b>let </b><font color="Maroon" title="c1">K</font> be   <a href="funcsdom.html#NM1" title="FUNCSDOM:NM.1">Ring</a>;<br/><b>let </b><font color="Maroon" title="c2">W</font> be   <a href="vectsp_2.html#NM7" title="VECTSP_2:NM.7">RightMod</a> of <font color="Maroon" title="c1">K</font>;<br/>



<b>set </b><font color="Maroon" title="c3">R</font> =  <a href="mod_4.html#K2" title="MOD_4:func.2">opp</a> <font color="Maroon" title="c1">K</font>;<br/>
<b>reconsider </b><font color="Maroon" title="c4">V</font> =  <a href="mod_4.html#K5" title="MOD_4:func.5">opp</a> <font color="Maroon" title="c2">W</font> as  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="vectsp_1.html#L4" title="VECTSP_1:struct.4">VectSpStr</a> of  <a href="mod_4.html#K2" title="MOD_4:func.2">opp</a> <font color="Maroon" title="c1">K</font> ;<br/>
<a NAME="E2:43"/><i><font color="Green" title="E20">A1</font></i>: 
 <a href="rlvect_1.html#G1" title="RLVECT_1:aggr.1">LoopStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="mod_4.html#K5" title="MOD_4:func.5">opp</a> <font color="Maroon" title="c2">W</font></span>)</span>,the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">add</a> of <span class="p1">(<span class="default"><a href="mod_4.html#K5" title="MOD_4:func.5">opp</a> <font color="Maroon" title="c2">W</font></span>)</span>,the <a href="struct_0.html#U2" title="STRUCT_0:sel.2">U2</a> of <span class="p1">(<span class="default"><a href="mod_4.html#K5" title="MOD_4:func.5">opp</a> <font color="Maroon" title="c2">W</font></span>)</span> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#G1" title="RLVECT_1:aggr.1">LoopStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">W</font>,the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">add</a> of <font color="Maroon" title="c2">W</font>,the <a href="struct_0.html#U2" title="STRUCT_0:sel.2">U2</a> of <font color="Maroon" title="c2">W</font> #)
 
<b>by </b><i><a class="ref" href="mod_4.html#T12" target="_self" title="MOD_4:th.12">Th12</a></i>;<br/>
<a NAME="E3:43"/><b>then </b><i><font color="Green" title="E21">A2</font></i>: 
for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="mod_4.html#K5" title="MOD_4:func.5">opp</a> <font color="Maroon" title="c2">W</font></span>)</span><br/> for <font color="Olive" title="b3">x</font>, <font color="Olive" title="b4">y</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">W</font>  st <font color="Olive" title="b3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> &amp; <font color="Olive" title="b2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">y</font> holds <br/><font color="Olive" title="b1">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Olive" title="b2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">x</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Olive" title="b4">y</font>
 
;<br/>
<a NAME="E4:43"/><i><font color="Green" title="E22">A3</font></i>: 
(  <a href="mod_4.html#K5" title="MOD_4:func.5">opp</a> <font color="Maroon" title="c2">W</font> is <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">Abelian</a> &amp;  <a href="mod_4.html#K5" title="MOD_4:func.5">opp</a> <font color="Maroon" title="c2">W</font> is <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">add-associative</a> &amp;  <a href="mod_4.html#K5" title="MOD_4:func.5">opp</a> <font color="Maroon" title="c2">W</font> is <a href="rlvect_1.html#V5" title="RLVECT_1:attr.5">right_zeroed</a> &amp;  <a href="mod_4.html#K5" title="MOD_4:func.5">opp</a> <font color="Maroon" title="c2">W</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="43_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:43_1"/>
 <a href="mod_4.html#K5" title="MOD_4:func.5">opp</a> <font color="Maroon" title="c2">W</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="43_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">a</font>, <font color="Maroon" title="c6">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="mod_4.html#K5" title="MOD_4:func.5">opp</a> <font color="Maroon" title="c2">W</font></span>)</span>; <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="c7">x</font> = <font color="Maroon" title="c5">a</font>, <font color="Maroon" title="c8">y</font> = <font color="Maroon" title="c6">b</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">W</font> <b>by </b><i><a class="ref" href="mod_4.html#T12" target="_self" title="MOD_4:th.12">Th12</a></i>;<br/>
<b>thus </b><font color="Maroon" title="c5">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Maroon" title="c6">b</font> = 
<font color="Maroon" title="c8">y</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c7">x</font>
<b>by </b><i><a class="txt" href="mod_4.html#E3:43"><i><font color="Green" title="E21">A2</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c6">b</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Maroon" title="c5">a</font>
<b>by </b><i><a class="txt" href="mod_4.html#E2:43"><i><font color="Green" title="E20">A1</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="c5">a</font>, <font color="Maroon" title="c6">b</font>, <font color="Maroon" title="c7">c</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="mod_4.html#K5" title="MOD_4:func.5">opp</a> <font color="Maroon" title="c2">W</font></span>)</span>;<br/><b>reconsider </b><font color="Maroon" title="c8">x</font> = <font color="Maroon" title="c5">a</font>, <font color="Maroon" title="c9">y</font> = <font color="Maroon" title="c6">b</font>, <font color="Maroon" title="c10">z</font> = <font color="Maroon" title="c7">c</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">W</font> <b>by </b><i><a class="ref" href="mod_4.html#T12" target="_self" title="MOD_4:th.12">Th12</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Maroon" title="c6">b</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Maroon" title="c7">c</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">x</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c9">y</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c10">z</font>
<b>by </b><i><a class="txt" href="mod_4.html#E2:43"><i><font color="Green" title="E20">A1</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c8">x</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">y</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c10">z</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D6" title="RLVECT_1:def.6">RLVECT_1:def 6</a></i>
<br/>.= 
<font color="Maroon" title="c5">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">b</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Maroon" title="c7">c</font></span>)</span>
<b>by </b><i><a class="txt" href="mod_4.html#E2:43"><i><font color="Green" title="E20">A1</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="c5">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="mod_4.html#K5" title="MOD_4:func.5">opp</a> <font color="Maroon" title="c2">W</font></span>)</span>;<br/><b>reconsider </b><font color="Maroon" title="c6">x</font> = <font color="Maroon" title="c5">a</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">W</font> <b>by </b><i><a class="ref" href="mod_4.html#T12" target="_self" title="MOD_4:th.12">Th12</a></i>;<br/><b>thus </b><font color="Maroon" title="c5">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> <span class="p2">(<span class="default"><a href="mod_4.html#K5" title="MOD_4:func.5">opp</a> <font color="Maroon" title="c2">W</font></span>)</span></span>)</span> = 
<font color="Maroon" title="c6">x</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">W</font></span>)</span>
<b>by </b><i><a class="txt" href="mod_4.html#E2:43"><i><font color="Green" title="E20">A1</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c5">a</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T10" title="RLVECT_1:th.10">RLVECT_1:10</a></i>
;<br/></div>
<b>end;</b></div>

<b>let </b><font color="Maroon" title="c5">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="mod_4.html#K5" title="MOD_4:func.5">opp</a> <font color="Maroon" title="c2">W</font></span>)</span>; <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>reconsider </b><font color="Maroon" title="c6">x</font> = <font color="Maroon" title="c5">a</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">W</font> <b>by </b><i><a class="ref" href="mod_4.html#T12" target="_self" title="MOD_4:th.12">Th12</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c7">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">W</font><b> such that </b><br/><a NAME="E6:43_1"/><i><font color="Green" title="E23">A4</font></i>: 
<font color="Maroon" title="c6">x</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c7">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">W</font>
 <b>by </b><i><a class="ref" href="rlvect_1.html#D8" title="RLVECT_1:def.8">RLVECT_1:def 8</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c8">b'</font> = <font color="Maroon" title="c7">b</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="mod_4.html#K5" title="MOD_4:func.5">opp</a> <font color="Maroon" title="c2">W</font></span>)</span> <b>by </b><i><a class="ref" href="mod_4.html#T12" target="_self" title="MOD_4:th.12">Th12</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c8">b'</font>
;<br/>

<b>thus </b><a NAME="E8:43_1"/>
<font color="Maroon" title="c5">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Maroon" title="c8">b'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="mod_4.html#K5" title="MOD_4:func.5">opp</a> <font color="Maroon" title="c2">W</font></span>)</span>
 <b>by </b><i><a class="txt" href="mod_4.html#E2:43"><i><font color="Green" title="E20">A1</font></i></a>, <a class="txt" href="mod_4.html#E6:43_1"><i><font color="Green" title="E23">A4</font></i></a></i>;<br/>


</div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">x</font>, <font color="Maroon" title="c6">y</font> be   <a href="vectsp_1.html#NM4" title="VECTSP_1:NM.4">Scalar</a> of <span class="p1">(<span class="default"><a href="mod_4.html#K2" title="MOD_4:func.2">opp</a> <font color="Maroon" title="c1">K</font></span>)</span>;<br/><b>let </b><font color="Maroon" title="c7">v</font>, <font color="Maroon" title="c8">w</font> be   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c4">V</font>;<br/><b>reconsider </b><font color="Maroon" title="c9">a</font> = <font color="Maroon" title="c5">x</font>, <font color="Maroon" title="c10">b</font> = <font color="Maroon" title="c6">y</font> as   <a href="vectsp_1.html#NM4" title="VECTSP_1:NM.4">Scalar</a> of <font color="Maroon" title="c1">K</font> ;<br/><b>reconsider </b><font color="Maroon" title="c11">p</font> = <font color="Maroon" title="c7">v</font>, <font color="Maroon" title="c12">q</font> = <font color="Maroon" title="c8">w</font> as   <a href="vectsp_2.html#NM5" title="VECTSP_2:NM.5">Vector</a> of <font color="Maroon" title="c2">W</font> <b>by </b><i><a class="ref" href="mod_4.html#T12" target="_self" title="MOD_4:th.12">Th12</a></i>;<br/><a NAME="E3:43_2"/><i><font color="Green" title="E23">A5</font></i>: 
<font color="Maroon" title="c7">v</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Maroon" title="c8">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">p</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c12">q</font>
 <b>by </b><i><a class="ref" href="mod_4.html#T22" target="_self" title="MOD_4:th.22">Th22</a></i>;<br/><a NAME="E4:43_2"/><i><font color="Green" title="E24">A6</font></i>: 
( <font color="Maroon" title="c5">x</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">b</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c9">a</font> &amp;  <a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <span class="p1">(<span class="default"><a href="mod_4.html#K2" title="MOD_4:func.2">opp</a> <font color="Maroon" title="c1">K</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c1">K</font> &amp; <font color="Maroon" title="c5">x</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">a</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c10">b</font> )
 <b>by </b><i><a class="txt" href="mod_4.html#E5"><i><font color="Green" title="E4">Lm3</font></i></a></i>;<br/><a NAME="E5:43_2"/><i><font color="Green" title="E25">A7</font></i>: 
( <font color="Maroon" title="c11">p</font> <a href="vectsp_2.html#K6" title="VECTSP_2:func.6">*</a> <font color="Maroon" title="c10">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">y</font> <a href="vectsp_1.html#K5" title="VECTSP_1:func.5">*</a> <font color="Maroon" title="c7">v</font> &amp; <font color="Maroon" title="c11">p</font> <a href="vectsp_2.html#K6" title="VECTSP_2:func.6">*</a> <font color="Maroon" title="c9">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x</font> <a href="vectsp_1.html#K5" title="VECTSP_1:func.5">*</a> <font color="Maroon" title="c7">v</font> &amp; <font color="Maroon" title="c11">p</font> <a href="vectsp_2.html#K6" title="VECTSP_2:func.6">*</a> <font color="Maroon" title="c10">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">y</font> <a href="vectsp_1.html#K5" title="VECTSP_1:func.5">*</a> <font color="Maroon" title="c7">v</font> &amp; <font color="Maroon" title="c12">q</font> <a href="vectsp_2.html#K6" title="VECTSP_2:func.6">*</a> <font color="Maroon" title="c9">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x</font> <a href="vectsp_1.html#K5" title="VECTSP_1:func.5">*</a> <font color="Maroon" title="c8">w</font> )
 <b>by </b><i><a class="ref" href="mod_4.html#T21" target="_self" title="MOD_4:th.21">Th21</a></i>;<br/><b>thus </b><font color="Maroon" title="c5">x</font> <a href="vectsp_1.html#K5" title="VECTSP_1:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Maroon" title="c8">w</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">p</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c12">q</font></span>)</span> <a href="vectsp_2.html#K6" title="VECTSP_2:func.6">*</a> <font color="Maroon" title="c9">a</font>
<b>by </b><i><a class="txt" href="mod_4.html#E3:43_2"><i><font color="Green" title="E23">A5</font></i></a>, <a class="ref" href="mod_4.html#T21" target="_self" title="MOD_4:th.21">Th21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">p</font> <a href="vectsp_2.html#K6" title="VECTSP_2:func.6">*</a> <font color="Maroon" title="c9">a</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">q</font> <a href="vectsp_2.html#K6" title="VECTSP_2:func.6">*</a> <font color="Maroon" title="c9">a</font></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_2.html#D23" title="VECTSP_2:def.23">VECTSP_2:def 23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="vectsp_1.html#K5" title="VECTSP_1:func.5">*</a> <font color="Maroon" title="c7">v</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="vectsp_1.html#K5" title="VECTSP_1:func.5">*</a> <font color="Maroon" title="c8">w</font></span>)</span>
<b>by </b><i><a class="txt" href="mod_4.html#E5:43_2"><i><font color="Green" title="E25">A7</font></i></a>, <a class="ref" href="mod_4.html#T22" target="_self" title="MOD_4:th.22">Th22</a></i>
;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c6">y</font></span>)</span> <a href="vectsp_1.html#K5" title="VECTSP_1:func.5">*</a> <font color="Maroon" title="c7">v</font> = 
<font color="Maroon" title="c11">p</font> <a href="vectsp_2.html#K6" title="VECTSP_2:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c10">b</font></span>)</span>
<b>by </b><i><a class="ref" href="mod_4.html#T21" target="_self" title="MOD_4:th.21">Th21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">p</font> <a href="vectsp_2.html#K6" title="VECTSP_2:func.6">*</a> <font color="Maroon" title="c9">a</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">p</font> <a href="vectsp_2.html#K6" title="VECTSP_2:func.6">*</a> <font color="Maroon" title="c10">b</font></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_2.html#D23" title="VECTSP_2:def.23">VECTSP_2:def 23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="vectsp_1.html#K5" title="VECTSP_1:func.5">*</a> <font color="Maroon" title="c7">v</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">y</font> <a href="vectsp_1.html#K5" title="VECTSP_1:func.5">*</a> <font color="Maroon" title="c7">v</font></span>)</span>
<b>by </b><i><a class="txt" href="mod_4.html#E5:43_2"><i><font color="Green" title="E25">A7</font></i></a>, <a class="ref" href="mod_4.html#T22" target="_self" title="MOD_4:th.22">Th22</a></i>
;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c6">y</font></span>)</span> <a href="vectsp_1.html#K5" title="VECTSP_1:func.5">*</a> <font color="Maroon" title="c7">v</font> = 
<font color="Maroon" title="c11">p</font> <a href="vectsp_2.html#K6" title="VECTSP_2:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">b</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c9">a</font></span>)</span>
<b>by </b><i><a class="txt" href="mod_4.html#E4:43_2"><i><font color="Green" title="E24">A6</font></i></a>, <a class="ref" href="mod_4.html#T21" target="_self" title="MOD_4:th.21">Th21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">p</font> <a href="vectsp_2.html#K6" title="VECTSP_2:func.6">*</a> <font color="Maroon" title="c10">b</font></span>)</span> <a href="vectsp_2.html#K6" title="VECTSP_2:func.6">*</a> <font color="Maroon" title="c9">a</font>
<b>by </b><i><a class="ref" href="vectsp_2.html#D23" title="VECTSP_2:def.23">VECTSP_2:def 23</a></i>
<br/>.= 
<font color="Maroon" title="c5">x</font> <a href="vectsp_1.html#K5" title="VECTSP_1:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">y</font> <a href="vectsp_1.html#K5" title="VECTSP_1:func.5">*</a> <font color="Maroon" title="c7">v</font></span>)</span>
<b>by </b><i><a class="txt" href="mod_4.html#E5:43_2"><i><font color="Green" title="E25">A7</font></i></a>, <a class="ref" href="mod_4.html#T21" target="_self" title="MOD_4:th.21">Th21</a></i>
;<br/><b>thus </b><span class="p1">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <span class="p2">(<span class="default"><a href="mod_4.html#K2" title="MOD_4:func.2">opp</a> <font color="Maroon" title="c1">K</font></span>)</span></span>)</span> <a href="vectsp_1.html#K5" title="VECTSP_1:func.5">*</a> <font color="Maroon" title="c7">v</font> = 
<font color="Maroon" title="c11">p</font> <a href="vectsp_2.html#K6" title="VECTSP_2:func.6">*</a> <span class="p1">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c1">K</font></span>)</span>
<b>by </b><i><a class="ref" href="mod_4.html#T21" target="_self" title="MOD_4:th.21">Th21</a></i>
<br/>.= 
<font color="Maroon" title="c7">v</font>
<b>by </b><i><a class="ref" href="vectsp_2.html#D23" title="VECTSP_2:def.23">VECTSP_2:def 23</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:43"/>
 <a href="mod_4.html#K5" title="MOD_4:func.5">opp</a> <font color="Maroon" title="c2">W</font> is  <a href="vectsp_1.html#V11" title="VECTSP_1:attr.11">strict</a> <a href="vectsp_2.html#NM6" title="VECTSP_2:NM.6">LeftMod</a> of  <a href="mod_4.html#K2" title="MOD_4:func.2">opp</a> <font color="Maroon" title="c1">K</font>
 <b>by </b><i><a class="txt" href="mod_4.html#E4:43"><i><font color="Green" title="E22">A3</font></i></a>, <a class="ref" href="vectsp_1.html#D26" title="VECTSP_1:def.26">VECTSP_1:def 26</a></i>;<br/>


</div>
