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

<span class="kw">let </span><font color="Maroon" title="c1">K</font> be   <a href="vectsp_1.html#NM3" title="VECTSP_1:NM.3">Ring</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">V</font> being   <a href="vectsp_2.html#NM5" title="VECTSP_2:NM.5">LeftMod</a> of <font color="Maroon" title="c1">K</font> holds   <a href="mod_4.html#K2" title="MOD_4:func.2">opp</a> <font color="Olive" title="b1">V</font> is   <a href="vectsp_2.html#V2" title="VECTSP_2:attr.2">strict</a>  <a href="vectsp_2.html#NM6" title="VECTSP_2:NM.6">RightMod</a> of  <a href="mod_4.html#K1" title="MOD_4:func.1">opp</a> <font color="Maroon" title="c1">K</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">V</font> be   <a href="vectsp_2.html#NM5" title="VECTSP_2:NM.5">LeftMod</a> of <font color="Maroon" title="c1">K</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="mod_4.html#K2" title="MOD_4:func.2">opp</a> <font color="Maroon" title="c2">V</font> is   <a href="vectsp_2.html#V2" title="VECTSP_2:attr.2">strict</a>  <a href="vectsp_2.html#NM6" title="VECTSP_2:NM.6">RightMod</a> of  <a href="mod_4.html#K1" title="MOD_4:func.1">opp</a> <font color="Maroon" title="c1">K</font></span><br/>



<span class="kw">set </span><font color="Maroon" title="c3">R</font> =  <a href="mod_4.html#K1" title="MOD_4:func.1">opp</a> <font color="Maroon" title="c1">K</font>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c4">W</font> =  <a href="mod_4.html#K2" title="MOD_4:func.2">opp</a> <font color="Maroon" title="c2">V</font> as   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="vectsp_2.html#L1" title="VECTSP_2:struct.1">RightModStr</a> over  <a href="mod_4.html#K1" title="MOD_4:func.1">opp</a> <font color="Maroon" title="c1">K</font> ;<br/>
<a NAME="E2:37"/><span class="lab"><font color="Green" title="E18">A1</font></span>: 
 <a href="algstr_0.html#G2" title="ALGSTR_0:aggr.2">addLoopStr</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#K2" title="MOD_4:func.2">opp</a> <font color="Maroon" title="c2">V</font></span>)</span>, the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</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="c2">V</font></span>)</span>, the <a href="struct_0.html#U2" title="STRUCT_0:sel.2">ZeroF</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="c2">V</font></span>)</span> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="algstr_0.html#G2" title="ALGSTR_0:aggr.2">addLoopStr</a>(#  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">V</font>, the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c2">V</font>, the <a href="struct_0.html#U2" title="STRUCT_0:sel.2">ZeroF</a> of <font color="Maroon" title="c2">V</font> #)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mod_4.html#T7" target="_self" onmouseover="rs('mod_4/T7')" onmouseout="rh()">Th7</a></span>;<br/>
<a NAME="E3:37"/><span class="kw">then </span><span class="lab"><font color="Green" title="E19">A2</font></span>: 
 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#K2" title="MOD_4:func.2">opp</a> <font color="Maroon" title="c2">V</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">V</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="algstr_0.html#K1" title="ALGSTR_0: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="E4:37"/><span class="lab"><font color="Green" title="E20">A3</font></span>: 
(  <a href="mod_4.html#K2" title="MOD_4:func.2">opp</a> <font color="Maroon" title="c2">V</font> is  <a href="rlvect_1.html#V2" title="RLVECT_1:attr.2">Abelian</a>  &amp;  <a href="mod_4.html#K2" title="MOD_4:func.2">opp</a> <font color="Maroon" title="c2">V</font> is  <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">add-associative</a>  &amp;  <a href="mod_4.html#K2" title="MOD_4:func.2">opp</a> <font color="Maroon" title="c2">V</font> is  <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a>  &amp;  <a href="mod_4.html#K2" title="MOD_4:func.2">opp</a> <font color="Maroon" title="c2">V</font> is  <a href="algstr_0.html#V13" title="ALGSTR_0:attr.13">right_complementable</a>  )
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="37_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">thus </span><a NAME="E1:37_1"/>
 <a href="mod_4.html#K2" title="MOD_4:func.2">opp</a> <font color="Maroon" title="c2">V</font> is  <a href="rlvect_1.html#V2" title="RLVECT_1:attr.2">Abelian</a> 
  <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="mod_4.html#K2" title="MOD_4:func.2">opp</a> <font color="Maroon" title="c2">V</font> is  <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">add-associative</a>  &amp;  <a href="mod_4.html#K2" title="MOD_4:func.2">opp</a> <font color="Maroon" title="c2">V</font> is  <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a>  &amp;  <a href="mod_4.html#K2" title="MOD_4:func.2">opp</a> <font color="Maroon" title="c2">V</font> is  <a href="algstr_0.html#V13" title="ALGSTR_0:attr.13">right_complementable</a>  )</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="37_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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#K2" title="MOD_4:func.2">opp</a> <font color="Maroon" title="c2">V</font></span>)</span>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="rlvect_1.html#D2" onmouseover="rs('rlvect_1/D2')" onmouseout="rh()">RLVECT_1:def 2</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">a</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c6">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c5">a</font></span><br/>

<span class="kw">reconsider </span><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">V</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="mod_4.html#T7" target="_self" onmouseover="rs('mod_4/T7')" onmouseout="rh()">Th7</a></span>;<br/>
<span class="kw">thus </span><font color="Maroon" title="c5">a</font> <a href="algstr_0.html#K1" title="ALGSTR_0: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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:37"><span class="lab"><font color="Green" title="E19">A2</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c6">b</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c5">a</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:37"><span class="lab"><font color="Green" title="E18">A1</font></span></a></span>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<div><span class="kw">hereby </span> <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="rlvect_1.html#D3" onmouseover="rs('rlvect_1/D3')" onmouseout="rh()">RLVECT_1:def 3</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="mod_4.html#K2" title="MOD_4:func.2">opp</a> <font color="Maroon" title="c2">V</font> is  <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a>  &amp;  <a href="mod_4.html#K2" title="MOD_4:func.2">opp</a> <font color="Maroon" title="c2">V</font> is  <a href="algstr_0.html#V13" title="ALGSTR_0:attr.13">right_complementable</a>  )</span>
<div class="add"><span class="kw">let </span><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#K2" title="MOD_4:func.2">opp</a> <font color="Maroon" title="c2">V</font></span>)</span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c6">b</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c7">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">b</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c7">c</font></span>)</span></span><br/><span class="kw">reconsider </span><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">V</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="mod_4.html#T7" target="_self" onmouseover="rs('mod_4/T7')" onmouseout="rh()">Th7</a></span>;<br/><span class="kw">thus </span><span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c6">b</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0: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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:37"><span class="lab"><font color="Green" title="E18">A1</font></span></a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#D3" onmouseover="rs('rlvect_1/D3')" onmouseout="rh()">RLVECT_1:def 3</a></span>
<br/>.= 
<font color="Maroon" title="c5">a</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">b</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c7">c</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:37"><span class="lab"><font color="Green" title="E18">A1</font></span></a></span>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div>
<span class="kw">end;</span></div>

<div><span class="kw">hereby </span> <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="rlvect_1.html#D4" onmouseover="rs('rlvect_1/D4')" onmouseout="rh()">RLVECT_1:def 4</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="mod_4.html#K2" title="MOD_4:func.2">opp</a> <font color="Maroon" title="c2">V</font> is  <a href="algstr_0.html#V13" title="ALGSTR_0:attr.13">right_complementable</a> </span>
<div class="add"><span class="kw">let </span><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#K2" title="MOD_4:func.2">opp</a> <font color="Maroon" title="c2">V</font></span>)</span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">a</font> <a href="algstr_0.html#K1" title="ALGSTR_0: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#K2" title="MOD_4:func.2">opp</a> <font color="Maroon" title="c2">V</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</font></span><br/><span class="kw">reconsider </span><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">V</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="mod_4.html#T7" target="_self" onmouseover="rs('mod_4/T7')" onmouseout="rh()">Th7</a></span>;<br/><span class="kw">thus </span><font color="Maroon" title="c5">a</font> <a href="algstr_0.html#K1" title="ALGSTR_0: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#K2" title="MOD_4:func.2">opp</a> <font color="Maroon" title="c2">V</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">V</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:37"><span class="lab"><font color="Green" title="E18">A1</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c5">a</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#T4" onmouseover="rs('rlvect_1/T4')" onmouseout="rh()">RLVECT_1:4</a></span>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div>
<span class="kw">end;</span></div>

<span class="kw">let </span><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#K2" title="MOD_4:func.2">opp</a> <font color="Maroon" title="c2">V</font></span>)</span>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="algstr_0.html#D16" onmouseover="rs('algstr_0/D16')" onmouseout="rh()">ALGSTR_0:def 16</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">a</font> is  <a href="algstr_0.html#V10" title="ALGSTR_0:attr.10">right_complementable</a> </span><br/>

<span class="kw">reconsider </span><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">V</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="mod_4.html#T7" target="_self" onmouseover="rs('mod_4/T7')" onmouseout="rh()">Th7</a></span>;<br/>
<span class="kw">consider </span><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">V</font><span class="kw"> such that </span><br/><a NAME="E6:37_1"/><span class="lab"><font color="Green" title="E21">A4</font></span>: 
<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">V</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="algstr_0.html#D11" onmouseover="rs('algstr_0/D11')" onmouseout="rh()">ALGSTR_0:def 11</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c8">b9</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#K2" title="MOD_4:func.2">opp</a> <font color="Maroon" title="c2">V</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="mod_4.html#T7" target="_self" onmouseover="rs('mod_4/T7')" onmouseout="rh()">Th7</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c8">b9</font>
; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="algstr_0.html#D11" onmouseover="rs('algstr_0/D11')" onmouseout="rh()">ALGSTR_0:def 11</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">a</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c8">b9</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#K2" title="MOD_4:func.2">opp</a> <font color="Maroon" title="c2">V</font></span>)</span></span><br/>

<span class="kw">thus </span><a NAME="E8:37_1"/>
<font color="Maroon" title="c5">a</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c8">b9</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#K2" title="MOD_4:func.2">opp</a> <font color="Maroon" title="c2">V</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:37"><span class="lab"><font color="Green" title="E18">A1</font></span></a>, <a class="txt" href="#E6:37_1"><span class="lab"><font color="Green" title="E21">A4</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="37_2"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Scalar</a> of <span class="p1">(<span class="default"><a href="mod_4.html#K1" title="MOD_4:func.1">opp</a> <font color="Maroon" title="c1">K</font></span>)</span><br/>  for <font color="Olive" title="b3">v</font>, <font color="Olive" title="b4">w</font> being   <a href="vectsp_2.html#NM4" title="VECTSP_2:NM.4">Vector</a> of <font color="Maroon" title="c4">W</font> holds <br/> ( <span class="p1">(<span class="default"><font color="Olive" title="b3">v</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Olive" title="b4">w</font></span>)</span> <a href="vectsp_2.html#K4" title="VECTSP_2:func.4">*</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">v</font> <a href="vectsp_2.html#K4" title="VECTSP_2:func.4">*</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b4">w</font> <a href="vectsp_2.html#K4" title="VECTSP_2:func.4">*</a> <font color="Olive" title="b1">x</font></span>)</span> &amp; <font color="Olive" title="b3">v</font> <a href="vectsp_2.html#K4" title="VECTSP_2:func.4">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">x</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Olive" title="b2">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">v</font> <a href="vectsp_2.html#K4" title="VECTSP_2:func.4">*</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">v</font> <a href="vectsp_2.html#K4" title="VECTSP_2:func.4">*</a> <font color="Olive" title="b2">y</font></span>)</span> &amp; <font color="Olive" title="b3">v</font> <a href="vectsp_2.html#K4" title="VECTSP_2:func.4">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">y</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">v</font> <a href="vectsp_2.html#K4" title="VECTSP_2:func.4">*</a> <font color="Olive" title="b2">y</font></span>)</span> <a href="vectsp_2.html#K4" title="VECTSP_2:func.4">*</a> <font color="Olive" title="b1">x</font> &amp; <font color="Olive" title="b3">v</font> <a href="vectsp_2.html#K4" title="VECTSP_2:func.4">*</a> <span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <span class="p2">(<span class="default"><a href="mod_4.html#K1" title="MOD_4:func.1">opp</a> <font color="Maroon" title="c1">K</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">v</font> )</span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c5">x</font>, <font color="Maroon" title="c6">y</font> be   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Scalar</a> of <span class="p1">(<span class="default"><a href="mod_4.html#K1" title="MOD_4:func.1">opp</a> <font color="Maroon" title="c1">K</font></span>)</span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">w</font> being   <a href="vectsp_2.html#NM4" title="VECTSP_2:NM.4">Vector</a> of <font color="Maroon" title="c4">W</font> holds <br/> ( <span class="p1">(<span class="default"><font color="Olive" title="b1">v</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Olive" title="b2">w</font></span>)</span> <a href="vectsp_2.html#K4" title="VECTSP_2:func.4">*</a> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">v</font> <a href="vectsp_2.html#K4" title="VECTSP_2:func.4">*</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">w</font> <a href="vectsp_2.html#K4" title="VECTSP_2:func.4">*</a> <font color="Maroon" title="c5">x</font></span>)</span> &amp; <font color="Olive" title="b1">v</font> <a href="vectsp_2.html#K4" title="VECTSP_2:func.4">*</a> <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="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">v</font> <a href="vectsp_2.html#K4" title="VECTSP_2:func.4">*</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">v</font> <a href="vectsp_2.html#K4" title="VECTSP_2:func.4">*</a> <font color="Maroon" title="c6">y</font></span>)</span> &amp; <font color="Olive" title="b1">v</font> <a href="vectsp_2.html#K4" title="VECTSP_2:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">y</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">v</font> <a href="vectsp_2.html#K4" title="VECTSP_2:func.4">*</a> <font color="Maroon" title="c6">y</font></span>)</span> <a href="vectsp_2.html#K4" title="VECTSP_2:func.4">*</a> <font color="Maroon" title="c5">x</font> &amp; <font color="Olive" title="b1">v</font> <a href="vectsp_2.html#K4" title="VECTSP_2:func.4">*</a> <span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <span class="p2">(<span class="default"><a href="mod_4.html#K1" title="MOD_4:func.1">opp</a> <font color="Maroon" title="c1">K</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">v</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c7">v</font>, <font color="Maroon" title="c8">w</font> be   <a href="vectsp_2.html#NM4" title="VECTSP_2:NM.4">Vector</a> of <font color="Maroon" title="c4">W</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c8">w</font></span>)</span> <a href="vectsp_2.html#K4" title="VECTSP_2:func.4">*</a> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="vectsp_2.html#K4" title="VECTSP_2:func.4">*</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">w</font> <a href="vectsp_2.html#K4" title="VECTSP_2:func.4">*</a> <font color="Maroon" title="c5">x</font></span>)</span> &amp; <font color="Maroon" title="c7">v</font> <a href="vectsp_2.html#K4" title="VECTSP_2:func.4">*</a> <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="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="vectsp_2.html#K4" title="VECTSP_2:func.4">*</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="vectsp_2.html#K4" title="VECTSP_2:func.4">*</a> <font color="Maroon" title="c6">y</font></span>)</span> &amp; <font color="Maroon" title="c7">v</font> <a href="vectsp_2.html#K4" title="VECTSP_2:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">y</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="vectsp_2.html#K4" title="VECTSP_2:func.4">*</a> <font color="Maroon" title="c6">y</font></span>)</span> <a href="vectsp_2.html#K4" title="VECTSP_2:func.4">*</a> <font color="Maroon" title="c5">x</font> &amp; <font color="Maroon" title="c7">v</font> <a href="vectsp_2.html#K4" title="VECTSP_2:func.4">*</a> <span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <span class="p2">(<span class="default"><a href="mod_4.html#K1" title="MOD_4:func.1">opp</a> <font color="Maroon" title="c1">K</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">v</font> )</span><br/><span class="kw">reconsider </span><font color="Maroon" title="c9">p</font> = <font color="Maroon" title="c7">v</font>, <font color="Maroon" title="c10">q</font> = <font color="Maroon" title="c8">w</font> as   <a href="vectsp_1.html#NM8" title="VECTSP_1:NM.8">Vector</a> of <font color="Maroon" title="c2">V</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="mod_4.html#T7" target="_self" onmouseover="rs('mod_4/T7')" onmouseout="rh()">Th7</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c11">a</font> = <font color="Maroon" title="c5">x</font>, <font color="Maroon" title="c12">b</font> = <font color="Maroon" title="c6">y</font> as   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Scalar</a> of <font color="Maroon" title="c1">K</font> ;<br/><a NAME="E3:37_2"/><span class="lab"><font color="Green" title="E21">A5</font></span>: 
<font color="Maroon" title="c12">b</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c9">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">v</font> <a href="vectsp_2.html#K4" title="VECTSP_2:func.4">*</a> <font color="Maroon" title="c6">y</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="mod_4.html#T12" target="_self" onmouseover="rs('mod_4/T12')" onmouseout="rh()">Th12</a></span>;<br/><a NAME="E4:37_2"/><span class="lab"><font color="Green" title="E22">A6</font></span>: 
<font color="Maroon" title="c11">a</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c10">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">w</font> <a href="vectsp_2.html#K4" title="VECTSP_2:func.4">*</a> <font color="Maroon" title="c5">x</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="mod_4.html#T12" target="_self" onmouseover="rs('mod_4/T12')" onmouseout="rh()">Th12</a></span>;<br/><a NAME="E5:37_2"/><span class="lab"><font color="Green" title="E23">A7</font></span>: 
<font color="Maroon" title="c11">a</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c9">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">v</font> <a href="vectsp_2.html#K4" title="VECTSP_2:func.4">*</a> <font color="Maroon" title="c5">x</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="mod_4.html#T12" target="_self" onmouseover="rs('mod_4/T12')" onmouseout="rh()">Th12</a></span>;<br/><a NAME="E6:37_2"/>
<font color="Maroon" title="c7">v</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c8">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">p</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c10">q</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="mod_4.html#T13" target="_self" onmouseover="rs('mod_4/T13')" onmouseout="rh()">Th13</a></span>;<br/><span class="kw">hence </span><span class="p1">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c8">w</font></span>)</span> <a href="vectsp_2.html#K4" title="VECTSP_2:func.4">*</a> <font color="Maroon" title="c5">x</font> = 
<font color="Maroon" title="c11">a</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">p</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c10">q</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="mod_4.html#T12" target="_self" onmouseover="rs('mod_4/T12')" onmouseout="rh()">Th12</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c9">p</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">a</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c10">q</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_1.html#D14" onmouseover="rs('vectsp_1/D14')" onmouseout="rh()">VECTSP_1:def 14</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="vectsp_2.html#K4" title="VECTSP_2:func.4">*</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">w</font> <a href="vectsp_2.html#K4" title="VECTSP_2:func.4">*</a> <font color="Maroon" title="c5">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:37_2"><span class="lab"><font color="Green" title="E23">A7</font></span></a>, <a class="txt" href="#E4:37_2"><span class="lab"><font color="Green" title="E22">A6</font></span></a>, <a class="ref" href="mod_4.html#T13" target="_self" onmouseover="rs('mod_4/T13')" onmouseout="rh()">Th13</a></span>
;<br/> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c7">v</font> <a href="vectsp_2.html#K4" title="VECTSP_2:func.4">*</a> <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="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="vectsp_2.html#K4" title="VECTSP_2:func.4">*</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="vectsp_2.html#K4" title="VECTSP_2:func.4">*</a> <font color="Maroon" title="c6">y</font></span>)</span> &amp; <font color="Maroon" title="c7">v</font> <a href="vectsp_2.html#K4" title="VECTSP_2:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">y</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="vectsp_2.html#K4" title="VECTSP_2:func.4">*</a> <font color="Maroon" title="c6">y</font></span>)</span> <a href="vectsp_2.html#K4" title="VECTSP_2:func.4">*</a> <font color="Maroon" title="c5">x</font> &amp; <font color="Maroon" title="c7">v</font> <a href="vectsp_2.html#K4" title="VECTSP_2:func.4">*</a> <span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <span class="p2">(<span class="default"><a href="mod_4.html#K1" title="MOD_4:func.1">opp</a> <font color="Maroon" title="c1">K</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">v</font> )</span><br/><span class="kw">thus </span><font color="Maroon" title="c7">v</font> <a href="vectsp_2.html#K4" title="VECTSP_2:func.4">*</a> <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> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c12">b</font></span>)</span> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c9">p</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="mod_4.html#T12" target="_self" onmouseover="rs('mod_4/T12')" onmouseout="rh()">Th12</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c9">p</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">b</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c9">p</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_1.html#D15" onmouseover="rs('vectsp_1/D15')" onmouseout="rh()">VECTSP_1:def 15</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="vectsp_2.html#K4" title="VECTSP_2:func.4">*</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="vectsp_2.html#K4" title="VECTSP_2:func.4">*</a> <font color="Maroon" title="c6">y</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:37_2"><span class="lab"><font color="Green" title="E21">A5</font></span></a>, <a class="txt" href="#E5:37_2"><span class="lab"><font color="Green" title="E23">A7</font></span></a>, <a class="ref" href="mod_4.html#T13" target="_self" onmouseover="rs('mod_4/T13')" onmouseout="rh()">Th13</a></span>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c7">v</font> <a href="vectsp_2.html#K4" title="VECTSP_2:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">y</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="vectsp_2.html#K4" title="VECTSP_2:func.4">*</a> <font color="Maroon" title="c6">y</font></span>)</span> <a href="vectsp_2.html#K4" title="VECTSP_2:func.4">*</a> <font color="Maroon" title="c5">x</font> &amp; <font color="Maroon" title="c7">v</font> <a href="vectsp_2.html#K4" title="VECTSP_2:func.4">*</a> <span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <span class="p2">(<span class="default"><a href="mod_4.html#K1" title="MOD_4:func.1">opp</a> <font color="Maroon" title="c1">K</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">v</font> )</span><br/><span class="kw">thus </span><font color="Maroon" title="c7">v</font> <a href="vectsp_2.html#K4" title="VECTSP_2:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">y</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">x</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c12">b</font></span>)</span> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c9">p</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="mod_4.html#E5"><span class="lab"><font color="Green" title="E3">Lm3</font></span></a>, <a class="ref" href="mod_4.html#T12" target="_self" onmouseover="rs('mod_4/T12')" onmouseout="rh()">Th12</a></span>
<br/>.= 
<font color="Maroon" title="c11">a</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">b</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c9">p</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_1.html#D16" onmouseover="rs('vectsp_1/D16')" onmouseout="rh()">VECTSP_1:def 16</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="vectsp_2.html#K4" title="VECTSP_2:func.4">*</a> <font color="Maroon" title="c6">y</font></span>)</span> <a href="vectsp_2.html#K4" title="VECTSP_2:func.4">*</a> <font color="Maroon" title="c5">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:37_2"><span class="lab"><font color="Green" title="E21">A5</font></span></a>, <a class="ref" href="mod_4.html#T12" target="_self" onmouseover="rs('mod_4/T12')" onmouseout="rh()">Th12</a></span>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">v</font> <a href="vectsp_2.html#K4" title="VECTSP_2:func.4">*</a> <span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <span class="p2">(<span class="default"><a href="mod_4.html#K1" title="MOD_4:func.1">opp</a> <font color="Maroon" title="c1">K</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">v</font></span><br/><span class="kw">thus </span><font color="Maroon" title="c7">v</font> <a href="vectsp_2.html#K4" title="VECTSP_2:func.4">*</a> <span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <span class="p2">(<span class="default"><a href="mod_4.html#K1" title="MOD_4:func.1">opp</a> <font color="Maroon" title="c1">K</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">K</font></span>)</span> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c9">p</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="mod_4.html#T12" target="_self" onmouseover="rs('mod_4/T12')" onmouseout="rh()">Th12</a></span>
<br/>.= 
<font color="Maroon" title="c7">v</font>

; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E6:37"/>
 <a href="mod_4.html#K2" title="MOD_4:func.2">opp</a> <font color="Maroon" title="c2">V</font> is   <a href="vectsp_2.html#V2" title="VECTSP_2:attr.2">strict</a>  <a href="vectsp_2.html#NM6" title="VECTSP_2:NM.6">RightMod</a> of  <a href="mod_4.html#K1" title="MOD_4:func.1">opp</a> <font color="Maroon" title="c1">K</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:37"><span class="lab"><font color="Green" title="E20">A3</font></span></a>, <a class="ref" href="vectsp_2.html#D9" onmouseover="rs('vectsp_2/D9')" onmouseout="rh()">VECTSP_2:def 9</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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