<?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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">V</font> being  <a href="vectsp_2.html#NM6" title="VECTSP_2:NM.6">LeftMod</a> of <font color="Maroon" title="c1">K</font><br/> for <font color="Olive" title="b2">W</font> being   <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Olive" title="b1">V</font> holds <br/> ( <font color="Olive" title="b1">V</font> <a href="lmod_7.html#K17" title="LMOD_7:func.17">/</a> <font color="Olive" title="b2">W</font> is <a href="rlvect_1.html#V2" title="RLVECT_1:attr.2">Abelian</a> &amp; <font color="Olive" title="b1">V</font> <a href="lmod_7.html#K17" title="LMOD_7:func.17">/</a> <font color="Olive" title="b2">W</font> is <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">add-associative</a> &amp; <font color="Olive" title="b1">V</font> <a href="lmod_7.html#K17" title="LMOD_7:func.17">/</a> <font color="Olive" title="b2">W</font> is <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a> &amp; <font color="Olive" title="b1">V</font> <a href="lmod_7.html#K17" title="LMOD_7:func.17">/</a> <font color="Olive" title="b2">W</font> is <a href="algstr_0.html#V13" title="ALGSTR_0:attr.13">right_complementable</a> )</span><br/><b>let </b><font color="Maroon" title="c2">V</font> be   <a href="vectsp_2.html#NM6" title="VECTSP_2:NM.6">LeftMod</a> of <font color="Maroon" title="c1">K</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">W</font> being   <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c2">V</font> holds <br/> ( <font color="Maroon" title="c2">V</font> <a href="lmod_7.html#K17" title="LMOD_7:func.17">/</a> <font color="Olive" title="b1">W</font> is <a href="rlvect_1.html#V2" title="RLVECT_1:attr.2">Abelian</a> &amp; <font color="Maroon" title="c2">V</font> <a href="lmod_7.html#K17" title="LMOD_7:func.17">/</a> <font color="Olive" title="b1">W</font> is <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">add-associative</a> &amp; <font color="Maroon" title="c2">V</font> <a href="lmod_7.html#K17" title="LMOD_7:func.17">/</a> <font color="Olive" title="b1">W</font> is <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a> &amp; <font color="Maroon" title="c2">V</font> <a href="lmod_7.html#K17" title="LMOD_7:func.17">/</a> <font color="Olive" title="b1">W</font> is <a href="algstr_0.html#V13" title="ALGSTR_0:attr.13">right_complementable</a> )</span><br/><b>let </b><font color="Maroon" title="c3">W</font> be    <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c2">V</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">V</font> <a href="lmod_7.html#K17" title="LMOD_7:func.17">/</a> <font color="Maroon" title="c3">W</font> is <a href="rlvect_1.html#V2" title="RLVECT_1:attr.2">Abelian</a> &amp; <font color="Maroon" title="c2">V</font> <a href="lmod_7.html#K17" title="LMOD_7:func.17">/</a> <font color="Maroon" title="c3">W</font> is <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">add-associative</a> &amp; <font color="Maroon" title="c2">V</font> <a href="lmod_7.html#K17" title="LMOD_7:func.17">/</a> <font color="Maroon" title="c3">W</font> is <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a> &amp; <font color="Maroon" title="c2">V</font> <a href="lmod_7.html#K17" title="LMOD_7:func.17">/</a> <font color="Maroon" title="c3">W</font> is <a href="algstr_0.html#V13" title="ALGSTR_0:attr.13">right_complementable</a> )</span><br/>





<a NAME="E1:57"/><i><font color="Green" title="E24">A1</font></i>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font>, <font color="Olive" title="b3">z</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">V</font> <a href="lmod_7.html#K13" title="LMOD_7:func.13">.</a> <font color="Maroon" title="c3">W</font></span>)</span><br/> for <font color="Olive" title="b4">x'</font>, <font color="Olive" title="b5">y'</font>, <font color="Olive" title="b6">z'</font> being  <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">V</font> <a href="lmod_7.html#K17" title="LMOD_7:func.17">/</a> <font color="Maroon" title="c3">W</font></span>)</span>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">x'</font> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">y'</font> &amp; <font color="Olive" title="b3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b6">z'</font> holds <br/><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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">x'</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Olive" title="b5">y'</font>
 
;<br/>
<b>thus </b><a NAME="E2:57"/>
<font color="Maroon" title="c2">V</font> <a href="lmod_7.html#K17" title="LMOD_7:func.17">/</a> <font color="Maroon" title="c3">W</font> is <a href="rlvect_1.html#V2" title="RLVECT_1:attr.2">Abelian</a>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">V</font> <a href="lmod_7.html#K17" title="LMOD_7:func.17">/</a> <font color="Maroon" title="c3">W</font> is <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">add-associative</a> &amp; <font color="Maroon" title="c2">V</font> <a href="lmod_7.html#K17" title="LMOD_7:func.17">/</a> <font color="Maroon" title="c3">W</font> is <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a> &amp; <font color="Maroon" title="c2">V</font> <a href="lmod_7.html#K17" title="LMOD_7:func.17">/</a> <font color="Maroon" title="c3">W</font> is <a href="algstr_0.html#V13" title="ALGSTR_0:attr.13">right_complementable</a> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">x</font>, <font color="Maroon" title="c5">y</font> be   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">V</font> <a href="lmod_7.html#K17" title="LMOD_7:func.17">/</a> <font color="Maroon" title="c3">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> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">y</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c4">x</font></span><br/>

<b>reconsider </b><font color="Maroon" title="c6">x'</font> = <font color="Maroon" title="c4">x</font>, <font color="Maroon" title="c7">y'</font> = <font color="Maroon" title="c5">y</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">V</font> <a href="lmod_7.html#K13" title="LMOD_7:func.13">.</a> <font color="Maroon" title="c3">W</font></span>)</span> ;<br/>
<b>thus </b><font color="Maroon" title="c4">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c5">y</font> = 
<font color="Maroon" title="c6">x'</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c7">y'</font>

<br/>.= 
<font color="Maroon" title="c5">y</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c4">x</font>
<b>by </b><i><a class="txt" href="lmod_7.html#E1:57"><i><font color="Green" title="E24">A1</font></i></a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><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> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">V</font> <a href="lmod_7.html#K17" title="LMOD_7:func.17">/</a> <font color="Maroon" title="c3">W</font> is <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a> &amp; <font color="Maroon" title="c2">V</font> <a href="lmod_7.html#K17" title="LMOD_7:func.17">/</a> <font color="Maroon" title="c3">W</font> is <a href="algstr_0.html#V13" title="ALGSTR_0:attr.13">right_complementable</a> )</span>
<div class="add"><b>let </b><font color="Maroon" title="c4">x</font>, <font color="Maroon" title="c5">y</font>, <font color="Maroon" title="c6">z</font> be   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">V</font> <a href="lmod_7.html#K17" title="LMOD_7:func.17">/</a> <font color="Maroon" title="c3">W</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c5">y</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c6">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">y</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c6">z</font></span>)</span></span><br/><b>reconsider </b><font color="Maroon" title="c7">x'</font> = <font color="Maroon" title="c4">x</font>, <font color="Maroon" title="c8">y'</font> = <font color="Maroon" title="c5">y</font>, <font color="Maroon" title="c9">z'</font> = <font color="Maroon" title="c6">z</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">V</font> <a href="lmod_7.html#K13" title="LMOD_7:func.13">.</a> <font color="Maroon" title="c3">W</font></span>)</span> ;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c5">y</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c6">z</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">x'</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c8">y'</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c9">z'</font>

<br/>.= 
<font color="Maroon" title="c7">x'</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">y'</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c9">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="c4">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">y</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c6">z</font></span>)</span>

; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><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> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">V</font> <a href="lmod_7.html#K17" title="LMOD_7:func.17">/</a> <font color="Maroon" title="c3">W</font> is <a href="algstr_0.html#V13" title="ALGSTR_0:attr.13">right_complementable</a></span>
<div class="add"><b>let </b><font color="Maroon" title="c4">x</font> be   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">V</font> <a href="lmod_7.html#K17" title="LMOD_7:func.17">/</a> <font color="Maroon" title="c3">W</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">x</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"><font color="Maroon" title="c2">V</font> <a href="lmod_7.html#K17" title="LMOD_7:func.17">/</a> <font color="Maroon" title="c3">W</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x</font></span><br/><b>reconsider </b><font color="Maroon" title="c5">x'</font> = <font color="Maroon" title="c4">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">V</font> <a href="lmod_7.html#K13" title="LMOD_7:func.13">.</a> <font color="Maroon" title="c3">W</font></span>)</span> ;<br/><b>thus </b><font color="Maroon" title="c4">x</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"><font color="Maroon" title="c2">V</font> <a href="lmod_7.html#K17" title="LMOD_7:func.17">/</a> <font color="Maroon" title="c3">W</font></span>)</span></span>)</span> = 
<font color="Maroon" title="c5">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> <span class="p2">(<span class="default"><font color="Maroon" title="c2">V</font> <a href="lmod_7.html#K13" title="LMOD_7:func.13">.</a> <font color="Maroon" title="c3">W</font></span>)</span></span>)</span>

<br/>.= 
<font color="Maroon" title="c4">x</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T10" title="RLVECT_1:th.10">RLVECT_1:10</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div>
<b>end;</b></div>

<b>let </b><font color="Maroon" title="c4">x</font> be   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">V</font> <a href="lmod_7.html#K17" title="LMOD_7:func.17">/</a> <font color="Maroon" title="c3">W</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="algstr_0.html#D16" title="ALGSTR_0:def.16">ALGSTR_0:def 16</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">x</font> is <a href="algstr_0.html#V10" title="ALGSTR_0:attr.10">right_complementable</a></span><br/>

<b>reconsider </b><font color="Maroon" title="c5">x'</font> = <font color="Maroon" title="c4">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">V</font> <a href="lmod_7.html#K13" title="LMOD_7:func.13">.</a> <font color="Maroon" title="c3">W</font></span>)</span> ;<br/>
<b>consider </b><font color="Maroon" title="c6">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">V</font> <a href="lmod_7.html#K13" title="LMOD_7:func.13">.</a> <font color="Maroon" title="c3">W</font></span>)</span><b> such that </b><br/><a NAME="E7:57"/><i><font color="Green" title="E25">A2</font></i>: 
<font color="Maroon" title="c5">x'</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c6">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"><font color="Maroon" title="c2">V</font> <a href="lmod_7.html#K13" title="LMOD_7:func.13">.</a> <font color="Maroon" title="c3">W</font></span>)</span>
 <b>by </b><i><a class="ref" href="algstr_0.html#D11" title="ALGSTR_0:def.11">ALGSTR_0:def 11</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">b'</font> = <font color="Maroon" title="c6">b</font> as   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">V</font> <a href="lmod_7.html#K17" title="LMOD_7:func.17">/</a> <font color="Maroon" title="c3">W</font></span>)</span> ;<br/>
<b>take </b>
<font color="Maroon" title="c7">b'</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="algstr_0.html#D11" title="ALGSTR_0:def.11">ALGSTR_0:def 11</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</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> <span class="p1">(<span class="default"><font color="Maroon" title="c2">V</font> <a href="lmod_7.html#K17" title="LMOD_7:func.17">/</a> <font color="Maroon" title="c3">W</font></span>)</span></span><br/>

<b>thus </b><a NAME="E9:57"/>
<font color="Maroon" title="c4">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</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> <span class="p1">(<span class="default"><font color="Maroon" title="c2">V</font> <a href="lmod_7.html#K17" title="LMOD_7:func.17">/</a> <font color="Maroon" title="c3">W</font></span>)</span>
 <b>by </b><i><a class="txt" href="lmod_7.html#E7:57"><i><font color="Green" title="E25">A2</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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