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

<b>let </b><font color="Maroon" title="c1">R</font> be   <a href="funcsdom.html#NM1" title="FUNCSDOM:NM.1">Ring</a>;<br/>

<div><i><font color="Green" title="E21">A1</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c2">x</font>, <font color="Maroon" title="c3">y</font>, <font color="Maroon" title="c4">z</font> be   <a href="vectsp_1.html#NM4" title="VECTSP_1:NM.4">Scalar</a> of <font color="Maroon" title="c1">R</font>;<br/><b>let </b><font color="Maroon" title="c5">x'</font>, <font color="Maroon" title="c6">y'</font>, <font color="Maroon" title="c7">z'</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c1">R</font>);<br/><b>assume </b><a NAME="E1:47_1"/>
( <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x'</font> &amp; <font color="Maroon" title="c3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">y'</font> &amp; <font color="Maroon" title="c4">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">z'</font> )
 ;<br/><b>hence </b><a NAME="E2:47_1"/>
<font color="Maroon" title="c2">x</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x'</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Maroon" title="c6">y'</font>
 ;<br/></div><b>end;</b></div>
<b>thus </b><a NAME="E2:47"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c1">R</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="47_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">x</font>, <font color="Maroon" title="c3">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c1">R</font>); <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="c4">x'</font> = <font color="Maroon" title="c2">x</font>, <font color="Maroon" title="c5">y'</font> = <font color="Maroon" title="c3">y</font> as   <a href="vectsp_1.html#NM4" title="VECTSP_1:NM.4">Scalar</a> of <font color="Maroon" title="c1">R</font> ;<br/>
<b>thus </b><font color="Maroon" title="c2">x</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Maroon" title="c3">y</font> = 
<font color="Maroon" title="c5">y'</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c4">x'</font>
<b>by </b><i><a class="txt" href="vectsp_2.html#E1:47"><i><font color="Green" title="E21">A1</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c3">y</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Maroon" title="c2">x</font>

;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E3:47"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c1">R</font>) is <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">add-associative</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">x</font>, <font color="Maroon" title="c3">y</font>, <font color="Maroon" title="c4">z</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c1">R</font>); <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><br/>

<b>reconsider </b><font color="Maroon" title="c5">x'</font> = <font color="Maroon" title="c2">x</font>, <font color="Maroon" title="c6">y'</font> = <font color="Maroon" title="c3">y</font>, <font color="Maroon" title="c7">z'</font> = <font color="Maroon" title="c4">z</font> as   <a href="vectsp_1.html#NM4" title="VECTSP_1:NM.4">Scalar</a> of <font color="Maroon" title="c1">R</font> ;<br/>
<b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Maroon" title="c4">z</font> = 
<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="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c7">z'</font>

<br/>.= 
<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"><font color="Maroon" title="c6">y'</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c7">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="c2">x</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">y</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Maroon" title="c4">z</font></span>)</span>

;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E4:47"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c1">R</font>) is <a href="rlvect_1.html#V5" title="RLVECT_1:attr.5">right_zeroed</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c1">R</font>); <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><br/>

<b>reconsider </b><font color="Maroon" title="c3">x'</font> = <font color="Maroon" title="c2">x</font> as   <a href="vectsp_1.html#NM4" title="VECTSP_1:NM.4">Scalar</a> of <font color="Maroon" title="c1">R</font> ;<br/>
<b>thus </b><font color="Maroon" title="c2">x</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> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c1">R</font>)</span>)</span> = 
<font color="Maroon" title="c3">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="c1">R</font></span>)</span>

<br/>.= 
<font color="Maroon" title="c2">x</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="c2">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c1">R</font>); <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="c3">x'</font> = <font color="Maroon" title="c2">x</font> as   <a href="vectsp_1.html#NM4" title="VECTSP_1:NM.4">Scalar</a> of <font color="Maroon" title="c1">R</font> ;<br/>
<b>consider </b><font color="Maroon" title="c4">b'</font> being   <a href="vectsp_1.html#NM4" title="VECTSP_1:NM.4">Scalar</a> of <font color="Maroon" title="c1">R</font><b> such that </b><br/><a NAME="E7:47"/><i><font color="Green" title="E22">A2</font></i>: 
<font color="Maroon" title="c3">x'</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c4">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="c1">R</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="c5">b</font> = <font color="Maroon" title="c4">b'</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c1">R</font>) ;<br/>
<b>take </b>
<font color="Maroon" title="c5">b</font>
;<br/>

<b>thus </b><a NAME="E9:47"/>
<font color="Maroon" title="c2">x</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Maroon" title="c5">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">H<sub>1</sub></font>(<font color="Maroon" title="c1">R</font>)
 <b>by </b><i><a class="txt" href="vectsp_2.html#E7:47"><i><font color="Green" title="E22">A2</font></i></a></i>;<br/>


</div>
