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

<b>let </b><font color="Maroon" title="c1">A</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<a NAME="E1:47"/><i><font color="Green" title="E29">A1</font></i>: 
 <a href="funcsdom.html#K12" title="FUNCSDOM:func.12">RRing</a> <font color="Maroon" title="c1">A</font> is <a href="group_1.html#V6" title="GROUP_1:attr.6">commutative</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1"><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 <span class="p1">(<span class="default"><a href="funcsdom.html#K12" title="FUNCSDOM:func.12">RRing</a> <font color="Maroon" title="c1">A</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="group_1.html#D16" title="GROUP_1:def.16">GROUP_1:def 16</a><br/>

<b>thus </b><a NAME="E1:47_1"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="funcsdom.html#K12" title="FUNCSDOM:func.12">RRing</a> <font color="Maroon" title="c1">A</font></span>)</span> holds  <font color="Maroon" title="c2">x</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c2">x</font>
 <b>by </b><i><a class="ref" href="funcsdom.html#T42" target="_self" title="FUNCSDOM:th.42">Th42</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:47"/><i><font color="Green" title="E30">A2</font></i>: 
 <a href="funcsdom.html#K12" title="FUNCSDOM:func.12">RRing</a> <font color="Maroon" title="c1">A</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> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="funcsdom.html#K12" title="FUNCSDOM:func.12">RRing</a> <font color="Maroon" title="c1">A</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>thus </b><a NAME="E1:47_2"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="funcsdom.html#K12" title="FUNCSDOM:func.12">RRing</a> <font color="Maroon" title="c1">A</font></span>)</span> holds  <font color="Maroon" title="c2">x</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Maroon" title="c2">x</font>
 <b>by </b><i><a class="ref" href="funcsdom.html#T42" target="_self" title="FUNCSDOM:th.42">Th42</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:47"/><i><font color="Green" title="E31">A3</font></i>: 
 <a href="funcsdom.html#K12" title="FUNCSDOM:func.12">RRing</a> <font color="Maroon" title="c1">A</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> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="funcsdom.html#K12" title="FUNCSDOM:func.12">RRing</a> <font color="Maroon" title="c1">A</font></span>)</span>; <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>thus </b><a NAME="E1:47_3"/>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="funcsdom.html#K12" title="FUNCSDOM:func.12">RRing</a> <font color="Maroon" title="c1">A</font></span>)</span> holds  <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="Olive">b<sub>1</sub></font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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="Olive">b<sub>1</sub></font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Olive">b<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="funcsdom.html#T42" target="_self" title="FUNCSDOM:th.42">Th42</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:47"/><i><font color="Green" title="E32">A4</font></i>: 
 <a href="funcsdom.html#K12" title="FUNCSDOM:func.12">RRing</a> <font color="Maroon" title="c1">A</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 <span class="p1">(<span class="default"><a href="funcsdom.html#K12" title="FUNCSDOM:func.12">RRing</a> <font color="Maroon" title="c1">A</font></span>)</span>; <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>thus </b><a NAME="E1:47_4"/>
<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> <span class="p2">(<span class="default"><a href="funcsdom.html#K12" title="FUNCSDOM:func.12">RRing</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font>
 <b>by </b><i><a class="ref" href="funcsdom.html#T42" target="_self" title="FUNCSDOM:th.42">Th42</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:47"/><i><font color="Green" title="E33">A5</font></i>: 
 <a href="funcsdom.html#K12" title="FUNCSDOM:func.12">RRing</a> <font color="Maroon" title="c1">A</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="47_5"><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 <span class="p1">(<span class="default"><a href="funcsdom.html#K12" title="FUNCSDOM:func.12">RRing</a> <font color="Maroon" title="c1">A</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>thus </b><a NAME="E1:47_5"/>
 ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="funcsdom.html#K12" title="FUNCSDOM:func.12">RRing</a> <font color="Maroon" title="c1">A</font></span>)</span> st <font color="Maroon" title="c2">x</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Olive">b<sub>1</sub></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="funcsdom.html#K12" title="FUNCSDOM:func.12">RRing</a> <font color="Maroon" title="c1">A</font></span>)</span>
 <b>by </b><i><a class="ref" href="funcsdom.html#T42" target="_self" title="FUNCSDOM:th.42">Th42</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:47"/><i><font color="Green" title="E34">A6</font></i>: 
 <a href="funcsdom.html#K12" title="FUNCSDOM:func.12">RRing</a> <font color="Maroon" title="c1">A</font> is <a href="group_1.html#V4" title="GROUP_1:attr.4">associative</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_6"><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 <span class="p1">(<span class="default"><a href="funcsdom.html#K12" title="FUNCSDOM:func.12">RRing</a> <font color="Maroon" title="c1">A</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="group_1.html#D4" title="GROUP_1:def.4">GROUP_1:def 4</a><br/>

<b>thus </b><a NAME="E1:47_6"/>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="funcsdom.html#K12" title="FUNCSDOM:func.12">RRing</a> <font color="Maroon" title="c1">A</font></span>)</span> holds  <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Olive">b<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="funcsdom.html#T42" target="_self" title="FUNCSDOM:th.42">Th42</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:47"/><i><font color="Green" title="E35">A7</font></i>: 
 <a href="funcsdom.html#K12" title="FUNCSDOM:func.12">RRing</a> <font color="Maroon" title="c1">A</font> is <a href="vectsp_1.html#V7" title="VECTSP_1:attr.7">well-unital</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_7"><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 <span class="p1">(<span class="default"><a href="funcsdom.html#K12" title="FUNCSDOM:func.12">RRing</a> <font color="Maroon" title="c1">A</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="vectsp_1.html#D16" title="VECTSP_1:def.16">VECTSP_1:def 16</a><br/>

<b>thus </b><a NAME="E1:47_7"/>
( <font color="Maroon" title="c2">x</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <span class="p2">(<span class="default"><a href="funcsdom.html#K12" title="FUNCSDOM:func.12">RRing</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font> &amp; <span class="p1">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <span class="p2">(<span class="default"><a href="funcsdom.html#K12" title="FUNCSDOM:func.12">RRing</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font> )
 <b>by </b><i><a class="ref" href="funcsdom.html#T42" target="_self" title="FUNCSDOM:th.42">Th42</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:47"/>
 <a href="funcsdom.html#K12" title="FUNCSDOM:func.12">RRing</a> <font color="Maroon" title="c1">A</font> is <a href="vectsp_1.html#V8" title="VECTSP_1:attr.8">distributive</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_8"><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 <span class="p1">(<span class="default"><a href="funcsdom.html#K12" title="FUNCSDOM:func.12">RRing</a> <font color="Maroon" title="c1">A</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="vectsp_1.html#D18" title="VECTSP_1:def.18">VECTSP_1:def 18</a><br/>

<b>thus </b><a NAME="E1:47_8"/>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="funcsdom.html#K12" title="FUNCSDOM:func.12">RRing</a> <font color="Maroon" title="c1">A</font></span>)</span> holds <br/> ( <font color="Maroon" title="c2">x</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Olive">b<sub>1</sub></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="c2">x</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Olive">b<sub>2</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c2">x</font></span>)</span> )
 <b>by </b><i><a class="ref" href="funcsdom.html#T42" target="_self" title="FUNCSDOM:th.42">Th42</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E9:47"/>
 <a href="funcsdom.html#K12" title="FUNCSDOM:func.12">RRing</a> <font color="Maroon" title="c1">A</font> is  <a href="group_1.html#V6" title="GROUP_1:attr.6">commutative</a> <a href="funcsdom.html#NM1" title="FUNCSDOM:NM.1">Ring</a>
 <b>by </b><i><a class="txt" href="funcsdom.html#E1:47"><i><font color="Green" title="E29">A1</font></i></a>, <a class="txt" href="funcsdom.html#E2:47"><i><font color="Green" title="E30">A2</font></i></a>, <a class="txt" href="funcsdom.html#E3:47"><i><font color="Green" title="E31">A3</font></i></a>, <a class="txt" href="funcsdom.html#E4:47"><i><font color="Green" title="E32">A4</font></i></a>, <a class="txt" href="funcsdom.html#E5:47"><i><font color="Green" title="E33">A5</font></i></a>, <a class="txt" href="funcsdom.html#E6:47"><i><font color="Green" title="E34">A6</font></i></a>, <a class="txt" href="funcsdom.html#E7:47"><i><font color="Green" title="E35">A7</font></i></a></i>;<br/>


</div>
