<?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/><b>let </b><font color="Maroon" title="c2">I</font> be   <a href="ideal_1.html#NM1" title="IDEAL_1:NM.1">Ideal</a> of <font color="Maroon" title="c1">R</font>;<br/>



<b>set </b><font color="Maroon" title="c3">E</font> =  <a href="ring_1.html#K1" title="RING_1:func.1">EqRel</a> <font color="Maroon" title="c1">R</font>,<font color="Maroon" title="c2">I</font>;<br/>
<b>assume </b><b>that </b><br/><a NAME="E1:38"/><i><font color="Green" title="E25">A1</font></i>: 
<font color="Maroon" title="c1">R</font> is <a href="group_1.html#V7" title="GROUP_1:attr.7">commutative</a>
 <b>and </b><br/><a NAME="E2:38"/><i><font color="Green" title="E26">A2</font></i>: 
<font color="Maroon" title="c2">I</font> is <a href="ring_1.html#V3" title="RING_1:attr.3">quasi-maximal</a>
 ;<br/>

<b>let </b><font color="Maroon" title="c4">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">R</font> <a href="ring_1.html#NK3" title="RING_1:NK.3">/</a> <font color="Maroon" title="c2">I</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="vectsp_1.html#D20" title="VECTSP_1:def.20">VECTSP_1:def 20</a><br/>

<b>assume </b><a NAME="E3:38"/><i><font color="Green" title="E27">A3</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">R</font> <a href="ring_1.html#NK3" title="RING_1:NK.3">/</a> <font color="Maroon" title="c2">I</font></span>)</span>
 ;<br/>

<b>consider </b><font color="Maroon" title="c5">a</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font><b> such that </b><br/><a NAME="E5:38"/><i><font color="Green" title="E28">A4</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K7" title="EQREL_1:func.7">Class</a> <span class="p1">(<span class="default"><a href="ring_1.html#K1" title="RING_1:func.1">EqRel</a> <font color="Maroon" title="c1">R</font>,<font color="Maroon" title="c2">I</font></span>)</span>,<font color="Maroon" title="c5">a</font>
 <b>by </b><i><a class="ref" href="ring_1.html#T11" target="_self" title="RING_1:th.11">Th11</a></i>;<br/>
<b>set </b><font color="Maroon" title="c6">M</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Olive" title="b1">r</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Olive" title="b2">s</font></span>)</span> where <font color="Olive" title="b1">r</font>, <font color="Olive" title="b2">s</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font> : <font color="Olive" title="b2">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font> </span>}</span> ;<br/>
<a NAME="E6:38"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Olive" title="b1">r</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Olive" title="b2">s</font></span>)</span> where <font color="Olive" title="b1">r</font>, <font color="Olive" title="b2">s</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font> : <font color="Olive" title="b2">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font> </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">k</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:38_1"/>
<font color="Maroon" title="c7">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Olive" title="b1">r</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Olive" title="b2">s</font></span>)</span> where <font color="Olive" title="b1">r</font>, <font color="Olive" title="b2">s</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font> : <font color="Olive" title="b2">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font> </span>}</span> 
 ;<br/>

<a NAME="E2:38_1"/><b>then </b>
 ex <font color="Olive" title="b1">r</font>, <font color="Olive" title="b2">s</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font> st <br/>( <font color="Maroon" title="c7">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Olive" title="b1">r</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Olive" title="b2">s</font> &amp; <font color="Olive" title="b2">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font> )
 
;<br/>
<b>hence </b><a NAME="E3:38_1"/>
<font color="Maroon" title="c7">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font>
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c7">M</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Olive" title="b1">r</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Olive" title="b2">s</font></span>)</span> where <font color="Olive" title="b1">r</font>, <font color="Olive" title="b2">s</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font> : <font color="Olive" title="b2">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font> </span>}</span>  as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">R</font> ;<br/>
<i><font color="Green" title="E29">A5</font></i>: <span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p2">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">R</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c1">R</font></span>)</span> = 
<font color="Maroon" title="c5">a</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c1">R</font></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_1.html#D13" title="VECTSP_1:def.13">VECTSP_1:def 13</a></i>
<br/>.= 
<font color="Maroon" title="c5">a</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#D7" title="RLVECT_1:def.7">RLVECT_1:def 7</a></i>
;<br/>
<a NAME="E9:38"/>
 <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c1">R</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font>
 
<b>by </b><i><a class="ref" href="ideal_1.html#T2" title="IDEAL_1:th.2">IDEAL_1:2</a></i>;<br/>
<a NAME="E10:38"/><b>then </b><i><font color="Green" title="E30">A6</font></i>: 
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">M</font>
 
<b>by </b><i><a class="txt" href="ring_1.html#E8:38"><i><font color="Green" title="E29">A5</font></i></a></i>;<br/>
<a NAME="E11:38"/><i><font color="Green" title="E31">A7</font></i>: 
<font color="Maroon" title="c7">M</font> is <a href="ideal_1.html#V1" title="IDEAL_1:attr.1">add-closed</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">c</font>, <font color="Maroon" title="c9">d</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="ideal_1.html#D1" title="IDEAL_1:def.1">IDEAL_1:def 1</a><br/>

<b>assume </b><a NAME="E1:38_3"/>
<font color="Maroon" title="c8">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">M</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c10">rc</font>, <font color="Maroon" title="c11">sc</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font><b> such that </b><br/><a NAME="E3:38_3"/><i><font color="Green" title="E32">A8</font></i>: 
<font color="Maroon" title="c8">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c10">rc</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c11">sc</font>
 <b>and </b><br/><a NAME="E4:38_3"/><i><font color="Green" title="E33">A9</font></i>: 
<font color="Maroon" title="c11">sc</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font>
 ;<br/>
<b>assume </b><a NAME="E5:38_3"/>
<font color="Maroon" title="c9">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">M</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c12">rd</font>, <font color="Maroon" title="c13">sd</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font><b> such that </b><br/><a NAME="E7:38_3"/><i><font color="Green" title="E34">A10</font></i>: 
<font color="Maroon" title="c9">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c12">rd</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c13">sd</font>
 <b>and </b><br/><a NAME="E8:38_3"/><i><font color="Green" title="E35">A11</font></i>: 
<font color="Maroon" title="c13">sd</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font>
 ;<br/>
<a NAME="E9:38_3"/><i><font color="Green" title="E36">A12</font></i>: 
<font color="Maroon" title="c11">sc</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c13">sd</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font>
 
<b>by </b><i><a class="txt" href="ring_1.html#E4:38_3"><i><font color="Green" title="E33">A9</font></i></a>, <a class="txt" href="ring_1.html#E8:38_3"><i><font color="Green" title="E35">A11</font></i></a>, <a class="ref" href="ideal_1.html#D1" title="IDEAL_1:def.1">IDEAL_1:def 1</a></i>;<br/>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">rc</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c12">rd</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">sc</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c13">sd</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c10">rc</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c12">rd</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">sc</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c13">sd</font></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_1.html#D11" title="VECTSP_1:def.11">VECTSP_1:def 11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c10">rc</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c12">rd</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c11">sc</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c13">sd</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#D6" title="RLVECT_1:def.6">RLVECT_1:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c10">rc</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c11">sc</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c12">rd</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c13">sd</font>
<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="c8">c</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c9">d</font>
<b>by </b><i><a class="txt" href="ring_1.html#E3:38_3"><i><font color="Green" title="E32">A8</font></i></a>, <a class="txt" href="ring_1.html#E7:38_3"><i><font color="Green" title="E34">A10</font></i></a>, <a class="ref" href="rlvect_1.html#D6" title="RLVECT_1:def.6">RLVECT_1:def 6</a></i>
;<br/>
<b>hence </b><a NAME="E11:38_3"/>
<font color="Maroon" title="c8">c</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c9">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">M</font>
 <b>by </b><i><a class="txt" href="ring_1.html#E9:38_3"><i><font color="Green" title="E36">A12</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E12:38"/><i><font color="Green" title="E32">A13</font></i>: 
<font color="Maroon" title="c7">M</font> is <a href="ideal_1.html#V2" title="IDEAL_1:attr.2">left-ideal</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">p</font>, <font color="Maroon" title="c9">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="ideal_1.html#D2" title="IDEAL_1:def.2">IDEAL_1:def 2</a><br/>

<b>assume </b><a NAME="E1:38_4"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">M</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c10">r</font>, <font color="Maroon" title="c11">s</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font><b> such that </b><br/><a NAME="E3:38_4"/><i><font color="Green" title="E33">A14</font></i>: 
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c10">r</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c11">s</font>
 <b>and </b><br/><a NAME="E4:38_4"/><i><font color="Green" title="E34">A15</font></i>: 
<font color="Maroon" title="c11">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font>
 ;<br/>
<a NAME="E5:38_4"/><i><font color="Green" title="E35">A16</font></i>: 
<font color="Maroon" title="c8">p</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c11">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font>
 
<b>by </b><i><a class="txt" href="ring_1.html#E4:38_4"><i><font color="Green" title="E34">A15</font></i></a>, <a class="ref" href="ideal_1.html#D2" title="IDEAL_1:def.2">IDEAL_1:def 2</a></i>;<br/>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">r</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c8">p</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">p</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c11">s</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c10">r</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c8">p</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">p</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c11">s</font></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#D4" title="GROUP_1:def.4">GROUP_1:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c10">r</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c8">p</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">s</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c8">p</font></span>)</span>
<b>by </b><i><a class="txt" href="ring_1.html#E1:38"><i><font color="Green" title="E25">A1</font></i></a>, <a class="ref" href="group_1.html#D16" title="GROUP_1:def.16">GROUP_1:def 16</a></i>
<br/>.= 
<font color="Maroon" title="c9">x</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c8">p</font>
<b>by </b><i><a class="txt" href="ring_1.html#E3:38_4"><i><font color="Green" title="E33">A14</font></i></a>, <a class="ref" href="vectsp_1.html#D12" title="VECTSP_1:def.12">VECTSP_1:def 12</a></i>
<br/>.= 
<font color="Maroon" title="c8">p</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c9">x</font>
<b>by </b><i><a class="txt" href="ring_1.html#E1:38"><i><font color="Green" title="E25">A1</font></i></a>, <a class="ref" href="group_1.html#D16" title="GROUP_1:def.16">GROUP_1:def 16</a></i>
;<br/>
<b>hence </b><a NAME="E7:38_4"/>
<font color="Maroon" title="c8">p</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">M</font>
 <b>by </b><i><a class="txt" href="ring_1.html#E5:38_4"><i><font color="Green" title="E35">A16</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E13:38"/><i><font color="Green" title="E33">A17</font></i>: 
<font color="Maroon" title="c7">M</font> is <a href="ideal_1.html#V3" title="IDEAL_1:attr.3">right-ideal</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">p</font>, <font color="Maroon" title="c9">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="ideal_1.html#D3" title="IDEAL_1:def.3">IDEAL_1:def 3</a><br/>

<b>assume </b><a NAME="E1:38_5"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">M</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c10">r</font>, <font color="Maroon" title="c11">s</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font><b> such that </b><br/><a NAME="E3:38_5"/><i><font color="Green" title="E34">A18</font></i>: 
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c10">r</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c11">s</font>
 <b>and </b><br/><a NAME="E4:38_5"/><i><font color="Green" title="E35">A19</font></i>: 
<font color="Maroon" title="c11">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font>
 ;<br/>
<a NAME="E5:38_5"/><i><font color="Green" title="E36">A20</font></i>: 
<font color="Maroon" title="c8">p</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c11">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font>
 
<b>by </b><i><a class="txt" href="ring_1.html#E4:38_5"><i><font color="Green" title="E35">A19</font></i></a>, <a class="ref" href="ideal_1.html#D2" title="IDEAL_1:def.2">IDEAL_1:def 2</a></i>;<br/>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">r</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c8">p</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">p</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c11">s</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c10">r</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c8">p</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">p</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c11">s</font></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#D4" title="GROUP_1:def.4">GROUP_1:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c10">r</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c8">p</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">s</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c8">p</font></span>)</span>
<b>by </b><i><a class="txt" href="ring_1.html#E1:38"><i><font color="Green" title="E25">A1</font></i></a>, <a class="ref" href="group_1.html#D16" title="GROUP_1:def.16">GROUP_1:def 16</a></i>
<br/>.= 
<font color="Maroon" title="c9">x</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c8">p</font>
<b>by </b><i><a class="txt" href="ring_1.html#E3:38_5"><i><font color="Green" title="E34">A18</font></i></a>, <a class="ref" href="vectsp_1.html#D12" title="VECTSP_1:def.12">VECTSP_1:def 12</a></i>
;<br/>
<b>hence </b><a NAME="E7:38_5"/>
<font color="Maroon" title="c9">x</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c8">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">M</font>
 <b>by </b><i><a class="txt" href="ring_1.html#E5:38_5"><i><font color="Green" title="E36">A20</font></i></a></i>;<br/>


</div><b>end;</b></div>
<div><i><font color="Green" title="E34">A21</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="38_6"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:38_6"/><i><font color="Green" title="E35">A22</font></i>: 
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font>
 ;<br/><a NAME="E2:38_6"/>
<font color="Maroon" title="c5">a</font> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c1">R</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</font>
 <b>by </b><i><a class="ref" href="rlvect_1.html#T26" title="RLVECT_1:th.26">RLVECT_1:26</a></i>;<br/><b>then </b> <a href="eqrel_1.html#K7" title="EQREL_1:func.7">Class</a> <span class="p1">(<span class="default"><a href="ring_1.html#K1" title="RING_1:func.1">EqRel</a> <font color="Maroon" title="c1">R</font>,<font color="Maroon" title="c2">I</font></span>)</span>,<font color="Maroon" title="c5">a</font> = 
 <a href="eqrel_1.html#K7" title="EQREL_1:func.7">Class</a> <span class="p1">(<span class="default"><a href="ring_1.html#K1" title="RING_1:func.1">EqRel</a> <font color="Maroon" title="c1">R</font>,<font color="Maroon" title="c2">I</font></span>)</span>,<span class="p1">(<span class="default"><a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c1">R</font></span>)</span>
<b>by </b><i><a class="txt" href="ring_1.html#E1:38_6"><i><font color="Green" title="E35">A22</font></i></a>, <a class="ref" href="ring_1.html#T6" target="_self" title="RING_1:th.6">Th6</a></i>
<br/>.= 
 <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">R</font> <a href="ring_1.html#NK3" title="RING_1:NK.3">/</a> <font color="Maroon" title="c2">I</font></span>)</span>
<b>by </b><i><a class="ref" href="ring_1.html#D6" target="_self" title="RING_1:def.6">Def6</a></i>
;<br/><b>hence </b><a NAME="E4:38_6"/>
contradiction
 <b>by </b><i><a class="txt" href="ring_1.html#E3:38"><i><font color="Green" title="E27">A3</font></i></a>, <a class="txt" href="ring_1.html#E5:38"><i><font color="Green" title="E28">A4</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E15:38"/>
<font color="Maroon" title="c2">I</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">M</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_7"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">i</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:38_7"/>
<font color="Maroon" title="c8">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c9">i</font> = <font color="Maroon" title="c8">i</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">I</font> ;<br/>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c1">R</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c9">i</font> = 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c1">R</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c9">i</font>
<b>by </b><i><a class="ref" href="vectsp_1.html#T36" title="VECTSP_1:th.36">VECTSP_1:36</a></i>
<br/>.= 
<font color="Maroon" title="c9">i</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#D7" title="RLVECT_1:def.7">RLVECT_1:def 7</a></i>
;<br/>
<b>hence </b><a NAME="E4:38_7"/>
<font color="Maroon" title="c8">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">M</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E16:38"/><b>then </b>
not <font color="Maroon" title="c7">M</font> is <a href="ideal_1.html#V5" title="IDEAL_1:attr.5">proper</a>
 
<b>by </b><i><a class="txt" href="ring_1.html#E10:38"><i><font color="Green" title="E30">A6</font></i></a>, <a class="txt" href="ring_1.html#E14:38"><i><font color="Green" title="E34">A21</font></i></a>, <a class="txt" href="ring_1.html#E2:38"><i><font color="Green" title="E26">A2</font></i></a>, <a class="txt" href="ring_1.html#E13:38"><i><font color="Green" title="E33">A17</font></i></a>, <a class="txt" href="ring_1.html#E11:38"><i><font color="Green" title="E31">A7</font></i></a>, <a class="txt" href="ring_1.html#E12:38"><i><font color="Green" title="E32">A13</font></i></a>, <a class="ref" href="ring_1.html#D3" target="_self" title="RING_1:def.3">Def3</a></i>;<br/>
<a NAME="E17:38"/><b>then </b>
<font color="Maroon" title="c7">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font>
 
<b>by </b><i><a class="ref" href="ideal_1.html#D5" title="IDEAL_1:def.5">IDEAL_1:def 5</a></i>;<br/>
<a NAME="E18:38"/><b>then </b>
 <a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">R</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">M</font>
 
;<br/>
<b>then consider </b><font color="Maroon" title="c8">b</font>, <font color="Maroon" title="c9">m</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font><b> such that </b><br/><a NAME="E20:38"/><i><font color="Green" title="E35">A23</font></i>: 
 <a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c8">b</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c9">m</font>
 <b>and </b><br/><a NAME="E21:38"/><i><font color="Green" title="E36">A24</font></i>: 
<font color="Maroon" title="c9">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font>
 ;<br/>
<b>reconsider </b><font color="Maroon" title="c10">y</font> =  <a href="eqrel_1.html#K7" title="EQREL_1:func.7">Class</a> <span class="p1">(<span class="default"><a href="ring_1.html#K1" title="RING_1:func.1">EqRel</a> <font color="Maroon" title="c1">R</font>,<font color="Maroon" title="c2">I</font></span>)</span>,<font color="Maroon" title="c8">b</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="c1">R</font> <a href="ring_1.html#NK3" title="RING_1:NK.3">/</a> <font color="Maroon" title="c2">I</font></span>)</span> <b>by </b><i><a class="ref" href="ring_1.html#T12" target="_self" title="RING_1:th.12">Th12</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c10">y</font>
;<br/>

<a NAME="E23:38"/><i><font color="Green" title="E37">A25</font></i>: 
<font color="Maroon" title="c9">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">R</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c8">b</font></span>)</span>
 
<b>by </b><i><a class="txt" href="ring_1.html#E20:38"><i><font color="Green" title="E35">A23</font></i></a>, <a class="ref" href="vectsp_2.html#T22" title="VECTSP_2:th.22">VECTSP_2:22</a></i>;<br/>
<a NAME="E24:38"/><i><font color="Green" title="E38">A26</font></i>: 
 <a href="eqrel_1.html#K7" title="EQREL_1:func.7">Class</a> <span class="p1">(<span class="default"><a href="ring_1.html#K1" title="RING_1:func.1">EqRel</a> <font color="Maroon" title="c1">R</font>,<font color="Maroon" title="c2">I</font></span>)</span>,<span class="p1">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">R</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">R</font> <a href="ring_1.html#NK3" title="RING_1:NK.3">/</a> <font color="Maroon" title="c2">I</font></span>)</span>
 
<b>by </b><i><a class="ref" href="ring_1.html#T15" target="_self" title="RING_1:th.15">Th15</a></i>;<br/>
<b>thus </b><font color="Maroon" title="c4">x</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c10">y</font> = 
 <a href="eqrel_1.html#K7" title="EQREL_1:func.7">Class</a> <span class="p1">(<span class="default"><a href="ring_1.html#K1" title="RING_1:func.1">EqRel</a> <font color="Maroon" title="c1">R</font>,<font color="Maroon" title="c2">I</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c8">b</font></span>)</span>
<b>by </b><i><a class="txt" href="ring_1.html#E5:38"><i><font color="Green" title="E28">A4</font></i></a>, <a class="ref" href="ring_1.html#T14" target="_self" title="RING_1:th.14">Th14</a></i>
<br/>.= 
 <a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">R</font> <a href="ring_1.html#NK3" title="RING_1:NK.3">/</a> <font color="Maroon" title="c2">I</font></span>)</span>
<b>by </b><i><a class="txt" href="ring_1.html#E21:38"><i><font color="Green" title="E36">A24</font></i></a>, <a class="txt" href="ring_1.html#E23:38"><i><font color="Green" title="E37">A25</font></i></a>, <a class="txt" href="ring_1.html#E24:38"><i><font color="Green" title="E38">A26</font></i></a>, <a class="ref" href="ring_1.html#T6" target="_self" title="RING_1:th.6">Th6</a></i>
;<br/>


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