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

<b>let </b><font color="Maroon" title="c1">L</font> be  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> <a href="group_1.html#V4" title="GROUP_1:attr.4">associative</a> <a href="group_1.html#V7" title="GROUP_1:attr.7">commutative</a> <a href="vectsp_1.html#V4" title="VECTSP_1:attr.4">right-distributive</a> <a href="vectsp_1.html#V8" title="VECTSP_1:attr.8">left_unital</a> <a href="algstr_1.html#V1" title="ALGSTR_1:attr.1">left_zeroed</a> <a href="algstr_1.html#V3" title="ALGSTR_1:attr.3">add-right-cancelable</a>  <a href="vectsp_1.html#L3" title="VECTSP_1:struct.3">doubleLoopStr</a> ;<br/>

<b>assume </b><a NAME="E1:3_1_1"/><i><font color="Green" title="E4">A1</font></i>: 
<font color="Maroon" title="c1">L</font> is <a href="vectsp_1.html#V9" title="VECTSP_1:attr.9">Field-like</a>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1_1_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> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>;<br/><b>assume </b><a NAME="E1:3_1_1_1"/><i><font color="Green" title="E5">A2</font></i>: 
<font color="Maroon" title="c2">x</font> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <font color="Maroon" title="c3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c1">L</font>
 ;<br/><b>thus </b><a NAME="E2:3_1_1_1"/>
( <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c1">L</font> or <font color="Maroon" title="c3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c1">L</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:3_1_1_1_1"/>
<font color="Maroon" title="c2">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> <font color="Maroon" title="c1">L</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c4">z</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font><b> such that </b><br/><a NAME="E3:3_1_1_1_1"/><i><font color="Green" title="E6">A3</font></i>: 
<font color="Maroon" title="c2">x</font> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <font color="Maroon" title="c4">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">L</font>
 <b>by </b><i><a class="txt" href="polynom4.html#E1:3_1_1"><i><font color="Green" title="E4">A1</font></i></a>, <a class="ref" href="vectsp_1.html#D20" title="VECTSP_1:def.20">VECTSP_1:def 20</a></i>;<br/>
<font color="Maroon" title="c4">z</font> <a href="group_1.html#K10" title="GROUP_1:func.10">*</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">L</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">L</font></span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <font color="Maroon" title="c3">y</font>
<b>by </b><i><a class="txt" href="polynom4.html#E1:3_1_1_1"><i><font color="Green" title="E5">A2</font></i></a>, <a class="txt" href="polynom4.html#E3:3_1_1_1_1"><i><font color="Green" title="E6">A3</font></i></a>, <a class="ref" href="group_1.html#D4" title="GROUP_1:def.4">GROUP_1:def 4</a></i>
<br/>.= 
<font color="Maroon" title="c3">y</font>
<b>by </b><i><a class="ref" href="vectsp_1.html#D19" title="VECTSP_1:def.19">VECTSP_1:def 19</a></i>
;<br/>
<b>hence </b><a NAME="E5:3_1_1_1_1"/>
<font color="Maroon" title="c3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c1">L</font>
 <b>by </b><i><a class="txt" href="polynom4.html#E5"><i><font color="Green" title="E3">Lm2</font></i></a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:3_1_1"/>
<font color="Maroon" title="c1">L</font> is <a href="vectsp_2.html#V2" title="VECTSP_2:attr.2">domRing-like</a>
 <b>by </b><i><a class="ref" href="vectsp_2.html#D5" title="VECTSP_2:def.5">VECTSP_2:def 5</a></i>;<br/>


</div>
