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

<b>set </b><font color="Maroon" title="c3">Pm</font> =  <a href="polynom1.html#K29" title="POLYNOM1:func.29">Polynom-Ring</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">L</font>;<br/>
<b>thus </b><a NAME="E1:187_1_1"/>
 <a href="polynom1.html#K29" title="POLYNOM1:func.29">Polynom-Ring</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">L</font> is <a href="vectsp_1.html#V4" title="VECTSP_1:attr.4">well-unital</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="187_1_1_1"><b>proof </b></a><div class="add">

<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"><a href="polynom1.html#K29" title="POLYNOM1:func.29">Polynom-Ring</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">L</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="E1:187_1_1_1"/>
( <font color="Maroon" title="c4">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</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="polynom1.html#K29" title="POLYNOM1:func.29">Polynom-Ring</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">L</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">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="polynom1.html#K29" title="POLYNOM1:func.29">Polynom-Ring</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">L</font></span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x</font> )
 <b>by </b><i><a class="txt" href="polynom1.html#E121"><i><font color="Green" title="E101">Lm5</font></i></a></i>;<br/>


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

<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="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K29" title="POLYNOM1:func.29">Polynom-Ring</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">L</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="vectsp_1.html#D11" title="VECTSP_1:def.11">VECTSP_1:def 11</a><br/>

<b>reconsider </b><font color="Maroon" title="c7">p</font> = <font color="Maroon" title="c4">x</font>, <font color="Maroon" title="c8">q</font> = <font color="Maroon" title="c5">y</font>, <font color="Maroon" title="c9">r</font> = <font color="Maroon" title="c6">z</font> as   <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">L</font> <b>by </b><i><a class="ref" href="polynom1.html#D27" target="_self" title="POLYNOM1:def.27">Def27</a></i>;<br/>
<a NAME="E3:187_1_1"/><i><font color="Green" title="E102">A1</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">p</font> <a href="polynom1.html#K27" title="POLYNOM1:func.27">*'</a> <font color="Maroon" title="c8">q</font>
 
<b>by </b><i><a class="ref" href="polynom1.html#D27" target="_self" title="POLYNOM1:def.27">Def27</a></i>;<br/>
<a NAME="E4:187_1_1"/><i><font color="Green" title="E103">A2</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c6">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">p</font> <a href="polynom1.html#K27" title="POLYNOM1:func.27">*'</a> <font color="Maroon" title="c9">r</font>
 
<b>by </b><i><a class="ref" href="polynom1.html#D27" target="_self" title="POLYNOM1:def.27">Def27</a></i>;<br/>
<a NAME="E5:187_1_1"/>
<font color="Maroon" title="c5">y</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c6">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">q</font> <a href="polynom1.html#K22" title="POLYNOM1:func.22">+</a> <font color="Maroon" title="c9">r</font>
 
<b>by </b><i><a class="ref" href="polynom1.html#D27" target="_self" title="POLYNOM1:def.27">Def27</a></i>;<br/>
<b>hence </b><font color="Maroon" title="c4">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">y</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c6">z</font></span>)</span> = 
<font color="Maroon" title="c7">p</font> <a href="polynom1.html#K27" title="POLYNOM1:func.27">*'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="polynom1.html#K22" title="POLYNOM1:func.22">+</a> <font color="Maroon" title="c9">r</font></span>)</span>
<b>by </b><i><a class="ref" href="polynom1.html#D27" target="_self" title="POLYNOM1:def.27">Def27</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">p</font> <a href="polynom1.html#K27" title="POLYNOM1:func.27">*'</a> <font color="Maroon" title="c8">q</font></span>)</span> <a href="polynom1.html#K22" title="POLYNOM1:func.22">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">p</font> <a href="polynom1.html#K27" title="POLYNOM1:func.27">*'</a> <font color="Maroon" title="c9">r</font></span>)</span>
<b>by </b><i><a class="ref" href="polynom1.html#T85" target="_self" title="POLYNOM1:th.85">Th85</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">y</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c6">z</font></span>)</span>
<b>by </b><i><a class="txt" href="polynom1.html#E3:187_1_1"><i><font color="Green" title="E102">A1</font></i></a>, <a class="txt" href="polynom1.html#E4:187_1_1"><i><font color="Green" title="E103">A2</font></i></a>, <a class="ref" href="polynom1.html#D27" target="_self" title="POLYNOM1:def.27">Def27</a></i>
;<br/><br/>


</div>
