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

<b>set </b><font color="Maroon" title="c3">Pm</font> =  <a href="polynom1.html#K30" title="POLYNOM1:func.30">Polynom-Ring</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">L</font>;<br/>
<b>thus </b><a NAME="E1:192_1_1"/>
 <a href="polynom1.html#K30" title="POLYNOM1:func.30">Polynom-Ring</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">L</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="192_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#K30" title="POLYNOM1:func.30">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="E1:192_1_1_1"/>
( <font color="Maroon" title="c4">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="polynom1.html#K30" title="POLYNOM1:func.30">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#K30" title="POLYNOM1:func.30">Polynom-Ring</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">L</font></span>)</span></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</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="E102">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#K30" title="POLYNOM1:func.30">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="E3:192_1_1"/><i><font color="Green" title="E103">A1</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</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#K28" title="POLYNOM1:func.28">*'</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="E4:192_1_1"/><i><font color="Green" title="E104">A2</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</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#K28" title="POLYNOM1:func.28">*'</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="E5:192_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#K23" title="POLYNOM1:func.23">+</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="group_1.html#K1" title="GROUP_1:func.1">*</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#K28" title="POLYNOM1:func.28">*'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="polynom1.html#K23" title="POLYNOM1:func.23">+</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#K28" title="POLYNOM1:func.28">*'</a> <font color="Maroon" title="c8">q</font></span>)</span> <a href="polynom1.html#K23" title="POLYNOM1:func.23">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">p</font> <a href="polynom1.html#K28" title="POLYNOM1:func.28">*'</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="group_1.html#K1" title="GROUP_1:func.1">*</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="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c6">z</font></span>)</span>
<b>by </b><i><a class="txt" href="polynom1.html#E3:192_1_1"><i><font color="Green" title="E103">A1</font></i></a>, <a class="txt" href="polynom1.html#E4:192_1_1"><i><font color="Green" title="E104">A2</font></i></a>, <a class="ref" href="polynom1.html#D27" target="_self" title="POLYNOM1:def.27">Def27</a></i>
;<br/><br/>


</div>
