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<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="E1:195_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="group_1.html#V2" title="GROUP_1:attr.2">unital</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="195_1_1_1"><b>proof </b></a><div class="add">

<b>take </b><font color="Maroon" title="c4">e</font> =  <a href="vectsp_1.html#K2" title="VECTSP_1:func.2">1_</a> <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="group_1.html#D2" title="GROUP_1:def.2">GROUP_1:def 2</a><br/>

<b>thus </b><a NAME="E1:195_1_1_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="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> holds <br/> ( <font color="Olive">b<sub>1</sub></font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c4">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon" title="c4">e</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> )
 <b>by </b><i><a class="txt" href="polynom1.html#E122"><i><font color="Green" title="E104">Lm6</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="E3:195_1_1"/><i><font color="Green" title="E105">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="E4:195_1_1"/><i><font color="Green" title="E106">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="E5:195_1_1"/>
<font color="Maroon" title="c5">y</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</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#K4" title="RLVECT_1:func.4">+</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#K4" title="RLVECT_1:func.4">+</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#E3:195_1_1"><i><font color="Green" title="E105">A1</font></i></a>, <a class="txt" href="polynom1.html#E4:195_1_1"><i><font color="Green" title="E106">A2</font></i></a>, <a class="ref" href="polynom1.html#D27" target="_self" title="POLYNOM1:def.27">Def27</a></i>
;<br/><br/>


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