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<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>; <i><font color="Red">:: according to </font></i><a class="ref" href="vectsp_1.html#D26" title="VECTSP_1:def.26">VECTSP_1:def 26</a><br/>

<b>reconsider </b><font color="Maroon" title="c4">x'</font> = <font color="Maroon" title="c2">x</font>, <font color="Maroon" title="c5">y'</font> = <font color="Maroon" title="c3">y</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> ;<br/>
<b>let </b><font color="Maroon" title="c6">v</font>, <font color="Maroon" title="c7">w</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="polyalg1.html#K1" title="POLYALG1:func.1">Formal-Series</a> <font color="Maroon" title="c1">L</font></span>)</span>;<br/>

<b>reconsider </b><font color="Maroon" title="c8">p</font> = <font color="Maroon" title="c6">v</font>, <font color="Maroon" title="c9">q</font> = <font color="Maroon" title="c7">w</font> as   <a href="normsp_1.html#NM2" title="NORMSP_1:NM.2">sequence</a> of <font color="Maroon" title="c1">L</font> <b>by </b><i><a class="ref" href="polyalg1.html#D2" target="_self" title="POLYALG1:def.2">Def2</a></i>;<br/>
<a NAME="E3:27_1_1"/><i><font color="Green" title="E13">A1</font></i>: 
<font color="Maroon" title="c2">x</font> <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <font color="Maroon" title="c6">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <font color="Maroon" title="c8">p</font>
 
<b>by </b><i><a class="ref" href="polyalg1.html#D2" target="_self" title="POLYALG1:def.2">Def2</a></i>;<br/>
<a NAME="E4:27_1_1"/><i><font color="Green" title="E14">A2</font></i>: 
<font color="Maroon" title="c2">x</font> <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <font color="Maroon" title="c7">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <font color="Maroon" title="c9">q</font>
 
<b>by </b><i><a class="ref" href="polyalg1.html#D2" target="_self" title="POLYALG1:def.2">Def2</a></i>;<br/>
<a NAME="E5:27_1_1"/><i><font color="Green" title="E15">A3</font></i>: 
<font color="Maroon" title="c3">y</font> <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <font color="Maroon" title="c6">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">y</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <font color="Maroon" title="c8">p</font>
 
<b>by </b><i><a class="ref" href="polyalg1.html#D2" target="_self" title="POLYALG1:def.2">Def2</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c10">k</font> = <font color="Maroon" title="c6">v</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c7">w</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="polyalg1.html#K1" title="POLYALG1:func.1">Formal-Series</a> <font color="Maroon" title="c1">L</font></span>)</span> ;<br/>
<b>reconsider </b><font color="Maroon" title="c11">r</font> = <font color="Maroon" title="c10">k</font> as   <a href="normsp_1.html#NM2" title="NORMSP_1:NM.2">sequence</a> of <font color="Maroon" title="c1">L</font> <b>by </b><i><a class="ref" href="polyalg1.html#D2" target="_self" title="POLYALG1:def.2">Def2</a></i>;<br/>
<a NAME="E8:27_1_1"/>
<font color="Maroon" title="c2">x</font> <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <font color="Maroon" title="c11">r</font>
 
<b>by </b><i><a class="ref" href="polyalg1.html#D2" target="_self" title="POLYALG1:def.2">Def2</a></i>;<br/>
<b>hence </b><font color="Maroon" title="c2">x</font> <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">v</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c7">w</font></span>)</span> = 
<font color="Maroon" title="c2">x</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">p</font> <a href="polynom3.html#K9" title="POLYNOM3:func.9">+</a> <font color="Maroon" title="c9">q</font></span>)</span>
<b>by </b><i><a class="ref" href="polyalg1.html#D2" target="_self" title="POLYALG1:def.2">Def2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">x'</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <font color="Maroon" title="c8">p</font></span>)</span> <a href="polynom3.html#K9" title="POLYNOM3:func.9">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">x'</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <font color="Maroon" title="c9">q</font></span>)</span>
<b>by </b><i><a class="ref" href="polyalg1.html#T6" target="_self" title="POLYALG1:th.6">Th6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <font color="Maroon" title="c6">v</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="c2">x</font> <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <font color="Maroon" title="c7">w</font></span>)</span>
<b>by </b><i><a class="txt" href="polyalg1.html#E3:27_1_1"><i><font color="Green" title="E13">A1</font></i></a>, <a class="txt" href="polyalg1.html#E4:27_1_1"><i><font color="Green" title="E14">A2</font></i></a>, <a class="ref" href="polyalg1.html#D2" target="_self" title="POLYALG1:def.2">Def2</a></i>
;<br/><br/>

<b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <font color="Maroon" title="c6">v</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">x'</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c5">y'</font></span>)</span> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <font color="Maroon" title="c8">p</font>
<b>by </b><i><a class="ref" href="polyalg1.html#D2" target="_self" title="POLYALG1:def.2">Def2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">x'</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <font color="Maroon" title="c8">p</font></span>)</span> <a href="polynom3.html#K9" title="POLYNOM3:func.9">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">y'</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <font color="Maroon" title="c8">p</font></span>)</span>
<b>by </b><i><a class="ref" href="polyalg1.html#T7" target="_self" title="POLYALG1:th.7">Th7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <font color="Maroon" title="c6">v</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="c3">y</font> <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <font color="Maroon" title="c6">v</font></span>)</span>
<b>by </b><i><a class="txt" href="polyalg1.html#E3:27_1_1"><i><font color="Green" title="E13">A1</font></i></a>, <a class="txt" href="polyalg1.html#E5:27_1_1"><i><font color="Green" title="E15">A3</font></i></a>, <a class="ref" href="polyalg1.html#D2" target="_self" title="POLYALG1:def.2">Def2</a></i>
;<br/>

<b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <font color="Maroon" title="c6">v</font> = 
<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="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <font color="Maroon" title="c8">p</font>
<b>by </b><i><a class="ref" href="polyalg1.html#D2" target="_self" title="POLYALG1:def.2">Def2</a></i>
<br/>.= 
<font color="Maroon" title="c4">x'</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">y'</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <font color="Maroon" title="c8">p</font></span>)</span>
<b>by </b><i><a class="ref" href="polyalg1.html#T8" target="_self" title="POLYALG1:th.8">Th8</a></i>
<br/>.= 
<font color="Maroon" title="c2">x</font> <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">y</font> <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <font color="Maroon" title="c6">v</font></span>)</span>
<b>by </b><i><a class="txt" href="polyalg1.html#E5:27_1_1"><i><font color="Green" title="E15">A3</font></i></a>, <a class="ref" href="polyalg1.html#D2" target="_self" title="POLYALG1:def.2">Def2</a></i>
;<br/>

<b>thus </b><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="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <font color="Maroon" title="c6">v</font> = 
<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="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <font color="Maroon" title="c8">p</font>
<b>by </b><i><a class="ref" href="polyalg1.html#D2" target="_self" title="POLYALG1:def.2">Def2</a></i>
<br/>.= 
<font color="Maroon" title="c6">v</font>
<b>by </b><i><a class="ref" href="polyalg1.html#T9" target="_self" title="POLYALG1:th.9">Th9</a></i>
;<br/>


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