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<b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>, <font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="vectsp_1.html#D26">VECTSP_1:def 26</a><br/>

<b>reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font>, <font color="Maroon">y'</font> = <font color="Maroon">y</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> ;<br/>
<b>let </b><font color="Maroon">v</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="polyalg1.html#K1" target="_self">Formal-Series</a> <font color="Maroon">L</font></span>)</span>, <font color="Maroon">w</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="polyalg1.html#K1" target="_self">Formal-Series</a> <font color="Maroon">L</font></span>)</span>;<br/>

<b>reconsider </b><font color="Maroon">p</font> = <font color="Maroon">v</font>, <font color="Maroon">q</font> = <font color="Maroon">w</font> as   <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">L</font> <b>by </b><i/>;<br/>
<a NAME="E3:27_1_1"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">x</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="polynom5.html#K3">*</a> <font color="Maroon">p</font>
 
<b>by </b><i/>;<br/>
<a NAME="E4:27_1_1"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">x</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">w</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="polynom5.html#K3">*</a> <font color="Maroon">q</font>
 
<b>by </b><i/>;<br/>
<a NAME="E5:27_1_1"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">y</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> <a href="polynom5.html#K3">*</a> <font color="Maroon">p</font>
 
<b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">k</font> = <font color="Maroon">v</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">w</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="polyalg1.html#K1" target="_self">Formal-Series</a> <font color="Maroon">L</font></span>)</span> ;<br/>
<b>reconsider </b><font color="Maroon">r</font> = <font color="Maroon">k</font> as   <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">L</font> <b>by </b><i/>;<br/>
<a NAME="E8:27_1_1"/>
<font color="Maroon">x</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="polynom5.html#K3">*</a> <font color="Maroon">r</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><font color="Maroon">x</font> <a href="vectsp_1.html#K6">*</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">w</font></span>)</span> = 
<font color="Maroon">x</font> <a href="polynom5.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="polynom3.html#K9">+</a> <font color="Maroon">q</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">x'</font> <a href="polynom5.html#K3">*</a> <font color="Maroon">p</font></span>)</span> <a href="polynom3.html#K9">+</a> <span class="p1">(<span class="default"><font color="Maroon">x'</font> <a href="polynom5.html#K3">*</a> <font color="Maroon">q</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">v</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">w</font></span>)</span>
<b>by </b><i>, , </i>
;<br/><br/>

<b>thus </b><span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">y</font></span>)</span> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">v</font> = 
<span class="p1">(<span class="default"><font color="Maroon">x'</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">y'</font></span>)</span> <a href="polynom5.html#K3">*</a> <font color="Maroon">p</font>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">x'</font> <a href="polynom5.html#K3">*</a> <font color="Maroon">p</font></span>)</span> <a href="polynom3.html#K9">+</a> <span class="p1">(<span class="default"><font color="Maroon">y'</font> <a href="polynom5.html#K3">*</a> <font color="Maroon">p</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">v</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">v</font></span>)</span>
<b>by </b><i>, , </i>
;<br/>

<b>thus </b><span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="group_1.html#K1">*</a> <font color="Maroon">y</font></span>)</span> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">v</font> = 
<span class="p1">(<span class="default"><font color="Maroon">x'</font> <a href="group_1.html#K1">*</a> <font color="Maroon">y'</font></span>)</span> <a href="polynom5.html#K3">*</a> <font color="Maroon">p</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">x'</font> <a href="polynom5.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">y'</font> <a href="polynom5.html#K3">*</a> <font color="Maroon">p</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">x</font> <a href="vectsp_1.html#K6">*</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">v</font></span>)</span>
<b>by </b><i>, </i>
;<br/>

<b>thus </b><span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">L</font></span>)</span> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">v</font> = 
<span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">L</font></span>)</span> <a href="polynom5.html#K3">*</a> <font color="Maroon">p</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">v</font>
<b>by </b><i/>
;<br/>


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