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<div class="add">

<b>let </b><font color="Maroon">L</font> be  non <a href="struct_0.html#V3">empty</a> <a href="rlvect_1.html#V4">add-associative</a> <a href="rlvect_1.html#V5">right_zeroed</a> <a href="rlvect_1.html#V6">right_complementable</a> <a href="vectsp_1.html#V7">distributive</a>  <a href="vectsp_1.html#L3">doubleLoopStr</a> ;<br/><b>let </b><font color="Maroon">A</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</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>assume </b><a NAME="E1:50"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom3.html#K16">Polynom-Ring</a> <font color="Maroon">L</font></span>)</span>
 ;<br/>

<b>set </b><font color="Maroon">B</font> =  <a href="polyalg1.html#K1" target="_self">Formal-Series</a> <font color="Maroon">L</font>;<br/>
<b>thus </b><a NAME="E2:50"/>
<font color="Maroon">A</font> is <a href="vectsp_4.html#V1">lineary-closed</a>
  <i><font color="Red">:: according to </font></i><a class="ref" href="polyalg1.html#D4" target="_self">POLYALG1:def 4</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:50_1"/>
for <font color="Olive">v</font>, <font color="Olive">u</font> being  <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>  st <font color="Olive">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Olive">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> holds <br/><font color="Olive">v</font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="vectsp_4.html#D1">VECTSP_4:def 1</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1_1"><b>proof </b></a><div class="add">

<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">u</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>assume </b><a NAME="E1:50_1_1"/><i><font color="Green">E44</font></i>: 
( <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> )
 ;<br/>

<b>reconsider </b><font color="Maroon">p</font> = <font color="Maroon">v</font>, <font color="Maroon">q</font> = <font color="Maroon">u</font> as   <a href="algseq_1.html#NM1">AlgSequence</a> of <font color="Maroon">L</font> <b>by </b><i>, , <a class="ref" href="polynom3.html#D12">POLYNOM3:def 12</a></i>;<br/>
<a NAME="E3:50_1_1"/>
<font color="Maroon">v</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="polynom3.html#K8">+</a> <font color="Maroon">q</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E4:50_1_1"/>
<font color="Maroon">v</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>by </b><i>, <a class="ref" href="polynom3.html#D12">POLYNOM3:def 12</a></i>;<br/>


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

<b>thus </b><a NAME="E2:50_1"/>
for <font color="Olive">a</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><br/> for <font color="Olive">v</font> being  <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>  st <font color="Olive">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> holds <br/><font color="Olive">a</font> <a href="vectsp_1.html#K6">*</a> <font color="Olive">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be   <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>;<br/>



<b>assume </b><a NAME="E1:50_1_2"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">a'</font> = <font color="Maroon">a</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> ;<br/>
<b>reconsider </b><font color="Maroon">p</font> = <font color="Maroon">v</font> as   <a href="algseq_1.html#NM1">AlgSequence</a> of <font color="Maroon">L</font> <b>by </b><i>, , <a class="ref" href="polynom3.html#D12">POLYNOM3:def 12</a></i>;<br/>
<a NAME="E4:50_1_2"/>
<font color="Maroon">a</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a'</font> <a href="polynom5.html#K3">*</a> <font color="Maroon">p</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E5:50_1_2"/>
<font color="Maroon">a</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>by </b><i>, <a class="ref" href="polynom3.html#D12">POLYNOM3:def 12</a></i>;<br/>


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


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

<b>thus </b><a NAME="E3:50"/>
for <font color="Olive">u</font>, <font color="Olive">v</font> being  <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>  st <font color="Olive">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Olive">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> holds <br/><font color="Olive">u</font> <a href="group_1.html#K1">*</a> <font color="Olive">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">u</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">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>;<br/>

<b>assume </b><a NAME="E1:50_2"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> )
 ;<br/>

<b>reconsider </b><font color="Maroon">p</font> = <font color="Maroon">u</font>, <font color="Maroon">q</font> = <font color="Maroon">v</font> as   <a href="algseq_1.html#NM1">AlgSequence</a> of <font color="Maroon">L</font> <b>by </b><i>, , <a class="ref" href="polynom3.html#D12">POLYNOM3:def 12</a></i>;<br/>
<a NAME="E3:50_2"/>
<font color="Maroon">u</font> <a href="group_1.html#K1">*</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="polynom3.html#K14">*'</a> <font color="Maroon">q</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E4:50_2"/>
<font color="Maroon">u</font> <a href="group_1.html#K1">*</a> <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>by </b><i>, <a class="ref" href="polynom3.html#D12">POLYNOM3:def 12</a></i>;<br/>


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

 <a href="vectsp_1.html#K2">1_</a> <span class="p1">(<span class="default"><a href="polyalg1.html#K1" target="_self">Formal-Series</a> <font color="Maroon">L</font></span>)</span> = 
 <a href="polynom3.html#K13">1_.</a> <font color="Maroon">L</font>
<b>by </b><i/>
<br/>.= 
 <a href="vectsp_1.html#K2">1_</a> <span class="p1">(<span class="default"><a href="polynom3.html#K16">Polynom-Ring</a> <font color="Maroon">L</font></span>)</span>
<b>by </b><i><a class="ref" href="polynom3.html#D12">POLYNOM3:def 12</a></i>
;<br/>
<b>hence </b><a NAME="E5:50"/>
 <a href="vectsp_1.html#K2">1_</a> <span class="p1">(<span class="default"><a href="polyalg1.html#K1" target="_self">Formal-Series</a> <font color="Maroon">L</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>by </b><i/>;<br/>

 <a href="rlvect_1.html#K1">0.</a> <span class="p1">(<span class="default"><a href="polyalg1.html#K1" target="_self">Formal-Series</a> <font color="Maroon">L</font></span>)</span> = 
 <a href="polynom3.html#K12">0_.</a> <font color="Maroon">L</font>
<b>by </b><i/>
<br/>.= 
 <a href="rlvect_1.html#K1">0.</a> <span class="p1">(<span class="default"><a href="polynom3.html#K16">Polynom-Ring</a> <font color="Maroon">L</font></span>)</span>
<b>by </b><i><a class="ref" href="polynom3.html#D12">POLYNOM3:def 12</a></i>
;<br/>
<b>hence </b><a NAME="E7:50"/>
 <a href="rlvect_1.html#K1">0.</a> <span class="p1">(<span class="default"><a href="polyalg1.html#K1" target="_self">Formal-Series</a> <font color="Maroon">L</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>by </b><i/>;<br/>


</div>
