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

<b>let </b><font color="Maroon">c<sub>1</sub></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">c<sub>2</sub></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">c<sub>1</sub></font></span>)</span>;<br/>



<b>assume </b><a NAME="E1:50"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>2</sub></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">c<sub>1</sub></font></span>)</span>
 ;<br/>

<b>set </b><font color="Maroon">c<sub>3</sub></font> =  <a href="polyalg1.html#K1" target="_self">Formal-Series</a> <font color="Maroon">c<sub>1</sub></font>;<br/>
<b>thus </b><a NAME="E2:50"/>
<font color="Maroon">c<sub>2</sub></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">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></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">c<sub>1</sub></font></span>)</span> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></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">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></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">c<sub>1</sub></font></span>)</span>;<br/>

<b>assume </b><a NAME="E1:50_1_1"/><i><font color="Green">E29</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> )
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>5</sub></font> as   <a href="algseq_1.html#NM1">AlgSequence</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="polyalg1.html#E1:50"><i><font color="Green">E28</font></i></a>, <a class="txt" href="polyalg1.html#E1:50_1_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="polynom3.html#D12">POLYNOM3:def 12</a></i>;<br/>
<a NAME="E3:50_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="polynom3.html#K8">+</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="ref" href="polyalg1.html#D2" target="_self">Def2</a></i>;<br/>
<b>hence </b><a NAME="E4:50_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="polyalg1.html#E1:50"><i><font color="Green">E28</font></i></a>, <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">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><br/>for <font color="Olive">b<sub>2</sub></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">c<sub>1</sub></font></span>)</span> st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="vectsp_1.html#K6">*</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></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">c<sub>4</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>5</sub></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">c<sub>1</sub></font></span>)</span>;<br/>



<b>assume </b><a NAME="E1:50_1_2"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>4</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>5</sub></font> as   <a href="algseq_1.html#NM1">AlgSequence</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="polyalg1.html#E1:50"><i><font color="Green">E28</font></i></a>, <a class="txt" href="polyalg1.html#E1:50_1_2"><i><font color="Green">E29</font></i></a>, <a class="ref" href="polynom3.html#D12">POLYNOM3:def 12</a></i>;<br/>
<a NAME="E4:50_1_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="polynom5.html#K3">*</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="ref" href="polyalg1.html#D2" target="_self">Def2</a></i>;<br/>
<b>hence </b><a NAME="E5:50_1_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="polyalg1.html#E1:50"><i><font color="Green">E28</font></i></a>, <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">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></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">c<sub>1</sub></font></span>)</span> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="group_1.html#K1">*</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></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">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></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">c<sub>1</sub></font></span>)</span>;<br/>

<b>assume </b><a NAME="E1:50_2"/><i><font color="Green">E29</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> )
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>5</sub></font> as   <a href="algseq_1.html#NM1">AlgSequence</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="polyalg1.html#E1:50"><i><font color="Green">E28</font></i></a>, <a class="txt" href="polyalg1.html#E1:50_2"><i><font color="Green">E29</font></i></a>, <a class="ref" href="polynom3.html#D12">POLYNOM3:def 12</a></i>;<br/>
<a NAME="E3:50_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="polynom3.html#K14">*'</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="ref" href="polyalg1.html#D2" target="_self">Def2</a></i>;<br/>
<b>hence </b><a NAME="E4:50_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="polyalg1.html#E1:50"><i><font color="Green">E28</font></i></a>, <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">c<sub>1</sub></font></span>)</span> = 
 <a href="polynom3.html#K13">1_.</a> <font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="ref" href="polyalg1.html#D2" target="_self">Def2</a></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">c<sub>1</sub></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">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="polyalg1.html#E1:50"><i><font color="Green">E28</font></i></a></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">c<sub>1</sub></font></span>)</span> = 
 <a href="polynom3.html#K12">0_.</a> <font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="ref" href="polyalg1.html#D2" target="_self">Def2</a></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">c<sub>1</sub></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">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="polyalg1.html#E1:50"><i><font color="Green">E28</font></i></a></i>;<br/>


</div>
