<?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>consider </b><font color="Maroon">c<sub>2</sub></font> being  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><b> such that </b><br/><a NAME="E2:51"/><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> &amp;  <a href="polyalg1.html#K3" target="_self">Polynom-Algebra</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="polyalg1.html#K2" target="_self">GenAlg</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="ref" href="polyalg1.html#D6" target="_self">Def6</a></i>;<br/>
<a NAME="E3:51"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is <a href="polyalg1.html#V3" target="_self">opers_closed</a>
 
<b>by </b><i><a class="txt" href="polyalg1.html#E2:51"><i><font color="Green">E28</font></i></a>, <a class="ref" href="polyalg1.html#T23" target="_self">Th23</a></i>;<br/>
<a NAME="E4:51"/><b>then </b><i><font color="Green">E30</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="polyalg1.html#K3" target="_self">Polynom-Algebra</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <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>
 
<b>by </b><i><a class="txt" href="polyalg1.html#E2:51"><i><font color="Green">E28</font></i></a>, <a class="ref" href="polyalg1.html#T22" target="_self">Th22</a></i>;<br/>
<a NAME="E5:51"/><i><font color="Green">E31</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="polyalg1.html#K3" target="_self">Polynom-Algebra</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</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>
 
<b>by </b><i><a class="txt" href="polyalg1.html#E2:51"><i><font color="Green">E28</font></i></a>, <a class="ref" href="polyalg1.html#D3" target="_self">Def3</a></i>;<br/>
<a NAME="E6:51"/><i><font color="Green">E32</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="polyalg1.html#K3" target="_self">Polynom-Algebra</a> <font color="Maroon">c<sub>1</sub></font></span>)</span><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="polynom3.html#K16">Polynom-Ring</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> holds the <a href="rlvect_1.html#U1">add</a> of <span class="p1">(<span class="default"><a href="polyalg1.html#K3" target="_self">Polynom-Algebra</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> the <a href="rlvect_1.html#U1">add</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> <a href="binop_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="polyalg1.html#K3" target="_self">Polynom-Algebra</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="struct_0.html#NM1">Element</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>reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>3</sub></font> as   <a href="struct_0.html#NM1">Element</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> <b>by </b><i><a class="txt" href="polyalg1.html#E2:51"><i><font color="Green">E28</font></i></a>, <a class="txt" href="polyalg1.html#E3:51"><i><font color="Green">E29</font></i></a>, <a class="ref" href="polyalg1.html#T22" target="_self">Th22</a></i>;<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 <span class="p1">(<span class="default"><a href="polyalg1.html#K3" target="_self">Polynom-Algebra</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <b>by </b><i><a class="txt" href="polyalg1.html#E2:51"><i><font color="Green">E28</font></i></a>, <a class="txt" href="polyalg1.html#E3:51"><i><font color="Green">E29</font></i></a>, <a class="ref" href="polyalg1.html#T22" target="_self">Th22</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>3</sub></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">c<sub>1</sub></font></span>)</span> <b>by </b><i><a class="txt" href="polyalg1.html#E5:51"><i><font color="Green">E31</font></i></a>, <a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>6</sub></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">c<sub>1</sub></font></span>)</span> <b>by </b><i><a class="txt" href="polyalg1.html#E2:51"><i><font color="Green">E28</font></i></a>, <a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>3</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#E4:51"><i><font color="Green">E30</font></i></a>, <a class="ref" href="polynom3.html#D12">POLYNOM3:def 12</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>4</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="ref" href="polynom3.html#D12">POLYNOM3:def 12</a></i>;<br/>
<b>thus </b>the <a href="rlvect_1.html#U1">add</a> of <span class="p1">(<span class="default"><a href="polyalg1.html#K3" target="_self">Polynom-Algebra</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> = 
<font color="Maroon">c<sub>3</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>6</sub></font>

<br/>.= 
<font color="Maroon">c<sub>7</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>8</sub></font>
<b>by </b><i><a class="txt" href="polyalg1.html#E2:51"><i><font color="Green">E28</font></i></a>, <a class="ref" href="polyalg1.html#T15" target="_self">Th15</a></i>
<br/>.= 
<font color="Maroon">c<sub>9</sub></font> <a href="polynom3.html#K8">+</a> <font color="Maroon">c<sub>10</sub></font>
<b>by </b><i><a class="ref" href="polyalg1.html#D2" target="_self">Def2</a></i>
<br/>.= 
<font color="Maroon">c<sub>5</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="ref" href="polynom3.html#D12">POLYNOM3:def 12</a></i>
<br/>.= 
the <a href="rlvect_1.html#U1">add</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> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>

;<br/>


</div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="polyalg1.html#K3" target="_self">Polynom-Algebra</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="struct_0.html#NM1">Element</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>reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>3</sub></font> as   <a href="struct_0.html#NM1">Element</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> <b>by </b><i><a class="txt" href="polyalg1.html#E2:51"><i><font color="Green">E28</font></i></a>, <a class="txt" href="polyalg1.html#E3:51"><i><font color="Green">E29</font></i></a>, <a class="ref" href="polyalg1.html#T22" target="_self">Th22</a></i>;<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 <span class="p1">(<span class="default"><a href="polyalg1.html#K3" target="_self">Polynom-Algebra</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <b>by </b><i><a class="txt" href="polyalg1.html#E2:51"><i><font color="Green">E28</font></i></a>, <a class="txt" href="polyalg1.html#E3:51"><i><font color="Green">E29</font></i></a>, <a class="ref" href="polyalg1.html#T22" target="_self">Th22</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>3</sub></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">c<sub>1</sub></font></span>)</span> <b>by </b><i><a class="txt" href="polyalg1.html#E5:51"><i><font color="Green">E31</font></i></a>, <a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>6</sub></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">c<sub>1</sub></font></span>)</span> <b>by </b><i><a class="txt" href="polyalg1.html#E2:51"><i><font color="Green">E28</font></i></a>, <a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>3</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#E4:51"><i><font color="Green">E30</font></i></a>, <a class="ref" href="polynom3.html#D12">POLYNOM3:def 12</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>4</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="ref" href="polynom3.html#D12">POLYNOM3:def 12</a></i>;<br/><b>thus </b>the <a href="group_1.html#U1">mult</a> of <span class="p1">(<span class="default"><a href="polyalg1.html#K3" target="_self">Polynom-Algebra</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> = 
<font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>6</sub></font>

<br/>.= 
<font color="Maroon">c<sub>7</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>8</sub></font>
<b>by </b><i><a class="txt" href="polyalg1.html#E2:51"><i><font color="Green">E28</font></i></a>, <a class="ref" href="polyalg1.html#T16" target="_self">Th16</a></i>
<br/>.= 
<font color="Maroon">c<sub>9</sub></font> <a href="polynom3.html#K14">*'</a> <font color="Maroon">c<sub>10</sub></font>
<b>by </b><i><a class="ref" href="polyalg1.html#D2" target="_self">Def2</a></i>
<br/>.= 
<font color="Maroon">c<sub>5</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="ref" href="polynom3.html#D12">POLYNOM3:def 12</a></i>
<br/>.= 
the <a href="group_1.html#U1">mult</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> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>

;<br/></div><b>end;</b></div>
<a NAME="E8:51"/><b>then </b><i><font color="Green">E33</font></i>: 
the <a href="group_1.html#U1">mult</a> of <span class="p1">(<span class="default"><a href="polyalg1.html#K3" target="_self">Polynom-Algebra</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#R1">=</a> the <a href="group_1.html#U1">mult</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>
 
<b>by </b><i><a class="txt" href="polyalg1.html#E4:51"><i><font color="Green">E30</font></i></a>, <a class="ref" href="binop_1.html#T2">BINOP_1:2</a></i>;<br/>
<i><font color="Green">E34</font></i>: the <a href="struct_0.html#U2">Zero</a> of <span class="p1">(<span class="default"><a href="polyalg1.html#K3" target="_self">Polynom-Algebra</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> = 
 <a href="rlvect_1.html#K1">0.</a> <span class="p1">(<span class="default"><a href="polyalg1.html#K3" target="_self">Polynom-Algebra</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>

<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>
<b>by </b><i><a class="txt" href="polyalg1.html#E2:51"><i><font color="Green">E28</font></i></a>, <a class="ref" href="polyalg1.html#D3" target="_self">Def3</a></i>
<br/>.= 
 <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/>.= 
the <a href="struct_0.html#U2">Zero</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/>
the <a href="vectsp_1.html#U1">unity</a> of <span class="p1">(<span class="default"><a href="polyalg1.html#K3" target="_self">Polynom-Algebra</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> = 
 <a href="vectsp_1.html#K2">1_</a> <span class="p1">(<span class="default"><a href="polyalg1.html#K3" target="_self">Polynom-Algebra</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>

<br/>.= 
 <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>
<b>by </b><i><a class="txt" href="polyalg1.html#E2:51"><i><font color="Green">E28</font></i></a>, <a class="ref" href="polyalg1.html#D3" target="_self">Def3</a></i>
<br/>.= 
 <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/>.= 
the <a href="vectsp_1.html#U1">unity</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>hence </b><a NAME="E11:51"/>
 <a href="vectsp_1.html#G3">doubleLoopStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="polyalg1.html#K3" target="_self">Polynom-Algebra</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,the <a href="rlvect_1.html#U1">add</a> of <span class="p1">(<span class="default"><a href="polyalg1.html#K3" target="_self">Polynom-Algebra</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,the <a href="group_1.html#U1">mult</a> of <span class="p1">(<span class="default"><a href="polyalg1.html#K3" target="_self">Polynom-Algebra</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,the <a href="vectsp_1.html#U1">unity</a> of <span class="p1">(<span class="default"><a href="polyalg1.html#K3" target="_self">Polynom-Algebra</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,the <a href="struct_0.html#U2">Zero</a> of <span class="p1">(<span class="default"><a href="polyalg1.html#K3" target="_self">Polynom-Algebra</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> #) <a href="hidden.html#R1">=</a>  <a href="polynom3.html#K16">Polynom-Ring</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="polyalg1.html#E4:51"><i><font color="Green">E30</font></i></a>, <a class="txt" href="polyalg1.html#E6:51"><i><font color="Green">E32</font></i></a>, <a class="txt" href="polyalg1.html#E8:51"><i><font color="Green">E33</font></i></a>, <a class="txt" href="polyalg1.html#E9:51"><i><font color="Green">E34</font></i></a>, <a class="ref" href="binop_1.html#T2">BINOP_1:2</a></i>;<br/>


</div>
