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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/><b>let </b><font color="Maroon">c<sub>2</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="group_1.html#V2">unital</a> <a href="group_1.html#V4">associative</a> <a href="group_1.html#V7">commutative</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>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be   <a href="polynom1.html#NM3">Series</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>







<b>set </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>5</sub></font> <a href="polynom7.html#K5">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="polynom1.html#K28">*'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>;<br/>
<b>set </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>3</sub></font> <a href="polynom1.html#K28">*'</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="polynom7.html#K5">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>;<br/>
<b>set </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>3</sub></font> <a href="polynom1.html#K28">*'</a> <font color="Maroon">c<sub>4</sub></font>;<br/>
<i><font color="Green">E21</font></i>:  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="polynom7.html#K5">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="polynom1.html#K28">*'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> = 
 <a href="polynom1.html#K14">Bags</a> <font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>
<br/>.= 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="polynom1.html#K28">*'</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="polynom7.html#K5">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>9</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:23_2"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="polynom7.html#K5">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="polynom1.html#K28">*'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>
 ;<br/><b>then reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>9</sub></font> as   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="polynom1.html#D14">POLYNOM1:def 14</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>11</sub></font> being    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E4:23_2"/><i><font color="Green">E22</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="polynom1.html#K28">*'</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="polynom7.html#K5">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="polynom1.html#K15">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">c<sub>11</sub></font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="polynom1.html#K21">decomp</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>11</sub></font> holds <br/>ex <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being  <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <span class="p1">(<span class="default"><a href="polynom1.html#K21">decomp</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font></span><a href="finseq_1.html#K10">*&gt;</a></span> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="polynom1.html#K15">.</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="polynom7.html#K5">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="polynom1.html#K15">.</a> <font color="Olive">b<sub>3</sub></font></span>)</span> ) ) )
 <b>by </b><i><a class="ref" href="polynom1.html#D26">POLYNOM1:def 26</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>12</sub></font> being    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E6:23_2"/><i><font color="Green">E23</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="polynom1.html#K28">*'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="polynom1.html#K15">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">c<sub>12</sub></font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="polynom1.html#K21">decomp</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>12</sub></font> holds <br/>ex <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being  <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <span class="p1">(<span class="default"><a href="polynom1.html#K21">decomp</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font></span><a href="finseq_1.html#K10">*&gt;</a></span> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="polynom1.html#K15">.</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="polynom1.html#K15">.</a> <font color="Olive">b<sub>3</sub></font></span>)</span> ) ) )
 <b>by </b><i><a class="ref" href="polynom1.html#D26">POLYNOM1:def 26</a></i>;<br/><i><font color="Green">E24</font></i>:  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>12</sub></font> = 
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>11</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="polyred.html#E4:23_2"><i><font color="Green">E22</font></i></a>, <a class="txt" href="polyred.html#E6:23_2"><i><font color="Green">E23</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
<br/>.= 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>11</sub></font>
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_2_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>13</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:23_2_2"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>12</sub></font>
 ;<br/><b>then reconsider </b><font color="Maroon">c<sub>14</sub></font> = <font color="Maroon">c<sub>13</sub></font> as   <a href="nat_1.html#NM1">Nat</a> ;<br/><b>consider </b><font color="Maroon">c<sub>15</sub></font>, <font color="Maroon">c<sub>16</sub></font> being   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E4:23_2_2"/><i><font color="Green">E26</font></i>: 
( <span class="p1">(<span class="default"><a href="polynom1.html#K21">decomp</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>16</sub></font></span><a href="finseq_1.html#K10">*&gt;</a></span> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="polynom1.html#K15">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="polynom1.html#K15">.</a> <font color="Maroon">c<sub>16</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="polyred.html#E6:23_2"><i><font color="Green">E23</font></i></a>, <a class="txt" href="polyred.html#E1:23_2_2"><i><font color="Green">E25</font></i></a></i>;<br/><b>consider </b><font color="Maroon">c<sub>17</sub></font>, <font color="Maroon">c<sub>18</sub></font> being   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E6:23_2_2"/><i><font color="Green">E27</font></i>: 
( <span class="p1">(<span class="default"><a href="polynom1.html#K21">decomp</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>18</sub></font></span><a href="finseq_1.html#K10">*&gt;</a></span> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="polynom1.html#K15">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="polynom7.html#K5">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="polynom1.html#K15">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="polyred.html#E4:23_2"><i><font color="Green">E22</font></i></a>, <a class="txt" href="polyred.html#E7:23_2"><i><font color="Green">E24</font></i></a>, <a class="txt" href="polyred.html#E1:23_2_2"><i><font color="Green">E25</font></i></a></i>;<br/><i><font color="Green">E28</font></i>: <font color="Maroon">c<sub>15</sub></font> = 
<span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>18</sub></font></span><a href="finseq_1.html#K10">*&gt;</a></span> <a href="funct_1.html#K1">.</a> 1
<b>by </b><i><a class="txt" href="polyred.html#E4:23_2_2"><i><font color="Green">E26</font></i></a>, <a class="txt" href="polyred.html#E6:23_2_2"><i><font color="Green">E27</font></i></a>, <a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>
<br/>.= 
<font color="Maroon">c<sub>17</sub></font>
<b>by </b><i><a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>
;<br/><font color="Maroon">c<sub>16</sub></font> = 
<span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>18</sub></font></span><a href="finseq_1.html#K10">*&gt;</a></span> <a href="funct_1.html#K1">.</a> 2
<b>by </b><i><a class="txt" href="polyred.html#E4:23_2_2"><i><font color="Green">E26</font></i></a>, <a class="txt" href="polyred.html#E6:23_2_2"><i><font color="Green">E27</font></i></a>, <a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>
<br/>.= 
<font color="Maroon">c<sub>18</sub></font>
<b>by </b><i><a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>
;<br/><b>hence </b><font color="Maroon">c<sub>11</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>13</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="polynom1.html#K15">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="polynom1.html#K15">.</a> <font color="Maroon">c<sub>16</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="polyred.html#E6:23_2_2"><i><font color="Green">E27</font></i></a>, <a class="txt" href="polyred.html#E7:23_2_2"><i><font color="Green">E28</font></i></a>, <a class="ref" href="polynom7.html#D10">POLYNOM7:def 10</a></i>
<br/>.= 
<font color="Maroon">c<sub>5</sub></font> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="polyred.html#E4:23_2_2"><i><font color="Green">E26</font></i></a>, <a class="ref" href="group_1.html#D4">GROUP_1:def 4</a></i>
;<br/><br/></div><b>end;</b></div><a NAME="E9:23_2"/><b>then </b>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="polynom1.html#K6">*</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="txt" href="polyred.html#E7:23_2"><i><font color="Green">E24</font></i></a>, <a class="ref" href="polynom1.html#D2">POLYNOM1:def 2</a></i>;<br/><b>then </b><span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="polynom1.html#K28">*'</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="polynom7.html#K5">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="polynom1.html#K15">.</a> <font color="Maroon">c<sub>10</sub></font> = 
<font color="Maroon">c<sub>5</sub></font> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">c<sub>12</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="polyred.html#E4:23_2"><i><font color="Green">E22</font></i></a>, <a class="ref" href="polynom1.html#T28">POLYNOM1:28</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="polynom7.html#K5">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="polynom1.html#K28">*'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="polynom1.html#K15">.</a> <font color="Maroon">c<sub>10</sub></font>
<b>by </b><i><a class="txt" href="polyred.html#E6:23_2"><i><font color="Green">E23</font></i></a>, <a class="ref" href="polynom7.html#D10">POLYNOM7:def 10</a></i>
;<br/><b>hence </b><a NAME="E11:23_2"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="polynom7.html#K5">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="polynom1.html#K28">*'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="polynom1.html#K28">*'</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="polynom7.html#K5">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:23"/>
<font color="Maroon">c<sub>5</sub></font> <a href="polynom7.html#K5">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="polynom1.html#K28">*'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_2.html#R2">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="polynom1.html#K28">*'</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="polynom7.html#K5">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="polyred.html#E1:23"><i><font color="Green">E21</font></i></a>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>


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