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

<b>set </b><font color="Maroon">c<sub>5</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/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>assume </b><a NAME="E1:7_1_1_1"/><i><font color="Green">E3</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="polynom1.html#K16">EmptyBag</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><b>consider </b><font color="Maroon">c<sub>7</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="E3:7_1_1_1"/><i><font color="Green">E4</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>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">c<sub>7</sub></font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>7</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>6</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>7</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>6</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>7</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#K1">*</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">E5</font></i>:  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>7</sub></font> = 
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="polynom1.html#K21">decomp</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="termord.html#E3:7_1_1_1"><i><font color="Green">E4</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> <span class="p1">(<span class="default"><a href="polynom1.html#K21">decomp</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
;<br/><div><i><font color="Green">E6</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>8</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:7_1_1_1_2"/><i><font color="Green">E7</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>9</sub></font>, <font color="Maroon">c<sub>10</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="E3:7_1_1_1_2"/><i><font color="Green">E8</font></i>: 
( <span class="p1">(<span class="default"><a href="polynom1.html#K21">decomp</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>8</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>9</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="finseq_1.html#K10">*&gt;</a></span> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>8</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>9</sub></font></span>)</span> <a href="group_1.html#K1">*</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>10</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="termord.html#E3:7_1_1_1"><i><font color="Green">E4</font></i></a></i>;<br/><b>consider </b><font color="Maroon">c<sub>11</sub></font>, <font color="Maroon">c<sub>12</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="E5:7_1_1_1_2"/><i><font color="Green">E9</font></i>: 
( <span class="p1">(<span class="default"><a href="polynom1.html#K21">decomp</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>8</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>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span><a href="finseq_1.html#K10">*&gt;</a></span> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="pboole.html#R6">=</a> <font color="Maroon">c<sub>11</sub></font> <a href="polynom1.html#K9">+</a> <font color="Maroon">c<sub>12</sub></font> )
 <b>by </b><i><a class="txt" href="termord.html#E4:7_1_1_1"><i><font color="Green">E5</font></i></a>, <a class="txt" href="termord.html#E1:7_1_1_1_2"><i><font color="Green">E7</font></i></a>, <a class="ref" href="polynom1.html#T72">POLYNOM1:72</a></i>;<br/><i><font color="Green">E10</font></i>: <font color="Maroon">c<sub>9</sub></font> = 
<span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</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="termord.html#E3:7_1_1_1_2"><i><font color="Green">E8</font></i></a>, <a class="txt" href="termord.html#E5:7_1_1_1_2"><i><font color="Green">E9</font></i></a>, <a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>
<br/>.= 
<font color="Maroon">c<sub>11</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>10</sub></font> = 
<span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</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="termord.html#E3:7_1_1_1_2"><i><font color="Green">E8</font></i></a>, <a class="txt" href="termord.html#E5:7_1_1_1_2"><i><font color="Green">E9</font></i></a>, <a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>
<br/>.= 
<font color="Maroon">c<sub>12</sub></font>
<b>by </b><i><a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>
;<br/><a NAME="E8:7_1_1_1_2"/><b>then </b><i><font color="Green">E11</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="polynom1.html#K16">EmptyBag</a> <font color="Maroon">c<sub>1</sub></font> or <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="polynom1.html#K16">EmptyBag</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="termord.html#E1:7_1_1_1"><i><font color="Green">E3</font></i></a>, <a class="txt" href="termord.html#E5:7_1_1_1_2"><i><font color="Green">E9</font></i></a>, <a class="txt" href="termord.html#E6:7_1_1_1_2"><i><font color="Green">E10</font></i></a>, <a class="ref" href="polynom1.html#T57">POLYNOM1:57</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_1_1_2_3"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_1_1_1_2_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:7_1_1_1_2_3_1"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="polynom1.html#K15">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>2</sub></font> or <font color="Maroon">c<sub>4</sub></font> <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#K1">0.</a> <font color="Maroon">c<sub>2</sub></font> )
 </span><b>by </b><i><a class="txt" href="termord.html#E8:7_1_1_1_2"><i><font color="Green">E11</font></i></a>, <a class="ref" href="polynom7.html#D8">POLYNOM7:def 8</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_1_1_1_2_3_1_1"><b>case </b></a><a NAME="E1:7_1_1_1_2_3_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="polynom1.html#K15">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:7_1_1_1_2_3_1_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="termord.html#E3:7_1_1_1_2"><i><font color="Green">E8</font></i></a>, <a class="ref" href="binom.html#T1">BINOM:1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_1_1_1_2_3_1_2"><b>case </b></a><a NAME="E1:7_1_1_1_2_3_1_2"/>
<font color="Maroon">c<sub>4</sub></font> <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#K1">0.</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:7_1_1_1_2_3_1_2"/>
<font color="Maroon">c<sub>7</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="termord.html#E3:7_1_1_1_2"><i><font color="Green">E8</font></i></a>, <a class="ref" href="binom.html#T2">BINOM:2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E10:7_1_1_1_2"/>
<font color="Maroon">c<sub>7</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_1_1_3"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_1_1_1_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:7_1_1_1_3_1"/>
(  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_1_1_1_3_1_1"><b>case </b></a><a NAME="E1:7_1_1_1_3_1_1"/>
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:7_1_1_1_3_1_1"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K6">&lt;*&gt;</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T26">FINSEQ_1:26</a></i>;<br/><b>hence </b><a NAME="E3:7_1_1_1_3_1_1"/>
<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>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="termord.html#E3:7_1_1_1"><i><font color="Green">E4</font></i></a>, <a class="ref" href="rlvect_1.html#T60">RLVECT_1:60</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_1_1_1_3_1_2"><b>case </b></a><a NAME="E1:7_1_1_1_3_1_2"/><i><font color="Green">E7</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><b>consider </b><font color="Maroon">c<sub>8</sub></font> being    <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>7</sub></font>;<br/><a NAME="E3:7_1_1_1_3_1_2"/><i><font color="Green">E8</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="termord.html#E1:7_1_1_1_3_1_2"><i><font color="Green">E7</font></i></a></i>;<br/><a NAME="E4:7_1_1_1_3_1_2"/>
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>7</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>8</sub></font> holds <br/><font color="Maroon">c<sub>7</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="termord.html#E5:7_1_1_1"><i><font color="Green">E6</font></i></a></i>;<br/><a NAME="E5:7_1_1_1_3_1_2"/><b>then </b>
 <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="termord.html#E3:7_1_1_1_3_1_2"><i><font color="Green">E8</font></i></a>, <a class="ref" href="polynom2.html#T5">POLYNOM2:5</a></i>;<br/><b>hence </b><a NAME="E6:7_1_1_1_3_1_2"/>
<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>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="termord.html#E3:7_1_1_1"><i><font color="Green">E4</font></i></a>, <a class="txt" href="termord.html#E5:7_1_1_1"><i><font color="Green">E6</font></i></a>, <a class="txt" href="termord.html#E3:7_1_1_1_3_1_2"><i><font color="Green">E8</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E7:7_1_1_1"/>
<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>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:7_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="polynom1.html#K28">*'</a> <font color="Maroon">c<sub>4</sub></font> is <a href="polynom7.html#V4">Constant</a>
 <b>by </b><i><a class="ref" href="polynom7.html#D8">POLYNOM7:def 8</a></i>;<br/>


</div>
