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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  <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="vectsp_1.html#V7">distributive</a> <a href="vectsp_2.html#V2">domRing-like</a> non <a href="realset2.html#V3">trivial</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#NM1">bag</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font> be  <a href="rlvect_1.html#V8">non-zero</a> <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>7</sub></font> =  <a href="polynom7.html#K1">Monom</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>3</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>8</sub></font> =  <a href="polynom7.html#K1">Monom</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>4</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>9</sub></font> =  <a href="polynom7.html#K1">Monom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="polynom1.html#K9">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>;<br/>
<b>set </b><font color="Maroon">c<sub>10</sub></font> = <span class="p1">(<span class="default"><a href="polynom7.html#K1">Monom</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="polynom1.html#K28">*'</a> <span class="p1">(<span class="default"><a href="polynom7.html#K1">Monom</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span>;<br/>
<a NAME="E1:8"/><i><font color="Green">E4</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="ref" href="rlvect_1.html#D13">RLVECT_1:def 13</a></i>;<br/>
<a NAME="E2:8"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="vectsp_2.html#D5">VECTSP_2:def 5</a></i>;<br/>
<a NAME="E3:8"/><b>then </b><i><font color="Green">E5</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>6</sub></font> is <a href="rlvect_1.html#V8">non-zero</a>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#D13">RLVECT_1:def 13</a></i>;<br/>
<i><font color="Green">E6</font></i>: <span class="p1">(<span class="default"><a href="polynom7.html#K1">Monom</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="polynom1.html#K15">.</a> <font color="Maroon">c<sub>3</sub></font> = 
<span class="p1">(<span class="default"><a href="polynom7.html#K1">Monom</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="polynom1.html#K15">.</a> <span class="p1">(<span class="default"><a href="polynom7.html#K2">term</a> <span class="p2">(<span class="default"><a href="polynom7.html#K1">Monom</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="polynom7.html#T10">POLYNOM7:10</a></i>
<br/>.= 
 <a href="polynom7.html#K3">coefficient</a> <span class="p1">(<span class="default"><a href="polynom7.html#K1">Monom</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="polynom7.html#D7">POLYNOM7:def 7</a></i>
<br/>.= 
<font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="ref" href="polynom7.html#T9">POLYNOM7:9</a></i>
;<br/>
<i><font color="Green">E7</font></i>: <span class="p1">(<span class="default"><a href="polynom7.html#K1">Monom</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="polynom1.html#K15">.</a> <font color="Maroon">c<sub>4</sub></font> = 
<span class="p1">(<span class="default"><a href="polynom7.html#K1">Monom</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="polynom1.html#K15">.</a> <span class="p1">(<span class="default"><a href="polynom7.html#K2">term</a> <span class="p2">(<span class="default"><a href="polynom7.html#K1">Monom</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="polynom7.html#T10">POLYNOM7:10</a></i>
<br/>.= 
 <a href="polynom7.html#K3">coefficient</a> <span class="p1">(<span class="default"><a href="polynom7.html#K1">Monom</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="polynom7.html#D7">POLYNOM7:def 7</a></i>
<br/>.= 
<font color="Maroon">c<sub>6</sub></font>
<b>by </b><i><a class="ref" href="polynom7.html#T9">POLYNOM7:9</a></i>
;<br/>
<div><i><font color="Green">E8</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="8_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>11</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="E1:8_3"/><i><font color="Green">E9</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><b>consider </b><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="E3:8_3"/><i><font color="Green">E10</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>12</sub></font> holds <br/><span class="p1">(<span class="default"><a href="polynom7.html#K1">Monom</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="polynom1.html#K15">.</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="ref" href="polynom7.html#D4">POLYNOM7:def 4</a></i>;<br/><b>assume </b><a NAME="E4:8_3"/>
<span class="p1">(<span class="default"><a href="polynom7.html#K1">Monom</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="polynom1.html#K15">.</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><a NAME="E5:8_3"/><b>then </b><i><font color="Green">E11</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="pboole.html#R6">=</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="txt" href="termord.html#E3:8_3"><i><font color="Green">E10</font></i></a></i>;<br/><a NAME="E6:8_3"/>
<span class="p1">(<span class="default"><a href="polynom7.html#K1">Monom</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="polynom1.html#K15">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</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#E4:8"><i><font color="Green">E6</font></i></a>, <a class="ref" href="rlvect_1.html#D13">RLVECT_1:def 13</a></i>;<br/><b>hence </b><a NAME="E7:8_3"/>
contradiction
 <b>by </b><i><a class="txt" href="termord.html#E1:8_3"><i><font color="Green">E9</font></i></a>, <a class="txt" href="termord.html#E3:8_3"><i><font color="Green">E10</font></i></a>, <a class="txt" href="termord.html#E5:8_3"><i><font color="Green">E11</font></i></a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E9</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="8_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>11</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="E1:8_4"/><i><font color="Green">E10</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><b>consider </b><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="E3:8_4"/><i><font color="Green">E11</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>12</sub></font> holds <br/><span class="p1">(<span class="default"><a href="polynom7.html#K1">Monom</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="polynom1.html#K15">.</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="ref" href="polynom7.html#D4">POLYNOM7:def 4</a></i>;<br/><b>assume </b><a NAME="E4:8_4"/>
<span class="p1">(<span class="default"><a href="polynom7.html#K1">Monom</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="polynom1.html#K15">.</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><a NAME="E5:8_4"/><b>then </b><i><font color="Green">E12</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="pboole.html#R6">=</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="txt" href="termord.html#E3:8_4"><i><font color="Green">E11</font></i></a></i>;<br/><a NAME="E6:8_4"/>
<span class="p1">(<span class="default"><a href="polynom7.html#K1">Monom</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="polynom1.html#K15">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</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#E5:8"><i><font color="Green">E7</font></i></a>, <a class="ref" href="rlvect_1.html#D13">RLVECT_1:def 13</a></i>;<br/><b>hence </b><a NAME="E7:8_4"/>
contradiction
 <b>by </b><i><a class="txt" href="termord.html#E1:8_4"><i><font color="Green">E10</font></i></a>, <a class="txt" href="termord.html#E3:8_4"><i><font color="Green">E11</font></i></a>, <a class="txt" href="termord.html#E5:8_4"><i><font color="Green">E12</font></i></a></i>;<br/></div><b>end;</b></div>
<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="E9:8"/><i><font color="Green">E10</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom7.html#K1">Monom</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="polynom1.html#K28">*'</a> <span class="p2">(<span class="default"><a href="polynom7.html#K1">Monom</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="polynom1.html#K15">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="polynom1.html#K9">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <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> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="polynom1.html#K9">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></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> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="polynom1.html#K9">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></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"><span class="p2">(<span class="default"><a href="polynom7.html#K1">Monom</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <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"><span class="p2">(<span class="default"><a href="polynom7.html#K1">Monom</a> <font color="Maroon">c<sub>6</sub></font>,<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/>
<i><font color="Green">E11</font></i>:  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>11</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> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="polynom1.html#K9">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="termord.html#E9:8"><i><font color="Green">E10</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> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="polynom1.html#K9">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
;<br/>
<b>set </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>3</sub></font> <a href="polynom1.html#K9">+</a> <font color="Maroon">c<sub>4</sub></font>;<br/>
<b>consider </b><font color="Maroon">c<sub>13</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E12:8"/><i><font color="Green">E12</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> <span class="p1">(<span class="default"><a href="polynom1.html#K21">decomp</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="polynom1.html#K9">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><a href="polynom1.html#K21">decomp</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="polynom1.html#K9">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>13</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>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="finseq_1.html#K10">*&gt;</a></span> )
 <b>by </b><i><a class="ref" href="polynom1.html#T73">POLYNOM1:73</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>14</sub></font>, <font color="Maroon">c<sub>15</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="E14:8"/><i><font color="Green">E13</font></i>: 
( <span class="p1">(<span class="default"><a href="polynom1.html#K21">decomp</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="polynom1.html#K9">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>13</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>14</sub></font>,<font color="Maroon">c<sub>15</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>13</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom7.html#K1">Monom</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="polynom1.html#K15">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom7.html#K1">Monom</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="polynom1.html#K15">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="termord.html#E9:8"><i><font color="Green">E10</font></i></a>, <a class="txt" href="termord.html#E10:8"><i><font color="Green">E11</font></i></a>, <a class="txt" href="termord.html#E12:8"><i><font color="Green">E12</font></i></a></i>;<br/>
<i><font color="Green">E14</font></i>: <font color="Maroon">c<sub>14</sub></font> = 
<span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</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#E12:8"><i><font color="Green">E12</font></i></a>, <a class="txt" href="termord.html#E14:8"><i><font color="Green">E13</font></i></a>, <a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>
<br/>.= 
<font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>
;<br/>
<i><font color="Green">E15</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>3</sub></font>,<font color="Maroon">c<sub>4</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#E12:8"><i><font color="Green">E12</font></i></a>, <a class="txt" href="termord.html#E14:8"><i><font color="Green">E13</font></i></a>, <a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>
<br/>.= 
<font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>
;<br/>
<div><i><font color="Green">E16</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="8_8"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>16</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:8_8"/><i><font color="Green">E17</font></i>: 
( <font color="Maroon">c<sub>16</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> &amp; <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>13</sub></font> )
 ;<br/><b>then 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="E3:8_8"/><i><font color="Green">E18</font></i>: 
( <span class="p1">(<span class="default"><a href="polynom1.html#K21">decomp</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="polynom1.html#K9">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>16</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>16</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom7.html#K1">Monom</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="polynom1.html#K15">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom7.html#K1">Monom</a> <font color="Maroon">c<sub>6</sub></font>,<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="termord.html#E9:8"><i><font color="Green">E10</font></i></a></i>;<br/><div><i><font color="Green">E19</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="8_8_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:8_8_1"/><i><font color="Green">E20</font></i>: 
( <font color="Maroon">c<sub>17</sub></font> <a href="pboole.html#R6">=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>18</sub></font> <a href="pboole.html#R6">=</a> <font color="Maroon">c<sub>4</sub></font> )
 ;<br/><span class="p1">(<span class="default"><a href="polynom1.html#K21">decomp</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="polynom1.html#K9">+</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>16</sub></font> = 
<span class="p1">(<span class="default"><a href="polynom1.html#K21">decomp</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="polynom1.html#K9">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>16</sub></font>
<b>by </b><i><a class="txt" href="termord.html#E10:8"><i><font color="Green">E11</font></i></a>, <a class="txt" href="termord.html#E1:8_8"><i><font color="Green">E17</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="polynom1.html#K21">decomp</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="polynom1.html#K9">+</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>13</sub></font>
<b>by </b><i><a class="txt" href="termord.html#E12:8"><i><font color="Green">E12</font></i></a>, <a class="txt" href="termord.html#E3:8_8"><i><font color="Green">E18</font></i></a>, <a class="txt" href="termord.html#E1:8_8_1"><i><font color="Green">E20</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
;<br/><b>hence </b><a NAME="E3:8_8_1"/>
contradiction
 <b>by </b><i><a class="txt" href="termord.html#E10:8"><i><font color="Green">E11</font></i></a>, <a class="txt" href="termord.html#E12:8"><i><font color="Green">E12</font></i></a>, <a class="txt" href="termord.html#E1:8_8"><i><font color="Green">E17</font></i></a>, <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_8_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="8_8_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:8_8_2_1"/>
( <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>3</sub></font> or <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>4</sub></font> )
 </span><b>by </b><i><a class="txt" href="termord.html#E4:8_8"><i><font color="Green">E19</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="8_8_2_1_1"><b>case </b></a><a NAME="E1:8_8_2_1_1"/>
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><div class="add"><a NAME="E2:8_8_2_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="polynom7.html#K1">Monom</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="polynom1.html#K15">.</a> <font color="Maroon">c<sub>17</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#E6:8"><i><font color="Green">E8</font></i></a></i>;<br/><b>hence </b><a NAME="E3:8_8_2_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom7.html#K1">Monom</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="polynom1.html#K15">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom7.html#K1">Monom</a> <font color="Maroon">c<sub>6</sub></font>,<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> <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="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="8_8_2_1_2"><b>case </b></a><a NAME="E1:8_8_2_1_2"/>
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><div class="add"><a NAME="E2:8_8_2_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="polynom7.html#K1">Monom</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="polynom1.html#K15">.</a> <font color="Maroon">c<sub>18</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#E7:8"><i><font color="Green">E9</font></i></a></i>;<br/><b>hence </b><a NAME="E3:8_8_2_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom7.html#K1">Monom</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="polynom1.html#K15">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom7.html#K1">Monom</a> <font color="Maroon">c<sub>6</sub></font>,<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> <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="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="E6:8_8"/>
<font color="Maroon">c<sub>11</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>16</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#E3:8_8"><i><font color="Green">E18</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E18:8"/><b>then </b>
 <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>13</sub></font>
 
<b>by </b><i><a class="txt" href="termord.html#E10:8"><i><font color="Green">E11</font></i></a>, <a class="txt" href="termord.html#E12:8"><i><font color="Green">E12</font></i></a>, <a class="ref" href="polynom2.html#T5">POLYNOM2:5</a></i>;<br/>
<a NAME="E19:8"/><b>then </b><i><font color="Green">E17</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom7.html#K1">Monom</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="polynom1.html#K28">*'</a> <span class="p2">(<span class="default"><a href="polynom7.html#K1">Monom</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="polynom1.html#K15">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="polynom1.html#K9">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</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#E1:8"><i><font color="Green">E4</font></i></a>, <a class="txt" href="termord.html#E4:8"><i><font color="Green">E6</font></i></a>, <a class="txt" href="termord.html#E5:8"><i><font color="Green">E7</font></i></a>, <a class="txt" href="termord.html#E9:8"><i><font color="Green">E10</font></i></a>, <a class="txt" href="termord.html#E14:8"><i><font color="Green">E13</font></i></a>, <a class="txt" href="termord.html#E15:8"><i><font color="Green">E14</font></i></a>, <a class="txt" href="termord.html#E16:8"><i><font color="Green">E15</font></i></a>, <a class="ref" href="vectsp_2.html#D5">VECTSP_2:def 5</a></i>;<br/>
<b>then </b><i><font color="Green">E18</font></i>:  <a href="polynom7.html#K2">term</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom7.html#K1">Monom</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="polynom1.html#K28">*'</a> <span class="p2">(<span class="default"><a href="polynom7.html#K1">Monom</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> = 
<font color="Maroon">c<sub>3</sub></font> <a href="polynom1.html#K9">+</a> <font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="ref" href="polynom7.html#D6">POLYNOM7:def 6</a></i>
<br/>.= 
 <a href="polynom7.html#K2">term</a> <span class="p1">(<span class="default"><a href="polynom7.html#K1">Monom</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="polynom1.html#K9">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="termord.html#E3:8"><i><font color="Green">E5</font></i></a>, <a class="ref" href="polynom7.html#T10">POLYNOM7:10</a></i>
;<br/>
<i><font color="Green">E19</font></i>:  <a href="polynom7.html#K3">coefficient</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom7.html#K1">Monom</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="polynom1.html#K28">*'</a> <span class="p2">(<span class="default"><a href="polynom7.html#K1">Monom</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom7.html#K1">Monom</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="polynom1.html#K28">*'</a> <span class="p2">(<span class="default"><a href="polynom7.html#K1">Monom</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="polynom1.html#K15">.</a> <span class="p1">(<span class="default"><a href="polynom7.html#K2">term</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom7.html#K1">Monom</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="polynom1.html#K28">*'</a> <span class="p3">(<span class="default"><a href="polynom7.html#K1">Monom</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="polynom7.html#D7">POLYNOM7:def 7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom7.html#K1">Monom</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="polynom1.html#K28">*'</a> <span class="p2">(<span class="default"><a href="polynom7.html#K1">Monom</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="polynom1.html#K15">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="polynom1.html#K9">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="termord.html#E19:8"><i><font color="Green">E17</font></i></a>, <a class="ref" href="polynom7.html#D6">POLYNOM7:def 6</a></i>
<br/>.= 
<font color="Maroon">c<sub>5</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>6</sub></font>
<b>by </b><i><a class="txt" href="termord.html#E4:8"><i><font color="Green">E6</font></i></a>, <a class="txt" href="termord.html#E5:8"><i><font color="Green">E7</font></i></a>, <a class="txt" href="termord.html#E9:8"><i><font color="Green">E10</font></i></a>, <a class="txt" href="termord.html#E10:8"><i><font color="Green">E11</font></i></a>, <a class="txt" href="termord.html#E12:8"><i><font color="Green">E12</font></i></a>, <a class="txt" href="termord.html#E14:8"><i><font color="Green">E13</font></i></a>, <a class="txt" href="termord.html#E15:8"><i><font color="Green">E14</font></i></a>, <a class="txt" href="termord.html#E16:8"><i><font color="Green">E15</font></i></a>, <a class="txt" href="termord.html#E17:8"><i><font color="Green">E16</font></i></a>, <a class="ref" href="polynom2.html#T5">POLYNOM2:5</a></i>
<br/>.= 
 <a href="polynom7.html#K3">coefficient</a> <span class="p1">(<span class="default"><a href="polynom7.html#K1">Monom</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="polynom1.html#K9">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="polynom7.html#T9">POLYNOM7:9</a></i>
;<br/>
<b>thus </b> <a href="polynom7.html#K1">Monom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="polynom1.html#K9">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> = 
 <a href="polynom7.html#K1">Monom</a> <span class="p1">(<span class="default"><a href="polynom7.html#K3">coefficient</a> <span class="p2">(<span class="default"><a href="polynom7.html#K1">Monom</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="polynom1.html#K9">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="polynom7.html#K2">term</a> <span class="p2">(<span class="default"><a href="polynom7.html#K1">Monom</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="polynom1.html#K9">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="polynom7.html#T11">POLYNOM7:11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="polynom7.html#K1">Monom</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="polynom1.html#K28">*'</a> <span class="p1">(<span class="default"><a href="polynom7.html#K1">Monom</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="termord.html#E20:8"><i><font color="Green">E18</font></i></a>, <a class="txt" href="termord.html#E21:8"><i><font color="Green">E19</font></i></a>, <a class="ref" href="polynom7.html#T11">POLYNOM7:11</a></i>
;<br/>


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