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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</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="E1:47"/>
 <a href="polynom2.html#K1">support</a> <font color="Maroon">c<sub>2</sub></font> <a href="xboole_0.html#R1">misses</a>  <a href="polynom2.html#K1">support</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<a NAME="E2:47"/><b>then </b><i><font color="Green">E22</font></i>: 
<span class="p1">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>2</sub></font> <a href="polynom1.html#K9">+</a> <font color="Maroon">c<sub>3</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>5</sub></font> =  <a href="nat_3.html#K8" target="_self">Product</a> <font color="Maroon">c<sub>2</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>6</sub></font> =  <a href="nat_3.html#K8" target="_self">Product</a> <font color="Maroon">c<sub>3</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>7</sub></font> =  <a href="nat_3.html#K8" target="_self">Product</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="polynom1.html#K9">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>;<br/>
<b>set </b><font color="Maroon">c<sub>8</sub></font> =  <a href="polynom2.html#K1">support</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="polynom1.html#K9">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>;<br/>
<b>set </b><font color="Maroon">c<sub>9</sub></font> =  <a href="polynom2.html#K1">support</a> <font color="Maroon">c<sub>2</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>10</sub></font> =  <a href="polynom2.html#K1">support</a> <font color="Maroon">c<sub>3</sub></font>;<br/>
<b>consider </b><font color="Maroon">c<sub>11</sub></font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K2">COMPLEX</a> <b> such that </b><br/><a NAME="E4:47"/><i><font color="Green">E23</font></i>: 
 <a href="nat_3.html#K8" target="_self">Product</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="polynom1.html#K9">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="rvsum_1.html#K17">Product</a> <font color="Maroon">c<sub>11</sub></font>
 <b>and </b><br/><a NAME="E5:47"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="polynom1.html#K9">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><a href="uproots.html#K1">canFS</a> <span class="p2">(<span class="default"><a href="polynom2.html#K1">support</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="polynom1.html#K9">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="nat_3.html#D5" target="_self">Def5</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>12</sub></font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K2">COMPLEX</a> <b> such that </b><br/><a NAME="E7:47"/><i><font color="Green">E25</font></i>: 
 <a href="nat_3.html#K8" target="_self">Product</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="rvsum_1.html#K17">Product</a> <font color="Maroon">c<sub>12</sub></font>
 <b>and </b><br/><a NAME="E8:47"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><a href="uproots.html#K1">canFS</a> <span class="p2">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="nat_3.html#D5" target="_self">Def5</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>13</sub></font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K2">COMPLEX</a> <b> such that </b><br/><a NAME="E10:47"/><i><font color="Green">E27</font></i>: 
 <a href="nat_3.html#K8" target="_self">Product</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="rvsum_1.html#K17">Product</a> <font color="Maroon">c<sub>13</sub></font>
 <b>and </b><br/><a NAME="E11:47"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><a href="uproots.html#K1">canFS</a> <span class="p2">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="nat_3.html#D5" target="_self">Def5</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>14</sub></font> = <font color="Maroon">c<sub>12</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>13</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>15</sub></font> =  <a href="uproots.html#K1">canFS</a> <span class="p1">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>;<br/>
<b>set </b><font color="Maroon">c<sub>16</sub></font> =  <a href="uproots.html#K1">canFS</a> <span class="p1">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>;<br/>
<b>set </b><font color="Maroon">c<sub>17</sub></font> = <span class="p1">(<span class="default"><a href="uproots.html#K1">canFS</a> <span class="p2">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="finseq_1.html#K7">^</a> <span class="p1">(<span class="default"><a href="uproots.html#K1">canFS</a> <span class="p2">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon">c<sub>18</sub></font> =  <a href="uproots.html#K1">canFS</a> <span class="p1">(<span class="default"><a href="polynom2.html#K1">support</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="polynom1.html#K9">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon">c<sub>19</sub></font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="uproots.html#K1">canFS</a> <span class="p4">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="finseq_1.html#K7">^</a> <span class="p3">(<span class="default"><a href="uproots.html#K1">canFS</a> <span class="p4">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><a href="uproots.html#K1">canFS</a> <span class="p2">(<span class="default"><a href="polynom2.html#K1">support</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="polynom1.html#K9">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span>;<br/>
<a NAME="E12:47"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="uproots.html#K1">canFS</a> <span class="p2">(<span class="default"><a href="polynom2.html#K1">support</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="polynom1.html#K9">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="polynom2.html#K1">support</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="polynom1.html#K9">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="uproots.html#D1">UPROOTS:def 1</a></i>;<br/>
<a NAME="E13:47"/><b>then </b><i><font color="Green">E29</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="uproots.html#K1">canFS</a> <span class="p2">(<span class="default"><a href="polynom2.html#K1">support</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="polynom1.html#K9">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><a href="polynom2.html#K1">support</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="polynom1.html#K9">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E14:47"/><i><font color="Green">E30</font></i>: 
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="uproots.html#K1">canFS</a> <span class="p2">(<span class="default"><a href="polynom2.html#K1">support</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="polynom1.html#K9">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="polynom2.html#K1">support</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="polynom1.html#K9">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="uproots.html#T5">UPROOTS:5</a></i>;<br/>
<a NAME="E15:47"/><i><font color="Green">E31</font></i>: 
 <a href="uproots.html#K1">canFS</a> <span class="p1">(<span class="default"><a href="polynom2.html#K1">support</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="polynom1.html#K9">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> is <a href="funct_1.html#V2">one-to-one</a>
 
<b>by </b><i><a class="ref" href="uproots.html#T4">UPROOTS:4</a></i>;<br/>
<a NAME="E16:47"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="polynom1.html#K9">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E17:47"/><b>then </b><i><font color="Green">E32</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><a href="polynom2.html#K1">support</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="polynom1.html#K9">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="nat_3.html#E5:47"><i><font color="Green">E24</font></i></a>, <a class="txt" href="nat_3.html#E13:47"><i><font color="Green">E29</font></i></a>, <a class="txt" href="nat_3.html#E14:47"><i><font color="Green">E30</font></i></a>, <a class="ref" href="relat_1.html#T46">RELAT_1:46</a></i>;<br/>
<a NAME="E18:47"/><i><font color="Green">E33</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="uproots.html#K1">canFS</a> <span class="p2">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="uproots.html#D1">UPROOTS:def 1</a></i>;<br/>
<a NAME="E19:47"/><i><font color="Green">E34</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="uproots.html#K1">canFS</a> <span class="p2">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="uproots.html#D1">UPROOTS:def 1</a></i>;<br/>
<a NAME="E20:47"/><i><font color="Green">E35</font></i>: 
 <a href="polynom2.html#K1">support</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="polynom1.html#K9">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="polynom1.html#T42">POLYNOM1:42</a></i>;<br/>
<a NAME="E21:47"/><i><font color="Green">E36</font></i>: 
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="uproots.html#K1">canFS</a> <span class="p2">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="polynom2.html#K1">support</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="uproots.html#T5">UPROOTS:5</a></i>;<br/>
<a NAME="E22:47"/><i><font color="Green">E37</font></i>: 
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="uproots.html#K1">canFS</a> <span class="p2">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="polynom2.html#K1">support</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="uproots.html#T5">UPROOTS:5</a></i>;<br/>
<a NAME="E23:47"/><i><font color="Green">E38</font></i>: 
 <a href="uproots.html#K1">canFS</a> <span class="p1">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> is <a href="funct_1.html#V2">one-to-one</a>
 
<b>by </b><i><a class="ref" href="uproots.html#T4">UPROOTS:4</a></i>;<br/>
<a NAME="E24:47"/><i><font color="Green">E39</font></i>: 
 <a href="uproots.html#K1">canFS</a> <span class="p1">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> is <a href="funct_1.html#V2">one-to-one</a>
 
<b>by </b><i><a class="ref" href="uproots.html#T4">UPROOTS:4</a></i>;<br/>
<i><font color="Green">E40</font></i>:  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1">canFS</a> <span class="p3">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="finseq_1.html#K7">^</a> <span class="p2">(<span class="default"><a href="uproots.html#K1">canFS</a> <span class="p3">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p3">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p2">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p3">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <a href="xboole_0.html#K1">{}</a> </span>)</span>
<b>by </b><i><a class="txt" href="nat_3.html#E18:47"><i><font color="Green">E33</font></i></a>, <a class="txt" href="nat_3.html#E19:47"><i><font color="Green">E34</font></i></a>, <a class="ref" href="card_1.html#T78">CARD_1:78</a>, <a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>
<br/>.= 
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="polynom2.html#K1">support</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="polynom1.html#K9">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="nat_3.html#E2:47"><i><font color="Green">E22</font></i></a>, <a class="txt" href="nat_3.html#E20:47"><i><font color="Green">E35</font></i></a>, <a class="ref" href="card_2.html#T64">CARD_2:64</a></i>
;<br/>
<a NAME="E26:47"/><b>then </b><i><font color="Green">E41</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1">canFS</a> <span class="p3">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="finseq_1.html#K7">^</a> <span class="p2">(<span class="default"><a href="uproots.html#K1">canFS</a> <span class="p3">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><a href="polynom2.html#K1">support</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="polynom1.html#K9">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E27:47"/><i><font color="Green">E42</font></i>: 
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1">canFS</a> <span class="p3">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="finseq_1.html#K7">^</a> <span class="p2">(<span class="default"><a href="uproots.html#K1">canFS</a> <span class="p3">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="polynom2.html#K1">support</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="polynom1.html#K9">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="nat_3.html#E20:47"><i><font color="Green">E35</font></i></a>, <a class="txt" href="nat_3.html#E21:47"><i><font color="Green">E36</font></i></a>, <a class="txt" href="nat_3.html#E22:47"><i><font color="Green">E37</font></i></a>, <a class="ref" href="finseq_1.html#T44">FINSEQ_1:44</a></i>;<br/>
<a NAME="E28:47"/><b>then </b><i><font color="Green">E43</font></i>: 
<span class="p1">(<span class="default"><a href="uproots.html#K1">canFS</a> <span class="p2">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="finseq_1.html#K7">^</a> <span class="p1">(<span class="default"><a href="uproots.html#K1">canFS</a> <span class="p2">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> is <a href="funct_1.html#V2">one-to-one</a>
 
<b>by </b><i><a class="txt" href="nat_3.html#E25:47"><i><font color="Green">E40</font></i></a>, <a class="ref" href="finseq_4.html#T77">FINSEQ_4:77</a></i>;<br/>
<a NAME="E29:47"/><b>then </b><i><font color="Green">E44</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="uproots.html#K1">canFS</a> <span class="p4">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="finseq_1.html#K7">^</a> <span class="p3">(<span class="default"><a href="uproots.html#K1">canFS</a> <span class="p4">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="polynom2.html#K1">support</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="polynom1.html#K9">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="nat_3.html#E27:47"><i><font color="Green">E42</font></i></a>, <a class="ref" href="funct_1.html#T55">FUNCT_1:55</a></i>;<br/>
<a NAME="E30:47"/><i><font color="Green">E45</font></i>: 
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="uproots.html#K1">canFS</a> <span class="p4">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="finseq_1.html#K7">^</a> <span class="p3">(<span class="default"><a href="uproots.html#K1">canFS</a> <span class="p4">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><a href="polynom2.html#K1">support</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="polynom1.html#K9">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="nat_3.html#E26:47"><i><font color="Green">E41</font></i></a>, <a class="txt" href="nat_3.html#E28:47"><i><font color="Green">E43</font></i></a>, <a class="ref" href="funct_1.html#T55">FUNCT_1:55</a></i>;<br/>
<a NAME="E31:47"/><i><font color="Green">E46</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1">canFS</a> <span class="p3">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="polynom2.html#K1">support</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="nat_3.html#E21:47"><i><font color="Green">E36</font></i></a>, <a class="txt" href="nat_3.html#E23:47"><i><font color="Green">E38</font></i></a>, <a class="ref" href="funct_1.html#T54">FUNCT_1:54</a></i>;<br/>
<a NAME="E32:47"/><i><font color="Green">E47</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="uproots.html#K1">canFS</a> <span class="p2">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="nat_3.html#E18:47"><i><font color="Green">E33</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E33:47"/><i><font color="Green">E48</font></i>: 
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1">canFS</a> <span class="p3">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="uproots.html#K1">canFS</a> <span class="p2">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="nat_3.html#E23:47"><i><font color="Green">E38</font></i></a>, <a class="ref" href="funct_1.html#T55">FUNCT_1:55</a></i>;<br/>
<a NAME="E34:47"/><i><font color="Green">E49</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1">canFS</a> <span class="p3">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="polynom2.html#K1">support</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="nat_3.html#E22:47"><i><font color="Green">E37</font></i></a>, <a class="txt" href="nat_3.html#E24:47"><i><font color="Green">E39</font></i></a>, <a class="ref" href="funct_1.html#T54">FUNCT_1:54</a></i>;<br/>
<a NAME="E35:47"/><i><font color="Green">E50</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="uproots.html#K1">canFS</a> <span class="p2">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="nat_3.html#E19:47"><i><font color="Green">E34</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E36:47"/><i><font color="Green">E51</font></i>: 
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1">canFS</a> <span class="p3">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="uproots.html#K1">canFS</a> <span class="p2">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="nat_3.html#E24:47"><i><font color="Green">E39</font></i></a>, <a class="ref" href="funct_1.html#T55">FUNCT_1:55</a></i>;<br/>
<a NAME="E37:47"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E38:47"/><b>then </b><i><font color="Green">E52</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="uproots.html#K1">canFS</a> <span class="p2">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>12</sub></font>
 
<b>by </b><i><a class="txt" href="nat_3.html#E8:47"><i><font color="Green">E26</font></i></a>, <a class="txt" href="nat_3.html#E22:47"><i><font color="Green">E37</font></i></a>, <a class="ref" href="relat_1.html#T46">RELAT_1:46</a></i>;<br/>
<a NAME="E39:47"/><b>then </b><i><font color="Green">E53</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="uproots.html#K1">canFS</a> <span class="p2">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>12</sub></font>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/>
<a NAME="E40:47"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E41:47"/><b>then </b>
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="uproots.html#K1">canFS</a> <span class="p2">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>13</sub></font>
 
<b>by </b><i><a class="txt" href="nat_3.html#E11:47"><i><font color="Green">E28</font></i></a>, <a class="txt" href="nat_3.html#E21:47"><i><font color="Green">E36</font></i></a>, <a class="ref" href="relat_1.html#T46">RELAT_1:46</a></i>;<br/>
<a NAME="E42:47"/><b>then </b><i><font color="Green">E54</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="uproots.html#K1">canFS</a> <span class="p2">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>13</sub></font>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/>
<a NAME="E43:47"/><i><font color="Green">E55</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1">canFS</a> <span class="p3">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="finseq_1.html#K7">^</a> <span class="p2">(<span class="default"><a href="uproots.html#K1">canFS</a> <span class="p3">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="uproots.html#K1">canFS</a> <span class="p3">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="uproots.html#K1">canFS</a> <span class="p3">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>;<br/>
<a NAME="E44:47"/><i><font color="Green">E56</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="uproots.html#K1">canFS</a> <span class="p5">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="finseq_1.html#K7">^</a> <span class="p4">(<span class="default"><a href="uproots.html#K1">canFS</a> <span class="p5">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="uproots.html#K1">canFS</a> <span class="p3">(<span class="default"><a href="polynom2.html#K1">support</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="polynom1.html#K9">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><a href="polynom2.html#K1">support</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="polynom1.html#K9">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="nat_3.html#E13:47"><i><font color="Green">E29</font></i></a>, <a class="txt" href="nat_3.html#E14:47"><i><font color="Green">E30</font></i></a>, <a class="txt" href="nat_3.html#E29:47"><i><font color="Green">E44</font></i></a>, <a class="ref" href="relat_1.html#T46">RELAT_1:46</a></i>;<br/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p3">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p2">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p3">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <a href="xboole_0.html#K1">{}</a> </span>)</span>
<b>by </b><i><a class="txt" href="nat_3.html#E18:47"><i><font color="Green">E33</font></i></a>, <a class="txt" href="nat_3.html#E19:47"><i><font color="Green">E34</font></i></a>, <a class="txt" href="nat_3.html#E39:47"><i><font color="Green">E53</font></i></a>, <a class="txt" href="nat_3.html#E42:47"><i><font color="Green">E54</font></i></a>, <a class="ref" href="card_1.html#T78">CARD_1:78</a>, <a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>
<br/>.= 
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="polynom2.html#K1">support</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="polynom1.html#K9">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="nat_3.html#E2:47"><i><font color="Green">E22</font></i></a>, <a class="txt" href="nat_3.html#E20:47"><i><font color="Green">E35</font></i></a>, <a class="ref" href="card_2.html#T64">CARD_2:64</a></i>
;<br/>
<a NAME="E46:47"/><b>then </b><i><font color="Green">E57</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><a href="polynom2.html#K1">support</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="polynom1.html#K9">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E47:47"/><i><font color="Green">E58</font></i>: 
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="uproots.html#K1">canFS</a> <span class="p5">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="finseq_1.html#K7">^</a> <span class="p4">(<span class="default"><a href="uproots.html#K1">canFS</a> <span class="p5">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="uproots.html#K1">canFS</a> <span class="p3">(<span class="default"><a href="polynom2.html#K1">support</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="polynom1.html#K9">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="uproots.html#K1">canFS</a> <span class="p4">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="finseq_1.html#K7">^</a> <span class="p3">(<span class="default"><a href="uproots.html#K1">canFS</a> <span class="p4">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K2">"</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="nat_3.html#E14:47"><i><font color="Green">E30</font></i></a>, <a class="txt" href="nat_3.html#E29:47"><i><font color="Green">E44</font></i></a>, <a class="ref" href="relat_1.html#T47">RELAT_1:47</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>20</sub></font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="uproots.html#K1">canFS</a> <span class="p4">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="finseq_1.html#K7">^</a> <span class="p3">(<span class="default"><a href="uproots.html#K1">canFS</a> <span class="p4">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><a href="uproots.html#K1">canFS</a> <span class="p2">(<span class="default"><a href="polynom2.html#K1">support</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="polynom1.html#K9">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> as   <a href="funct_2.html#NM1">Function</a> of  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>13</sub></font></span>)</span>, <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <b>by </b><i><a class="txt" href="nat_3.html#E30:47"><i><font color="Green">E45</font></i></a>, <a class="txt" href="nat_3.html#E44:47"><i><font color="Green">E56</font></i></a>, <a class="txt" href="nat_3.html#E46:47"><i><font color="Green">E57</font></i></a>, <a class="ref" href="funct_2.html#T3">FUNCT_2:3</a></i>;<br/>
<a NAME="E49:47"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1">canFS</a> <span class="p3">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="finseq_1.html#K7">^</a> <span class="p2">(<span class="default"><a href="uproots.html#K1">canFS</a> <span class="p3">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K2">"</a>  is <a href="funct_1.html#V2">one-to-one</a>
 
<b>by </b><i><a class="txt" href="nat_3.html#E28:47"><i><font color="Green">E43</font></i></a>, <a class="ref" href="funct_1.html#T62">FUNCT_1:62</a></i>;<br/>
<a NAME="E50:47"/><b>then </b><i><font color="Green">E59</font></i>: 
<font color="Maroon">c<sub>20</sub></font> is <a href="funct_1.html#V2">one-to-one</a>
 
<b>by </b><i><a class="txt" href="nat_3.html#E15:47"><i><font color="Green">E31</font></i></a>, <a class="ref" href="funct_1.html#T46">FUNCT_1:46</a></i>;<br/>
<a NAME="E51:47"/>
<font color="Maroon">c<sub>20</sub></font> is <a href="funct_2.html#V2">onto</a>
 
<b>by </b><i><a class="txt" href="nat_3.html#E30:47"><i><font color="Green">E45</font></i></a>, <a class="txt" href="nat_3.html#E46:47"><i><font color="Green">E57</font></i></a>, <a class="txt" href="nat_3.html#E47:47"><i><font color="Green">E58</font></i></a>, <a class="ref" href="funct_2.html#D3">FUNCT_2:def 3</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>21</sub></font> = <font color="Maroon">c<sub>20</sub></font> as   <a href="funct_2.html#NM2">Permutation</a> of  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <b>by </b><i><a class="txt" href="nat_3.html#E50:47"><i><font color="Green">E59</font></i></a>, <a class="ref" href="funct_2.html#D4">FUNCT_2:def 4</a></i>;<br/>
<a NAME="E53:47"/><i><font color="Green">E60</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><a href="polynom2.html#K1">support</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="polynom1.html#K9">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="nat_3.html#E30:47"><i><font color="Green">E45</font></i></a>, <a class="txt" href="nat_3.html#E44:47"><i><font color="Green">E56</font></i></a>, <a class="txt" href="nat_3.html#E46:47"><i><font color="Green">E57</font></i></a>, <a class="txt" href="nat_3.html#E47:47"><i><font color="Green">E58</font></i></a>, <a class="ref" href="relat_1.html#T46">RELAT_1:46</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>22</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:47_3"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">c<sub>22</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>
 ;<br/><b>set </b><font color="Maroon">c<sub>23</sub></font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="uproots.html#K1">canFS</a> <span class="p4">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="finseq_1.html#K7">^</a> <span class="p3">(<span class="default"><a href="uproots.html#K1">canFS</a> <span class="p4">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1">canFS</a> <span class="p3">(<span class="default"><a href="polynom2.html#K1">support</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="polynom1.html#K9">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>22</sub></font></span>)</span>;<br/><i><font color="Green">E62</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>22</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>22</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="nat_3.html#E17:47"><i><font color="Green">E32</font></i></a>, <a class="txt" href="nat_3.html#E53:47"><i><font color="Green">E60</font></i></a>, <a class="txt" href="nat_3.html#E1:47_3"><i><font color="Green">E61</font></i></a>, <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="uproots.html#K1">canFS</a> <span class="p5">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="finseq_1.html#K7">^</a> <span class="p4">(<span class="default"><a href="uproots.html#K1">canFS</a> <span class="p5">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="uproots.html#K1">canFS</a> <span class="p4">(<span class="default"><a href="polynom2.html#K1">support</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="polynom1.html#K9">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>22</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="nat_3.html#E13:47"><i><font color="Green">E29</font></i></a>, <a class="txt" href="nat_3.html#E17:47"><i><font color="Green">E32</font></i></a>, <a class="txt" href="nat_3.html#E1:47_3"><i><font color="Green">E61</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
;<br/><a NAME="E3:47_3"/><i><font color="Green">E63</font></i>: 
<span class="p1">(<span class="default"><a href="uproots.html#K1">canFS</a> <span class="p2">(<span class="default"><a href="polynom2.html#K1">support</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="polynom1.html#K9">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R2">in</a>  <a href="polynom2.html#K1">support</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="polynom1.html#K9">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="nat_3.html#E13:47"><i><font color="Green">E29</font></i></a>, <a class="txt" href="nat_3.html#E14:47"><i><font color="Green">E30</font></i></a>, <a class="txt" href="nat_3.html#E17:47"><i><font color="Green">E32</font></i></a>, <a class="txt" href="nat_3.html#E1:47_3"><i><font color="Green">E61</font></i></a>, <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/><a NAME="E4:47_3"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="uproots.html#K1">canFS</a> <span class="p4">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="finseq_1.html#K7">^</a> <span class="p3">(<span class="default"><a href="uproots.html#K1">canFS</a> <span class="p4">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1">canFS</a> <span class="p3">(<span class="default"><a href="polynom2.html#K1">support</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="polynom1.html#K9">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>22</sub></font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><a href="polynom2.html#K1">support</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="polynom1.html#K9">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="nat_3.html#E29:47"><i><font color="Green">E44</font></i></a>, <a class="txt" href="nat_3.html#E30:47"><i><font color="Green">E45</font></i></a>, <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>24</sub></font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="uproots.html#K1">canFS</a> <span class="p4">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="finseq_1.html#K7">^</a> <span class="p3">(<span class="default"><a href="uproots.html#K1">canFS</a> <span class="p4">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1">canFS</a> <span class="p3">(<span class="default"><a href="polynom2.html#K1">support</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="polynom1.html#K9">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>22</sub></font></span>)</span> as  <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  <b>by </b><i><a class="txt" href="nat_3.html#E1"><i><font color="Green">Lemma1</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>25</sub></font> = <font color="Maroon">c<sub>24</sub></font> as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>26</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E8:47_3"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1">canFS</a> <span class="p3">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="finseq_1.html#K7">^</a> <span class="p2">(<span class="default"><a href="uproots.html#K1">canFS</a> <span class="p3">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E9:47_3"/><i><font color="Green">E65</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1">canFS</a> <span class="p3">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="finseq_1.html#K7">^</a> <span class="p2">(<span class="default"><a href="uproots.html#K1">canFS</a> <span class="p3">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="uproots.html#K1">canFS</a> <span class="p2">(<span class="default"><a href="polynom2.html#K1">support</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="polynom1.html#K9">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>22</sub></font>
 <b>by </b><i><a class="txt" href="nat_3.html#E27:47"><i><font color="Green">E42</font></i></a>, <a class="txt" href="nat_3.html#E3:47_3"><i><font color="Green">E63</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E10:47_3"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>26</sub></font>
 <b>by </b><i><a class="txt" href="nat_3.html#E28:47"><i><font color="Green">E43</font></i></a>, <a class="txt" href="nat_3.html#E8:47_3"><i><font color="Green">E64</font></i></a>, <a class="txt" href="nat_3.html#E9:47_3"><i><font color="Green">E65</font></i></a>, <a class="ref" href="funct_1.html#T56">FUNCT_1:56</a></i>;<br/><a NAME="E11:47_3"/><b>then </b><i><font color="Green">E67</font></i>: 
<font color="Maroon">c<sub>25</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>13</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="nat_3.html#E39:47"><i><font color="Green">E53</font></i></a>, <a class="txt" href="nat_3.html#E42:47"><i><font color="Green">E54</font></i></a>, <a class="txt" href="nat_3.html#E43:47"><i><font color="Green">E55</font></i></a>, <a class="txt" href="nat_3.html#E8:47_3"><i><font color="Green">E64</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E12:47_3"/><i><font color="Green">E68</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>25</sub></font>
 <b>by </b><i><a class="txt" href="nat_3.html#E8:47_3"><i><font color="Green">E64</font></i></a>, <a class="txt" href="nat_3.html#E10:47_3"><i><font color="Green">E66</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_3_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:47_3_2"/>
( <span class="p1">(<span class="default"><a href="uproots.html#K1">canFS</a> <span class="p2">(<span class="default"><a href="polynom2.html#K1">support</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="polynom1.html#K9">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R2">in</a>  <a href="polynom2.html#K1">support</a> <font color="Maroon">c<sub>2</sub></font> or <span class="p1">(<span class="default"><a href="uproots.html#K1">canFS</a> <span class="p2">(<span class="default"><a href="polynom2.html#K1">support</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="polynom1.html#K9">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R2">in</a>  <a href="polynom2.html#K1">support</a> <font color="Maroon">c<sub>3</sub></font> )
 </span><b>by </b><i><a class="txt" href="nat_3.html#E20:47"><i><font color="Green">E35</font></i></a>, <a class="txt" href="nat_3.html#E3:47_3"><i><font color="Green">E63</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_3_2_1"><b>suppose </b></a><a NAME="E1:47_3_2_1"/><i><font color="Green">E69</font></i>: 
<span class="p1">(<span class="default"><a href="uproots.html#K1">canFS</a> <span class="p2">(<span class="default"><a href="polynom2.html#K1">support</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="polynom1.html#K9">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R2">in</a>  <a href="polynom2.html#K1">support</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><div class="add"><a NAME="E2:47_3_2_1"/><b>then </b><i><font color="Green">E70</font></i>: 
not <span class="p1">(<span class="default"><a href="uproots.html#K1">canFS</a> <span class="p2">(<span class="default"><a href="polynom2.html#K1">support</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="polynom1.html#K9">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R2">in</a>  <a href="polynom2.html#K1">support</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="nat_3.html#E2:47"><i><font color="Green">E22</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><b>set </b><font color="Maroon">c<sub>27</sub></font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1">canFS</a> <span class="p3">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1">canFS</a> <span class="p3">(<span class="default"><a href="polynom2.html#K1">support</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="polynom1.html#K9">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>22</sub></font></span>)</span>;<br/><a NAME="E3:47_3_2_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1">canFS</a> <span class="p3">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1">canFS</a> <span class="p3">(<span class="default"><a href="polynom2.html#K1">support</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="polynom1.html#K9">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>22</sub></font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1">canFS</a> <span class="p3">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K2">"</a> </span>)</span>
 <b>by </b><i><a class="txt" href="nat_3.html#E34:47"><i><font color="Green">E49</font></i></a>, <a class="txt" href="nat_3.html#E1:47_3_2_1"><i><font color="Green">E69</font></i></a>, <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>28</sub></font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1">canFS</a> <span class="p3">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1">canFS</a> <span class="p3">(<span class="default"><a href="polynom2.html#K1">support</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="polynom1.html#K9">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>22</sub></font></span>)</span> as  <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  <b>by </b><i><a class="txt" href="nat_3.html#E35:47"><i><font color="Green">E50</font></i></a>, <a class="txt" href="nat_3.html#E36:47"><i><font color="Green">E51</font></i></a>, <a class="txt" href="nat_3.html#E1"><i><font color="Green">Lemma1</font></i></a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_3_2_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:47_3_2_1_1"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>24</sub></font>
 ;<br/><a NAME="E2:47_3_2_1_1"/><b>then </b><i><font color="Green">E71</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="binop_2.html#K23">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>24</sub></font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E3:47_3_2_1_1"/><b>then </b><i><font color="Green">E72</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1">canFS</a> <span class="p3">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="finseq_1.html#K7">^</a> <span class="p2">(<span class="default"><a href="uproots.html#K1">canFS</a> <span class="p3">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="uproots.html#K1">canFS</a> <span class="p2">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>24</sub></font> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="uproots.html#K1">canFS</a> <span class="p4">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="nat_3.html#E39:47"><i><font color="Green">E53</font></i></a>, <a class="txt" href="nat_3.html#E42:47"><i><font color="Green">E54</font></i></a>, <a class="txt" href="nat_3.html#E11:47_3"><i><font color="Green">E67</font></i></a>, <a class="ref" href="finseq_1.html#T36">FINSEQ_1:36</a></i>;<br/><a NAME="E4:47_3_2_1_1"/><i><font color="Green">E73</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="uproots.html#K1">canFS</a> <span class="p4">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="uproots.html#K1">canFS</a> <span class="p3">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>24</sub></font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="uproots.html#K1">canFS</a> <span class="p3">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="nat_3.html#E39:47"><i><font color="Green">E53</font></i></a>, <a class="txt" href="nat_3.html#E2:47_3_2_1_1"><i><font color="Green">E71</font></i></a>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E5:47_3_2_1_1"/><b>then </b>
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>24</sub></font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="uproots.html#K1">canFS</a> <span class="p3">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span>
 ;<br/><a NAME="E6:47_3_2_1_1"/><b>then </b><i><font color="Green">E74</font></i>: 
<font color="Maroon">c<sub>24</sub></font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="uproots.html#K1">canFS</a> <span class="p3">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span> is   <a href="nat_1.html#NM1">Nat</a>
 <b>by </b><i><a class="ref" href="int_1.html#T16">INT_1:16</a></i>;<br/><a NAME="E7:47_3_2_1_1"/>
<font color="Maroon">c<sub>24</sub></font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="uproots.html#K1">canFS</a> <span class="p3">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="uproots.html#K1">canFS</a> <span class="p4">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="uproots.html#K1">canFS</a> <span class="p4">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="uproots.html#K1">canFS</a> <span class="p3">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="nat_3.html#E39:47"><i><font color="Green">E53</font></i></a>, <a class="txt" href="nat_3.html#E42:47"><i><font color="Green">E54</font></i></a>, <a class="txt" href="nat_3.html#E11:47_3"><i><font color="Green">E67</font></i></a>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E8:47_3_2_1_1"/><b>then </b>
<font color="Maroon">c<sub>24</sub></font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="uproots.html#K1">canFS</a> <span class="p3">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="uproots.html#K1">canFS</a> <span class="p2">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="nat_3.html#E4:47_3_2_1_1"><i><font color="Green">E73</font></i></a>, <a class="txt" href="nat_3.html#E6:47_3_2_1_1"><i><font color="Green">E74</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><b>hence </b><a NAME="E9:47_3_2_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="nat_3.html#E21:47"><i><font color="Green">E36</font></i></a>, <a class="txt" href="nat_3.html#E9:47_3"><i><font color="Green">E65</font></i></a>, <a class="txt" href="nat_3.html#E10:47_3"><i><font color="Green">E66</font></i></a>, <a class="txt" href="nat_3.html#E2:47_3_2_1"><i><font color="Green">E70</font></i></a>, <a class="txt" href="nat_3.html#E3:47_3_2_1_1"><i><font color="Green">E72</font></i></a>, <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/></div><b>end;</b></div><a NAME="E6:47_3_2_1"/><b>then </b><i><font color="Green">E71</font></i>: 
<font color="Maroon">c<sub>25</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>
 <b>by </b><i><a class="txt" href="nat_3.html#E12:47_3"><i><font color="Green">E68</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E7:47_3_2_1"/><b>then </b><i><font color="Green">E72</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1">canFS</a> <span class="p3">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="finseq_1.html#K7">^</a> <span class="p2">(<span class="default"><a href="uproots.html#K1">canFS</a> <span class="p3">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="uproots.html#K1">canFS</a> <span class="p2">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>24</sub></font>
 <b>by </b><i><a class="txt" href="nat_3.html#E38:47"><i><font color="Green">E52</font></i></a>, <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>;<br/><i><font color="Green">E73</font></i>: <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>24</sub></font> = 
<font color="Maroon">c<sub>12</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>25</sub></font>
<b>by </b><i><a class="txt" href="nat_3.html#E6:47_3_2_1"><i><font color="Green">E71</font></i></a>, <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>
<br/>.= 
<font color="Maroon">c<sub>2</sub></font> <a href="polynom1.html#K8">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1">canFS</a> <span class="p3">(<span class="default"><a href="polynom2.html#K1">support</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="polynom1.html#K9">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>22</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="nat_3.html#E8:47"><i><font color="Green">E26</font></i></a>, <a class="txt" href="nat_3.html#E38:47"><i><font color="Green">E52</font></i></a>, <a class="txt" href="nat_3.html#E9:47_3"><i><font color="Green">E65</font></i></a>, <a class="txt" href="nat_3.html#E10:47_3"><i><font color="Green">E66</font></i></a>, <a class="txt" href="nat_3.html#E6:47_3_2_1"><i><font color="Green">E71</font></i></a>, <a class="txt" href="nat_3.html#E7:47_3_2_1"><i><font color="Green">E72</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
;<br/><b>thus </b><font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>22</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="polynom1.html#K9">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="polynom1.html#K8">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1">canFS</a> <span class="p3">(<span class="default"><a href="polynom2.html#K1">support</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="polynom1.html#K9">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>22</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="nat_3.html#E5:47"><i><font color="Green">E24</font></i></a>, <a class="txt" href="nat_3.html#E1:47_3"><i><font color="Green">E61</font></i></a>, <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="polynom1.html#K8">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="uproots.html#K1">canFS</a> <span class="p4">(<span class="default"><a href="polynom2.html#K1">support</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="polynom1.html#K9">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>22</sub></font></span>)</span></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="polynom1.html#K8">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="uproots.html#K1">canFS</a> <span class="p4">(<span class="default"><a href="polynom2.html#K1">support</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="polynom1.html#K9">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>22</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="polynom1.html#D5">POLYNOM1:def 5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="polynom1.html#K8">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="uproots.html#K1">canFS</a> <span class="p4">(<span class="default"><a href="polynom2.html#K1">support</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="polynom1.html#K9">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>22</sub></font></span>)</span></span>)</span> <a href="binop_2.html#K23">+</a> 0
<b>by </b><i><a class="txt" href="nat_3.html#E2:47_3_2_1"><i><font color="Green">E70</font></i></a>, <a class="ref" href="polynom1.html#D7">POLYNOM1:def 7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>22</sub></font>
<b>by </b><i><a class="txt" href="nat_3.html#E2:47_3"><i><font color="Green">E62</font></i></a>, <a class="txt" href="nat_3.html#E8:47_3_2_1"><i><font color="Green">E73</font></i></a></i>
;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_3_2_2"><b>suppose </b></a><a NAME="E1:47_3_2_2"/><i><font color="Green">E69</font></i>: 
<span class="p1">(<span class="default"><a href="uproots.html#K1">canFS</a> <span class="p2">(<span class="default"><a href="polynom2.html#K1">support</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="polynom1.html#K9">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R2">in</a>  <a href="polynom2.html#K1">support</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><div class="add"><a NAME="E2:47_3_2_2"/><b>then </b><i><font color="Green">E70</font></i>: 
not <span class="p1">(<span class="default"><a href="uproots.html#K1">canFS</a> <span class="p2">(<span class="default"><a href="polynom2.html#K1">support</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="polynom1.html#K9">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R2">in</a>  <a href="polynom2.html#K1">support</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="nat_3.html#E2:47"><i><font color="Green">E22</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><b>set </b><font color="Maroon">c<sub>27</sub></font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1">canFS</a> <span class="p3">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1">canFS</a> <span class="p3">(<span class="default"><a href="polynom2.html#K1">support</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="polynom1.html#K9">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>22</sub></font></span>)</span>;<br/><a NAME="E3:47_3_2_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1">canFS</a> <span class="p3">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1">canFS</a> <span class="p3">(<span class="default"><a href="polynom2.html#K1">support</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="polynom1.html#K9">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>22</sub></font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1">canFS</a> <span class="p3">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K2">"</a> </span>)</span>
 <b>by </b><i><a class="txt" href="nat_3.html#E31:47"><i><font color="Green">E46</font></i></a>, <a class="txt" href="nat_3.html#E1:47_3_2_2"><i><font color="Green">E69</font></i></a>, <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>28</sub></font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1">canFS</a> <span class="p3">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1">canFS</a> <span class="p3">(<span class="default"><a href="polynom2.html#K1">support</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="polynom1.html#K9">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>22</sub></font></span>)</span> as  <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  <b>by </b><i><a class="txt" href="nat_3.html#E32:47"><i><font color="Green">E47</font></i></a>, <a class="txt" href="nat_3.html#E33:47"><i><font color="Green">E48</font></i></a>, <a class="txt" href="nat_3.html#E1"><i><font color="Green">Lemma1</font></i></a></i>;<br/><div><i><font color="Green">E71</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="47_3_2_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:47_3_2_2_1"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="binop_2.html#K23">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>24</sub></font>
 ;<br/><a NAME="E2:47_3_2_2_1"/><b>then </b>
<font color="Maroon">c<sub>24</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E3:47_3_2_2_1"/><b>then </b><i><font color="Green">E72</font></i>: 
<font color="Maroon">c<sub>24</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="uproots.html#K1">canFS</a> <span class="p2">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="nat_3.html#E38:47"><i><font color="Green">E52</font></i></a>, <a class="txt" href="nat_3.html#E12:47_3"><i><font color="Green">E68</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E4:47_3_2_2_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="uproots.html#K1">canFS</a> <span class="p2">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R2">in</a>  <a href="polynom2.html#K1">support</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="nat_3.html#E22:47"><i><font color="Green">E37</font></i></a>, <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/><a NAME="E5:47_3_2_2_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1">canFS</a> <span class="p3">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="finseq_1.html#K7">^</a> <span class="p2">(<span class="default"><a href="uproots.html#K1">canFS</a> <span class="p3">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R2">in</a>  <a href="polynom2.html#K1">support</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="nat_3.html#E3:47_3_2_2_1"><i><font color="Green">E72</font></i></a>, <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>;<br/><b>hence </b><a NAME="E6:47_3_2_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="nat_3.html#E2:47"><i><font color="Green">E22</font></i></a>, <a class="txt" href="nat_3.html#E9:47_3"><i><font color="Green">E65</font></i></a>, <a class="txt" href="nat_3.html#E10:47_3"><i><font color="Green">E66</font></i></a>, <a class="txt" href="nat_3.html#E1:47_3_2_2"><i><font color="Green">E69</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/></div><b>end;</b></div><a NAME="E6:47_3_2_2"/><b>then </b><i><font color="Green">E72</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1">canFS</a> <span class="p3">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="finseq_1.html#K7">^</a> <span class="p2">(<span class="default"><a href="uproots.html#K1">canFS</a> <span class="p3">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="uproots.html#K1">canFS</a> <span class="p2">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>24</sub></font> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="uproots.html#K1">canFS</a> <span class="p4">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="nat_3.html#E39:47"><i><font color="Green">E53</font></i></a>, <a class="txt" href="nat_3.html#E42:47"><i><font color="Green">E54</font></i></a>, <a class="txt" href="nat_3.html#E11:47_3"><i><font color="Green">E67</font></i></a>, <a class="ref" href="finseq_1.html#T36">FINSEQ_1:36</a></i>;<br/><a NAME="E7:47_3_2_2"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">c<sub>24</sub></font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="uproots.html#K1">canFS</a> <span class="p3">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="uproots.html#K1">canFS</a> <span class="p4">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="uproots.html#K1">canFS</a> <span class="p4">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="uproots.html#K1">canFS</a> <span class="p3">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="nat_3.html#E39:47"><i><font color="Green">E53</font></i></a>, <a class="txt" href="nat_3.html#E42:47"><i><font color="Green">E54</font></i></a>, <a class="txt" href="nat_3.html#E11:47_3"><i><font color="Green">E67</font></i></a>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E8:47_3_2_2"/><i><font color="Green">E74</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="uproots.html#K1">canFS</a> <span class="p4">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="uproots.html#K1">canFS</a> <span class="p3">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>24</sub></font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="uproots.html#K1">canFS</a> <span class="p3">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="nat_3.html#E39:47"><i><font color="Green">E53</font></i></a>, <a class="txt" href="nat_3.html#E5:47_3_2_2"><i><font color="Green">E71</font></i></a>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E9:47_3_2_2"/><b>then </b>
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>25</sub></font> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="uproots.html#K1">canFS</a> <span class="p3">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span>
 ;<br/><a NAME="E10:47_3_2_2"/><b>then </b>
<font color="Maroon">c<sub>25</sub></font> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="uproots.html#K1">canFS</a> <span class="p3">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span> is   <a href="nat_1.html#NM1">Nat</a>
 <b>by </b><i><a class="ref" href="int_1.html#T16">INT_1:16</a></i>;<br/><a NAME="E11:47_3_2_2"/><b>then </b><i><font color="Green">E75</font></i>: 
<font color="Maroon">c<sub>25</sub></font> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="uproots.html#K1">canFS</a> <span class="p3">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="uproots.html#K1">canFS</a> <span class="p2">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="nat_3.html#E7:47_3_2_2"><i><font color="Green">E73</font></i></a>, <a class="txt" href="nat_3.html#E8:47_3_2_2"><i><font color="Green">E74</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><i><font color="Green">E76</font></i>: <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>24</sub></font> = 
<font color="Maroon">c<sub>13</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>25</sub></font> <a href="binop_2.html#K10">-</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="nat_3.html#E11:47_3"><i><font color="Green">E67</font></i></a>, <a class="txt" href="nat_3.html#E5:47_3_2_2"><i><font color="Green">E71</font></i></a>, <a class="ref" href="finseq_1.html#T36">FINSEQ_1:36</a></i>
<br/>.= 
<font color="Maroon">c<sub>3</sub></font> <a href="polynom1.html#K8">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1">canFS</a> <span class="p3">(<span class="default"><a href="polynom2.html#K1">support</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="polynom1.html#K9">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>22</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="nat_3.html#E11:47"><i><font color="Green">E28</font></i></a>, <a class="txt" href="nat_3.html#E39:47"><i><font color="Green">E53</font></i></a>, <a class="txt" href="nat_3.html#E9:47_3"><i><font color="Green">E65</font></i></a>, <a class="txt" href="nat_3.html#E10:47_3"><i><font color="Green">E66</font></i></a>, <a class="txt" href="nat_3.html#E6:47_3_2_2"><i><font color="Green">E72</font></i></a>, <a class="txt" href="nat_3.html#E11:47_3_2_2"><i><font color="Green">E75</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
;<br/><b>thus </b><font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>22</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="polynom1.html#K9">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="polynom1.html#K8">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1">canFS</a> <span class="p3">(<span class="default"><a href="polynom2.html#K1">support</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="polynom1.html#K9">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>22</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="nat_3.html#E5:47"><i><font color="Green">E24</font></i></a>, <a class="txt" href="nat_3.html#E1:47_3"><i><font color="Green">E61</font></i></a>, <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="polynom1.html#K8">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="uproots.html#K1">canFS</a> <span class="p4">(<span class="default"><a href="polynom2.html#K1">support</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="polynom1.html#K9">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>22</sub></font></span>)</span></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="polynom1.html#K8">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="uproots.html#K1">canFS</a> <span class="p4">(<span class="default"><a href="polynom2.html#K1">support</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="polynom1.html#K9">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>22</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="polynom1.html#D5">POLYNOM1:def 5</a></i>
<br/>.= 
0 <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="polynom1.html#K8">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="uproots.html#K1">canFS</a> <span class="p4">(<span class="default"><a href="polynom2.html#K1">support</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="polynom1.html#K9">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>22</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="nat_3.html#E2:47_3_2_2"><i><font color="Green">E70</font></i></a>, <a class="ref" href="polynom1.html#D7">POLYNOM1:def 7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>22</sub></font>
<b>by </b><i><a class="txt" href="nat_3.html#E2:47_3"><i><font color="Green">E62</font></i></a>, <a class="txt" href="nat_3.html#E12:47_3_2_2"><i><font color="Green">E76</font></i></a></i>
;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E55:47"/><b>then </b><i><font color="Green">E61</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">c<sub>21</sub></font>
 
<b>by </b><i><a class="txt" href="nat_3.html#E17:47"><i><font color="Green">E32</font></i></a>, <a class="txt" href="nat_3.html#E53:47"><i><font color="Green">E60</font></i></a>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>
<b>thus </b> <a href="nat_3.html#K8" target="_self">Product</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="polynom1.html#K9">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> = 
<a href="binop_2.html#K29">multcomplex</a>  <a href="finsop_1.html#K2">$$</a> <font color="Maroon">c<sub>11</sub></font>
<b>by </b><i><a class="txt" href="nat_3.html#E4:47"><i><font color="Green">E23</font></i></a>, <a class="ref" href="rvsum_1.html#D12">RVSUM_1:def 12</a></i>
<br/>.= 
<a href="binop_2.html#K29">multcomplex</a>  <a href="finsop_1.html#K2">$$</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>13</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="nat_3.html#E55:47"><i><font color="Green">E61</font></i></a>, <a class="ref" href="finsop_1.html#T8">FINSOP_1:8</a></i>
<br/>.= 
 <a href="rvsum_1.html#K17">Product</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>13</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="rvsum_1.html#D12">RVSUM_1:def 12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="nat_3.html#K8" target="_self">Product</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="binop_2.html#K24">*</a> <span class="p1">(<span class="default"><a href="nat_3.html#K8" target="_self">Product</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="nat_3.html#E7:47"><i><font color="Green">E25</font></i></a>, <a class="txt" href="nat_3.html#E10:47"><i><font color="Green">E27</font></i></a>, <a class="ref" href="rvsum_1.html#T127">RVSUM_1:127</a></i>
;<br/>


</div>
