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

<b>let </b><font color="Maroon" title="c1">A</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c2">S</font> be  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">A</font>;<br/><b>let </b><font color="Maroon" title="c3">b</font> be   <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">A</font>;<br/>





<b>set </b><font color="Maroon" title="c4">cS</font> =  <a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c2">S</font>;<br/>
<b>set </b><font color="Maroon" title="c5">f</font> = <font color="Maroon" title="c3">b</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c2">S</font></span>)</span>;<br/>
<a NAME="E1:22"/><i><font color="Green" title="E18">A1</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">S</font>
 
<b>by </b><i><a class="ref" href="uproots.html#T5" target="_self" title="UPROOTS:th.5">Th5</a></i>;<br/>
<a NAME="E2:22"/><i><font color="Green" title="E19">A2</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">A</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E3:22"/><b>then </b><i><font color="Green" title="E20">A3</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finsop_1.html#K5" title="FINSOP_1:func.5">dom</a> <span class="p1">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c2">S</font></span>)</span>
 
<b>by </b><i><a class="txt" href="uproots.html#E1:22"><i><font color="Green" title="E18">A1</font></i></a>, <a class="ref" href="relat_1.html#T46" title="RELAT_1:th.46">RELAT_1:46</a></i>;<br/>
<a NAME="E4:22"/><b>then </b><i><font color="Green" title="E21">A4</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c6">f</font> = <font color="Maroon" title="c3">b</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c2">S</font></span>)</span> as   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> <b>by </b><i><a class="ref" href="finseq_1.html#D2" title="FINSEQ_1:def.2">FINSEQ_1:def 2</a></i>;<br/>
<a NAME="E6:22"/><i><font color="Green" title="E22">A5</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c2">S</font>
 
<b>by </b><i><a class="ref" href="uproots.html#D1" target="_self" title="UPROOTS:def.1">Def1</a></i>;<br/>
<a NAME="E7:22"/><i><font color="Green" title="E23">A6</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">S</font>
 
<b>by </b><i><a class="ref" href="uproots.html#T5" target="_self" title="UPROOTS:th.5">Th5</a></i>;<br/>
<a NAME="E8:22"/><i><font color="Green" title="E24">A7</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font>
 
<b>by </b><i><a class="txt" href="uproots.html#E3:22"><i><font color="Green" title="E20">A3</font></i></a>, <a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></i>;<br/>
<a NAME="E9:22"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:22_1"/>
<font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">f</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c8">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:22_1"/><i><font color="Green" title="E25">A8</font></i>: 
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finsop_1.html#K5" title="FINSOP_1:func.5">dom</a> <font color="Maroon" title="c6">f</font>
 <b>and </b><br/><a NAME="E4:22_1"/><i><font color="Green" title="E26">A9</font></i>: 
<font color="Maroon" title="c7">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E5:22_1"/>
<font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font></span>)</span>
 
<b>by </b><i><a class="txt" href="uproots.html#E3:22_1"><i><font color="Green" title="E25">A8</font></i></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></i>;<br/>
<b>hence </b><a NAME="E6:22_1"/>
<font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <b>by </b><i><a class="txt" href="uproots.html#E4:22_1"><i><font color="Green" title="E26">A9</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c7">f</font> = <font color="Maroon" title="c6">f</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></i>;<br/>
<div><b>hereby </b>
<div class="add"><b>assume </b><b>that </b><br/><a NAME="E1:22_2"/><i><font color="Green" title="E25">A10</font></i>: 
<font color="Maroon" title="c2">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom1.html#K11" title="POLYNOM1:func.11">support</a> <font color="Maroon" title="c3">b</font>
 <b>and </b><br/><a NAME="E2:22_2"/><i><font color="Green" title="E26">A11</font></i>: 
 <a href="uproots.html#K4" title="UPROOTS:func.4">degree</a> <font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c2">S</font>
 ;<br/><b>consider </b><font color="Maroon" title="c8">F</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E4:22_2"/><i><font color="Green" title="E27">A12</font></i>: 
 <a href="uproots.html#K4" title="UPROOTS:func.4">degree</a> <font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="wsierp_1.html#K5" title="WSIERP_1:func.5">Sum</a> <font color="Maroon" title="c8">F</font>
 <b>and </b><br/><a NAME="E5:22_2"/><i><font color="Green" title="E28">A13</font></i>: 
<font color="Maroon" title="c8">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c2">S</font></span>)</span>
 <b>by </b><i><a class="txt" href="uproots.html#E1:22_2"><i><font color="Green" title="E25">A10</font></i></a>, <a class="ref" href="uproots.html#D4" target="_self" title="UPROOTS:def.4">Def4</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c9">i</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>assume </b><a NAME="E1:22_2_1"/><i><font color="Green" title="E29">A14</font></i>: 
<font color="Maroon" title="c9">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finsop_1.html#K5" title="FINSOP_1:func.5">dom</a> <font color="Maroon" title="c8">F</font>
 ;<br/><a NAME="E2:22_2_1"/><b>then </b><i><font color="Green" title="E30">A15</font></i>: 
<font color="Maroon" title="c8">F</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <font color="Maroon" title="c9">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i</font></span>)</span>
 <b>by </b><i><a class="txt" href="uproots.html#E5:22_2"><i><font color="Green" title="E28">A13</font></i></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></i>;<br/><a NAME="E3:22_2_1"/>
<span class="p1">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c2">S</font></span>)</span>
 <b>by </b><i><a class="txt" href="uproots.html#E3:22"><i><font color="Green" title="E20">A3</font></i></a>, <a class="txt" href="uproots.html#E5:22_2"><i><font color="Green" title="E28">A13</font></i></a>, <a class="txt" href="uproots.html#E1:22_2_1"><i><font color="Green" title="E29">A14</font></i></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></i>;<br/><b>hence </b><a NAME="E4:22_2_1"/>
<font color="Maroon" title="c8">F</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <font color="Maroon" title="c9">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0
 <b>by </b><i><a class="txt" href="uproots.html#E1:22_2"><i><font color="Green" title="E25">A10</font></i></a>, <a class="txt" href="uproots.html#E2:22_2_1"><i><font color="Green" title="E30">A15</font></i></a>, <a class="ref" href="polynom1.html#D7" title="POLYNOM1:def.7">POLYNOM1:def 7</a></i>;<br/></div><b>end;</b></div><a NAME="E7:22_2"/><b>then </b><i><font color="Green" title="E29">A16</font></i>: 
<font color="Maroon" title="c8">F</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">F</font></span>)</span> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">|-&gt;</a> 1
 <b>by </b><i><a class="txt" href="uproots.html#E6:22"><i><font color="Green" title="E22">A5</font></i></a>, <a class="txt" href="uproots.html#E8:22"><i><font color="Green" title="E24">A7</font></i></a>, <a class="txt" href="uproots.html#E2:22_2"><i><font color="Green" title="E26">A11</font></i></a>, <a class="txt" href="uproots.html#E4:22_2"><i><font color="Green" title="E27">A12</font></i></a>, <a class="txt" href="uproots.html#E5:22_2"><i><font color="Green" title="E28">A13</font></i></a>, <a class="ref" href="uproots.html#T2" target="_self" title="UPROOTS:th.2">Th2</a></i>;<br/><b>set </b><font color="Maroon" title="c9">sb</font> = <font color="Maroon" title="c2">S</font>,1 <a href="uproots.html#K2" title="UPROOTS:func.2">-bag</a> ;<br/><a NAME="E8:22_2"/><i><font color="Green" title="E30">A17</font></i>: 
 <a href="polynom1.html#K11" title="POLYNOM1:func.11">support</a> <font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom1.html#K11" title="POLYNOM1:func.11">support</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font>,1 <a href="uproots.html#K2" title="UPROOTS:func.2">-bag</a> </span>)</span>
 <b>by </b><i><a class="txt" href="uproots.html#E1:22_2"><i><font color="Green" title="E25">A10</font></i></a>, <a class="ref" href="uproots.html#T10" target="_self" title="UPROOTS:th.10">Th10</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_2_2"><b>now </b></a><div class="add"><b>thus </b><a NAME="E1:22_2_2"/>
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font>,1 <a href="uproots.html#K2" title="UPROOTS:func.2">-bag</a> </span>)</span>
 <b>by </b><i><a class="txt" href="uproots.html#E2:22"><i><font color="Green" title="E19">A2</font></i></a>, <a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/><b>let </b><font color="Maroon" title="c10">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E2:22_2_2"/>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c3">b</font>
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_2_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:22_2_2_1"/>
( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K11" title="POLYNOM1:func.11">support</a> <font color="Maroon" title="c3">b</font> or  not <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K11" title="POLYNOM1:func.11">support</a> <font color="Maroon" title="c3">b</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_2_2_1_1"><b>suppose </b></a><a NAME="E1:22_2_2_1_1"/><i><font color="Green" title="E31">A18</font></i>: 
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K11" title="POLYNOM1:func.11">support</a> <font color="Maroon" title="c3">b</font>
 ;<br/><div class="add"><a NAME="E2:22_2_2_1_1"/><i><font color="Green" title="E32">A19</font></i>: 
 <a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c2">S</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 <b>by </b><i><a class="ref" href="uproots.html#T4" target="_self" title="UPROOTS:th.4">Th4</a></i>;<br/><a NAME="E3:22_2_2_1_1"/><i><font color="Green" title="E33">A20</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom1.html#K11" title="POLYNOM1:func.11">support</a> <font color="Maroon" title="c3">b</font>
 <b>by </b><i><a class="txt" href="uproots.html#E1:22_2"><i><font color="Green" title="E25">A10</font></i></a>, <a class="ref" href="uproots.html#T5" target="_self" title="UPROOTS:th.5">Th5</a></i>;<br/><a NAME="E4:22_2_2_1_1"/><i><font color="Green" title="E34">A21</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finsop_1.html#K5" title="FINSOP_1:func.5">dom</a> <span class="p1">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c2">S</font></span>)</span>
 <b>by </b><i><a class="txt" href="uproots.html#E1:22"><i><font color="Green" title="E18">A1</font></i></a>, <a class="txt" href="uproots.html#E1:22_2"><i><font color="Green" title="E25">A10</font></i></a>, <a class="txt" href="uproots.html#E1:22_2_2_1_1"><i><font color="Green" title="E31">A18</font></i></a>, <a class="txt" href="uproots.html#E2:22_2_2_1_1"><i><font color="Green" title="E32">A19</font></i></a>, <a class="ref" href="funct_1.html#T54" title="FUNCT_1:th.54">FUNCT_1:54</a></i>;<br/><a NAME="E5:22_2_2_1_1"/><b>then </b><i><font color="Green" title="E35">A22</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">F</font></span>)</span>
 <b>by </b><i><a class="txt" href="uproots.html#E3:22"><i><font color="Green" title="E20">A3</font></i></a>, <a class="txt" href="uproots.html#E5:22_2"><i><font color="Green" title="E28">A13</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><b>thus </b><font color="Maroon" title="c3">b</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c10">x</font> = 
<font color="Maroon" title="c3">b</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="uproots.html#E1:22_2_2_1_1"><i><font color="Green" title="E31">A18</font></i></a>, <a class="txt" href="uproots.html#E2:22_2_2_1_1"><i><font color="Green" title="E32">A19</font></i></a>, <a class="txt" href="uproots.html#E3:22_2_2_1_1"><i><font color="Green" title="E33">A20</font></i></a>, <a class="ref" href="funct_1.html#T57" title="FUNCT_1:th.57">FUNCT_1:57</a></i>
<br/>.= 
<font color="Maroon" title="c8">F</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font></span>)</span>
<b>by </b><i><a class="txt" href="uproots.html#E5:22_2"><i><font color="Green" title="E28">A13</font></i></a>, <a class="txt" href="uproots.html#E4:22_2_2_1_1"><i><font color="Green" title="E34">A21</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>
<br/>.= 
1
<b>by </b><i><a class="txt" href="uproots.html#E7:22_2"><i><font color="Green" title="E29">A16</font></i></a>, <a class="txt" href="uproots.html#E5:22_2_2_1_1"><i><font color="Green" title="E35">A22</font></i></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font>,1 <a href="uproots.html#K2" title="UPROOTS:func.2">-bag</a> </span>)</span> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c10">x</font>
<b>by </b><i><a class="txt" href="uproots.html#E1:22_2"><i><font color="Green" title="E25">A10</font></i></a>, <a class="txt" href="uproots.html#E1:22_2_2_1_1"><i><font color="Green" title="E31">A18</font></i></a>, <a class="ref" href="uproots.html#T9" target="_self" title="UPROOTS:th.9">Th9</a></i>
;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_2_2_1_2"><b>suppose </b></a><a NAME="E1:22_2_2_1_2"/><i><font color="Green" title="E31">A23</font></i>: 
not <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K11" title="POLYNOM1:func.11">support</a> <font color="Maroon" title="c3">b</font>
 ;<br/><div class="add"><b>hence </b><font color="Maroon" title="c3">b</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c10">x</font> = 
0
<b>by </b><i><a class="ref" href="polynom1.html#D7" title="POLYNOM1:def.7">POLYNOM1:def 7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font>,1 <a href="uproots.html#K2" title="UPROOTS:func.2">-bag</a> </span>)</span> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c10">x</font>
<b>by </b><i><a class="txt" href="uproots.html#E8:22_2"><i><font color="Green" title="E30">A17</font></i></a>, <a class="txt" href="uproots.html#E1:22_2_2_1_2"><i><font color="Green" title="E31">A23</font></i></a>, <a class="ref" href="polynom1.html#D7" title="POLYNOM1:def.7">POLYNOM1:def 7</a></i>
;<br/><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E10:22_2"/>
<font color="Maroon" title="c3">b</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c2">S</font>,1 <a href="uproots.html#K2" title="UPROOTS:func.2">-bag</a> 
 <b>by </b><i><a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E12:22"/><i><font color="Green" title="E25">A24</font></i>: 
<font color="Maroon" title="c3">b</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c2">S</font>,1 <a href="uproots.html#K2" title="UPROOTS:func.2">-bag</a> 
 ;<br/>

<b>hence </b><a NAME="E13:22"/>
<font color="Maroon" title="c2">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom1.html#K11" title="POLYNOM1:func.11">support</a> <font color="Maroon" title="c3">b</font>
 <b>by </b><i><a class="ref" href="uproots.html#T10" target="_self" title="UPROOTS:th.10">Th10</a></i>;<br/>

<a NAME="E14:22"/><b>then </b><i><font color="Green" title="E26">A25</font></i>: 
 <a href="uproots.html#K4" title="UPROOTS:func.4">degree</a> <font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="wsierp_1.html#K5" title="WSIERP_1:func.5">Sum</a> <font color="Maroon" title="c7">f</font>
 
<b>by </b><i><a class="ref" href="uproots.html#D4" target="_self" title="UPROOTS:def.4">Def4</a></i>;<br/>
<b>set </b><font color="Maroon" title="c8">g</font> = <span class="p1">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">|-&gt;</a> 1;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_3"><b>now </b></a><div class="add"><b>thus </b><a NAME="E1:22_3"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c2">S</font>
 <b>by </b><i><a class="txt" href="uproots.html#E3:22"><i><font color="Green" title="E20">A3</font></i></a>, <a class="txt" href="uproots.html#E6:22"><i><font color="Green" title="E22">A5</font></i></a>, <a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></i>;<br/><b>thus </b><a NAME="E2:22_3"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">|-&gt;</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c2">S</font>
 <b>by </b><i><a class="ref" href="finseq_2.html#T69" title="FINSEQ_2:th.69">FINSEQ_2:69</a></i>;<br/><b>let </b><font color="Maroon" title="c9">i</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>assume </b><a NAME="E3:22_3"/><i><font color="Green" title="E27">A26</font></i>: 
<font color="Maroon" title="c9">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c2">S</font></span>)</span>
 ;<br/><a NAME="E4:22_3"/><b>then </b><i><font color="Green" title="E28">A27</font></i>: 
<span class="p1">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font>
 <b>by </b><i><a class="txt" href="uproots.html#E3:22"><i><font color="Green" title="E20">A3</font></i></a>, <a class="txt" href="uproots.html#E4:22"><i><font color="Green" title="E21">A4</font></i></a>, <a class="txt" href="uproots.html#E6:22"><i><font color="Green" title="E22">A5</font></i></a>, <a class="txt" href="uproots.html#E7:22"><i><font color="Green" title="E23">A6</font></i></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></i>;<br/><b>thus </b><font color="Maroon" title="c7">f</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <font color="Maroon" title="c9">i</font> = 
<font color="Maroon" title="c3">b</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i</font></span>)</span>
<b>by </b><i><a class="txt" href="uproots.html#E4:22"><i><font color="Green" title="E21">A4</font></i></a>, <a class="txt" href="uproots.html#E6:22"><i><font color="Green" title="E22">A5</font></i></a>, <a class="txt" href="uproots.html#E3:22_3"><i><font color="Green" title="E27">A26</font></i></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></i>
<br/>.= 
1
<b>by </b><i><a class="txt" href="uproots.html#E12:22"><i><font color="Green" title="E25">A24</font></i></a>, <a class="txt" href="uproots.html#E4:22_3"><i><font color="Green" title="E28">A27</font></i></a>, <a class="ref" href="uproots.html#T9" target="_self" title="UPROOTS:th.9">Th9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">|-&gt;</a> 1</span>)</span> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">.</a> <font color="Maroon" title="c9">i</font>
<b>by </b><i><a class="txt" href="uproots.html#E3:22_3"><i><font color="Green" title="E27">A26</font></i></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>
;<br/></div><b>end;</b></div>
<a NAME="E16:22"/><b>then </b>
<font color="Maroon" title="c7">f</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">|-&gt;</a> 1
 
<b>by </b><i><a class="ref" href="finseq_2.html#T10" title="FINSEQ_2:th.10">FINSEQ_2:10</a></i>;<br/>
<b>hence </b> <a href="uproots.html#K4" title="UPROOTS:func.4">degree</a> <font color="Maroon" title="c3">b</font> = 
<span class="p1">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> 1
<b>by </b><i><a class="txt" href="uproots.html#E14:22"><i><font color="Green" title="E26">A25</font></i></a>, <a class="ref" href="rvsum_1.html#T110" title="RVSUM_1:th.110">RVSUM_1:110</a></i>
<br/>.= 
 <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c2">S</font>

;<br/><br/>


</div>
