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

<a NAME="E1:72_1_1"/><i><font color="Green" title="E56">A1</font></i>: 
 <a href="algseq_1.html#K3" title="ALGSEQ_1:func.3">len</a> <font color="Maroon" title="c3">p</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 
<b>by </b><i><a class="ref" href="uproots.html#T19" target="_self" title="UPROOTS:th.19">Th19</a></i>;<br/>
<b>set </b><font color="Maroon" title="c4">r</font> = <span class="p1"><a href="polynom5.html#K4" title="POLYNOM5:func.4">&lt;%</a><span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <font color="Maroon" title="c2">x</font></span>)</span>,<span class="p2">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">L</font></span>)</span></span><a href="polynom5.html#K4" title="POLYNOM5:func.4">%&gt;</a></span>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <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> ] <b>means </b> ex <font color="Olive" title="b1">q</font> being  <a href="polynom3.html#NM1" title="POLYNOM3:NM.1">Polynomial</a> of <font color="Maroon" title="c1">L</font> st <font color="Maroon" title="c3">p</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="polynom5.html#K4" title="POLYNOM5:func.4">&lt;%</a><span class="default"><span class="p3">(<span class="default"><a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <font color="Maroon" title="c2">x</font></span>)</span>,<span class="p3">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">L</font></span>)</span></span><a href="polynom5.html#K4" title="POLYNOM5:func.4">%&gt;</a></span> <a href="polynom5.html#K2" title="POLYNOM5:func.2">`^</a> $1</span>)</span> <a href="polynom3.html#K15" title="POLYNOM3:func.15">*'</a> <font color="Olive" title="b1">q</font>;<br/>
<b>set </b><font color="Maroon" title="c5">F</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">k</font> where <font color="Olive" title="b1">k</font> is    <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>  : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>] </span>}</span> ;<br/>
<font color="Maroon" title="c3">p</font> = 
<span class="p1">(<span class="default"><a href="polynom3.html#K13" title="POLYNOM3:func.13">1_.</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="polynom3.html#K15" title="POLYNOM3:func.15">*'</a> <font color="Maroon" title="c3">p</font>
<b>by </b><i><a class="ref" href="polynom3.html#T36" title="POLYNOM3:th.36">POLYNOM3:36</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="polynom5.html#K4" title="POLYNOM5:func.4">&lt;%</a><span class="default"><span class="p3">(<span class="default"><a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <font color="Maroon" title="c2">x</font></span>)</span>,<span class="p3">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">L</font></span>)</span></span><a href="polynom5.html#K4" title="POLYNOM5:func.4">%&gt;</a></span> <a href="polynom5.html#K2" title="POLYNOM5:func.2">`^</a> 0</span>)</span> <a href="polynom3.html#K15" title="POLYNOM3:func.15">*'</a> <font color="Maroon" title="c3">p</font>
<b>by </b><i><a class="ref" href="polynom5.html#T16" title="POLYNOM5:th.16">POLYNOM5:16</a></i>
;<br/>
<a NAME="E3:72_1_1"/><b>then </b><i><font color="Green" title="E57">A2</font></i>: 
0 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">k</font> where <font color="Olive" title="b1">k</font> is    <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>  : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>] </span>}</span> 
 
;<br/>
<a NAME="E4:72_1_1"/><i><font color="Green" title="E58">A3</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive" title="b1">k</font> where <font color="Olive" title="b1">k</font> is    <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>  : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>] </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="algseq_1.html#K3" title="ALGSEQ_1:func.3">len</a> <font color="Maroon" title="c3">p</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="72_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">a</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="E1:72_1_1_2"/>
<font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">k</font> where <font color="Olive" title="b1">k</font> is    <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>  : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>] </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon" title="c7">k</font> being    <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> <b> such that </b><br/><a NAME="E3:72_1_1_2"/><i><font color="Green" title="E59">A4</font></i>: 
<font color="Maroon" title="c6">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">k</font>
 <b>and </b><br/><a NAME="E4:72_1_1_2"/><i><font color="Green" title="E60">A5</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c7">k</font>]
 ;<br/>
<b>consider </b><font color="Maroon" title="c8">q</font> being   <a href="polynom3.html#NM1" title="POLYNOM3:NM.1">Polynomial</a> of <font color="Maroon" title="c1">L</font><b> such that </b><br/><a NAME="E6:72_1_1_2"/><i><font color="Green" title="E61">A6</font></i>: 
<font color="Maroon" title="c3">p</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="polynom5.html#K4" title="POLYNOM5:func.4">&lt;%</a><span class="default"><span class="p3">(<span class="default"><a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <font color="Maroon" title="c2">x</font></span>)</span>,<span class="p3">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">L</font></span>)</span></span><a href="polynom5.html#K4" title="POLYNOM5:func.4">%&gt;</a></span> <a href="polynom5.html#K2" title="POLYNOM5:func.2">`^</a> <font color="Maroon" title="c7">k</font></span>)</span> <a href="polynom3.html#K15" title="POLYNOM3:func.15">*'</a> <font color="Maroon" title="c8">q</font>
 <b>by </b><i><a class="txt" href="uproots.html#E4:72_1_1_2"><i><font color="Green" title="E60">A5</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="72_1_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:72_1_1_2_1"/>
 <a href="algseq_1.html#K3" title="ALGSEQ_1:func.3">len</a> <font color="Maroon" title="c8">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 ;<br/><a NAME="E2:72_1_1_2_1"/><b>then </b>
<font color="Maroon" title="c8">q</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="polynom3.html#K12" title="POLYNOM3:func.12">0_.</a> <font color="Maroon" title="c1">L</font>
 <b>by </b><i><a class="ref" href="polynom4.html#T8" title="POLYNOM4:th.8">POLYNOM4:8</a></i>;<br/><a NAME="E3:72_1_1_2_1"/><b>then </b>
<font color="Maroon" title="c3">p</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="polynom3.html#K12" title="POLYNOM3:func.12">0_.</a> <font color="Maroon" title="c1">L</font>
 <b>by </b><i><a class="txt" href="uproots.html#E6:72_1_1_2"><i><font color="Green" title="E61">A6</font></i></a>, <a class="ref" href="polynom4.html#T5" title="POLYNOM4:th.5">POLYNOM4:5</a></i>;<br/><b>hence </b><a NAME="E4:72_1_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="uproots.html#E1:72_1_1"><i><font color="Green" title="E56">A1</font></i></a>, <a class="ref" href="polynom4.html#T6" title="POLYNOM4:th.6">POLYNOM4:6</a></i>;<br/></div><b>end;</b></div>
<a NAME="E8:72_1_1_2"/><b>then </b><i><font color="Green" title="E62">A7</font></i>: 
 <a href="algseq_1.html#K3" title="ALGSEQ_1:func.3">len</a> <font color="Maroon" title="c8">q</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 
;<br/>
<b>then reconsider </b><font color="Maroon" title="c9">q</font> = <font color="Maroon" title="c8">q</font> as  <a href="uproots.html#V1" title="UPROOTS:attr.1">non-zero</a> <a href="polynom3.html#NM1" title="POLYNOM3:NM.1">Polynomial</a> of <font color="Maroon" title="c1">L</font> <b>by </b><i><a class="ref" href="uproots.html#T19" target="_self" title="UPROOTS:th.19">Th19</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="72_1_1_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:72_1_1_2_2"/>
<font color="Maroon" title="c7">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="algseq_1.html#K3" title="ALGSEQ_1:func.3">len</a> <font color="Maroon" title="c3">p</font>
 ;<br/><a NAME="E2:72_1_1_2_2"/><b>then </b>
<font color="Maroon" title="c7">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><a href="algseq_1.html#K3" title="ALGSEQ_1:func.3">len</a> <font color="Maroon" title="c9">q</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><a href="algseq_1.html#K3" title="ALGSEQ_1:func.3">len</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 0
 <b>by </b><i><a class="txt" href="uproots.html#E8:72_1_1_2"><i><font color="Green" title="E62">A7</font></i></a>, <a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></i>;<br/><b>hence </b><a NAME="E3:72_1_1_2_2"/>
contradiction
 <b>by </b><i><a class="txt" href="uproots.html#E6:72_1_1_2"><i><font color="Green" title="E61">A6</font></i></a>, <a class="ref" href="uproots.html#T42" target="_self" title="UPROOTS:th.42">Th42</a></i>;<br/></div><b>end;</b></div>
<a NAME="E11:72_1_1_2"/><b>then </b>
<font color="Maroon" title="c7">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">i</font> where <font color="Olive" title="b1">i</font> is    <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>  : <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="algseq_1.html#K3" title="ALGSEQ_1:func.3">len</a> <font color="Maroon" title="c3">p</font> </span>}</span> 
 
;<br/>
<b>hence </b><a NAME="E12:72_1_1_2"/>
<font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="algseq_1.html#K3" title="ALGSEQ_1:func.3">len</a> <font color="Maroon" title="c3">p</font>
 <b>by </b><i><a class="txt" href="uproots.html#E3:72_1_1_2"><i><font color="Green" title="E59">A4</font></i></a>, <a class="ref" href="axioms.html#T30" title="AXIOMS:th.30">AXIOMS:30</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:72_1_1"/>
<span class="p1">{<span class="default"> <font color="Olive" title="b1">k</font> where <font color="Olive" title="b1">k</font> is    <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>  : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>] </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<b>from </b><i><a class="ref" href="fraenkel.html#S10" title="FRAENKEL:sch.10">FRAENKEL:sch 10</a>();<br/></i>
<b>then reconsider </b><font color="Maroon" title="c6">F</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">k</font> where <font color="Olive" title="b1">k</font> is    <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>  : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>] </span>}</span>  as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <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 <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="uproots.html#E3:72_1_1"><i><font color="Green" title="E57">A2</font></i></a>, <a class="txt" href="uproots.html#E4:72_1_1"><i><font color="Green" title="E58">A3</font></i></a>, <a class="ref" href="finset_1.html#T13" title="FINSET_1:th.13">FINSET_1:13</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">FF</font> = <font color="Maroon" title="c6">F</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="membered.html#V5" title="MEMBERED:attr.5">natural-membered</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ;<br/>
<b>reconsider </b><font color="Maroon" title="c8">m</font> =  <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">max</a> <font color="Maroon" title="c7">FF</font> as    <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>  <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c8">m</font>
;<br/>

<b>thus </b><a NAME="E9:72_1_1"/>
 ex <font color="Olive" title="b1">F</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <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 <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Olive" title="b1">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b2">k</font> where <font color="Olive" title="b2">k</font> is    <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>  :  ex <font color="Olive" title="b3">q</font> being  <a href="polynom3.html#NM1" title="POLYNOM3:NM.1">Polynomial</a> of <font color="Maroon" title="c1">L</font> st <font color="Maroon" title="c3">p</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="polynom5.html#K4" title="POLYNOM5:func.4">&lt;%</a><span class="default"><span class="p3">(<span class="default"><a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <font color="Maroon" title="c2">x</font></span>)</span>,<span class="p3">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">L</font></span>)</span></span><a href="polynom5.html#K4" title="POLYNOM5:func.4">%&gt;</a></span> <a href="polynom5.html#K2" title="POLYNOM5:func.2">`^</a> <font color="Olive" title="b2">k</font></span>)</span> <a href="polynom3.html#K15" title="POLYNOM3:func.15">*'</a> <font color="Olive" title="b3">q</font> </span>}</span>  &amp; <font color="Maroon" title="c8">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">max</a> <font color="Olive" title="b1">F</font> )
 ;<br/>


</div>
