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<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>( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> $1 &amp; $1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon">F<sub>1</sub></font>() &amp; <font color="Maroon">P<sub>1</sub></font>[$1,<font color="Maroon">F<sub>1</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1] );<br/>
<b>assume </b><a NAME="E1:3_1"/>
 ex <font color="Olive" title="b1">i</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>  st <br/>( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon">F<sub>1</sub></font>() &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">i</font>,<font color="Maroon">F<sub>1</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font>] )
 ;<br/>

<a NAME="E2:3_1"/><b>then </b><i><font color="Green" title="E5">A3</font></i>: 
 ex <font color="Olive" title="b1">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>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>]
 
;<br/>
<b>consider </b><font color="Maroon" title="c1">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="E4:3_1"/><i><font color="Green" title="E6">A4</font></i>: 
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c1">k</font>] &amp; ( for <font color="Olive" title="b1">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>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k'</font>] holds <br/><font color="Maroon" title="c1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k'</font> ) )
 <b>from </b><i><a class="ref" href="nat_1.html#S5" title="NAT_1:sch.5">NAT_1:sch 5</a>(<a class="txt" href="uproots.html#E2:3_1"><i><font color="Green" title="E5">A3</font></i></a>);<br/></i>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:3_1_1"/>
( <font color="Maroon" title="c1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c1">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="3_1_1_1"><b>suppose </b></a><a NAME="E1:3_1_1_1"/>
<font color="Maroon" title="c1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:3_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="uproots.html#E1:3"><i><font color="Green" title="E3">A1</font></i></a>, <a class="txt" href="uproots.html#E4:3_1"><i><font color="Green" title="E6">A4</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="3_1_1_2"><b>suppose </b></a><a NAME="E1:3_1_1_2"/><i><font color="Green" title="E7">A5</font></i>: 
<font color="Maroon" title="c1">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 ;<br/><div class="add"><a NAME="E2:3_1_1_2"/>
1 <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">k</font> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> 1
 <b>by </b><i><a class="txt" href="uproots.html#E4:3_1"><i><font color="Green" title="E6">A4</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c2">k'</font> = <font color="Maroon" title="c1">k</font> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> 1 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="int_1.html#T16" title="INT_1:th.16">INT_1:16</a></i>;<br/><a NAME="E4:3_1_1_2"/><i><font color="Green" title="E8">A6</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> 1</span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 0
 ;<br/><a NAME="E5:3_1_1_2"/><i><font color="Green" title="E9">A7</font></i>: 
<font color="Maroon" title="c2">k'</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">k'</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/><a NAME="E6:3_1_1_2"/>
<font color="Maroon" title="c2">k'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">k'</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1
 ;<br/><a NAME="E7:3_1_1_2"/><b>then </b><i><font color="Green" title="E10">A8</font></i>: 
<font color="Maroon" title="c2">k'</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c1">k</font>
 <b>by </b><i><a class="txt" href="uproots.html#E5:3_1_1_2"><i><font color="Green" title="E9">A7</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><a NAME="E8:3_1_1_2"/><b>then </b><i><font color="Green" title="E11">A9</font></i>: 
 not <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">k'</font>]
 <b>by </b><i><a class="txt" href="uproots.html#E4:3_1"><i><font color="Green" title="E6">A4</font></i></a></i>;<br/><a NAME="E9:3_1_1_2"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c1">k</font>
 <b>by </b><i><a class="txt" href="uproots.html#E4:3_1"><i><font color="Green" title="E6">A4</font></i></a>, <a class="txt" href="uproots.html#E1:3_1_1_2"><i><font color="Green" title="E7">A5</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><a NAME="E10:3_1_1_2"/><b>then </b><i><font color="Green" title="E12">A10</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">k'</font>
 <b>by </b><i><a class="txt" href="uproots.html#E4:3_1_1_2"><i><font color="Green" title="E8">A6</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E11:3_1_1_2"/>
<font color="Maroon" title="c2">k'</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon">F<sub>1</sub></font>()
 <b>by </b><i><a class="txt" href="uproots.html#E4:3_1"><i><font color="Green" title="E6">A4</font></i></a>, <a class="txt" href="uproots.html#E7:3_1_1_2"><i><font color="Green" title="E10">A8</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E12:3_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="uproots.html#E2:3"><i><font color="Green" title="E4">A2</font></i></a>, <a class="txt" href="uproots.html#E4:3_1"><i><font color="Green" title="E6">A4</font></i></a>, <a class="txt" href="uproots.html#E4:3_1_1_2"><i><font color="Green" title="E8">A6</font></i></a>, <a class="txt" href="uproots.html#E8:3_1_1_2"><i><font color="Green" title="E11">A9</font></i></a>, <a class="txt" href="uproots.html#E10:3_1_1_2"><i><font color="Green" title="E12">A10</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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