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<b>reconsider </b><font color="Maroon" title="c3">d1</font> = <font color="Maroon" title="c2">d</font>, <font color="Maroon" title="c4">n1</font> = <font color="Maroon" title="c1">n</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/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:52_1_1_1"/>
( <font color="Maroon" title="c2">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 or <font color="Maroon" title="c2">d</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_1_1_1"><b>suppose </b></a><a NAME="E1:52_1_1_1_1"/><i><font color="Green" title="E25">A3</font></i>: 
<font color="Maroon" title="c2">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 ;<br/><div class="add"><b>take </b>
0
;<br/><a NAME="E2:52_1_1_1_1"/>
0 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> 0 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="ref" href="newton.html#T9" title="NEWTON:th.9">NEWTON:9</a></i>;<br/><b>hence </b><a NAME="E3:52_1_1_1_1"/>
<font color="Maroon" title="c2">d</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> 0 <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c1">n</font>
 <b>by </b><i><a class="txt" href="nat_3.html#E1:52_1_1_1_1"><i><font color="Green" title="E25">A3</font></i></a>, <a class="ref" href="nat_d.html#T6" title="NAT_D:th.6">NAT_D:6</a></i>;<br/><a NAME="E4:52_1_1_1_1"/>
not 0 <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c4">n1</font>
 <b>by </b><i><a class="txt" href="nat_3.html#E2:52"><i><font color="Green" title="E24">A2</font></i></a>, <a class="ref" href="int_2.html#T3" title="INT_2:th.3">INT_2:3</a></i>;<br/><b>hence </b><a NAME="E5:52_1_1_1_1"/>
not <font color="Maroon" title="c2">d</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p1">(<span class="default">0 <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c1">n</font>
 <b>by </b><i><a class="txt" href="nat_3.html#E1:52_1_1_1_1"><i><font color="Green" title="E25">A3</font></i></a>, <a class="ref" href="newton.html#T16" title="NEWTON:th.16">NEWTON:16</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_1_1_2"><b>suppose </b></a><a NAME="E1:52_1_1_1_2"/><i><font color="Green" title="E25">A4</font></i>: 
<font color="Maroon" title="c2">d</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:52_1_1_1_2"/><i><font color="Green" title="E26">A5</font></i>: 
<font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 <b>by </b><i><a class="txt" href="nat_3.html#E2:52"><i><font color="Green" title="E24">A2</font></i></a></i>;<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><font color="Maroon" title="c2">d</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> $1 <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c1">n</font>;<br/><a NAME="E3:52_1_1_1_2"/><i><font color="Green" title="E27">A6</font></i>: 
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="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">n1</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">k</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="E1:52_1_1_1_2_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">k</font>]
 ;<br/>

<a NAME="E2:52_1_1_1_2_1"/><b>then </b><i><font color="Green" title="E28">A7</font></i>: 
<font color="Maroon" title="c2">d</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c5">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font>
 
<b>by </b><i><a class="txt" href="nat_3.html#E2:52_1_1_1_2"><i><font color="Green" title="E26">A5</font></i></a>, <a class="ref" href="nat_d.html#T7" title="NAT_D:th.7">NAT_D:7</a></i>;<br/>
<a NAME="E3:52_1_1_1_2_1"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c2">d</font>
 
<b>by </b><i><a class="txt" href="nat_3.html#E1:52"><i><font color="Green" title="E23">A1</font></i></a>, <a class="txt" href="nat_3.html#E1:52_1_1_1_2"><i><font color="Green" title="E25">A4</font></i></a>, <a class="ref" href="nat_1.html#T26" title="NAT_1:th.26">NAT_1:26</a></i>;<br/>
<a NAME="E4:52_1_1_1_2_1"/><b>then </b>
<font color="Maroon" title="c5">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">d</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c5">k</font>
 
<b>by </b><i><a class="ref" href="nat_3.html#T2" target="_self" title="NAT_3:th.2">Th2</a></i>;<br/>
<b>hence </b><a NAME="E5:52_1_1_1_2_1"/>
<font color="Maroon" title="c5">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">n1</font>
 <b>by </b><i><a class="txt" href="nat_3.html#E2:52_1_1_1_2_1"><i><font color="Green" title="E28">A7</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>


</div><b>end;</b></div><a NAME="E4:52_1_1_1_2"/><i><font color="Green" title="E28">A8</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>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1_1_1_2_2"><b>proof </b></a><div class="add">

<b>take </b>
0
;<br/>

<a NAME="E1:52_1_1_1_2_2"/>
<font color="Maroon" title="c2">d</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> 0 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="ref" href="newton.html#T9" title="NEWTON:th.9">NEWTON:9</a></i>;<br/>
<b>hence </b><a NAME="E2:52_1_1_1_2_2"/>
<font color="Maroon" title="c2">d</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> 0 <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c1">n</font>
 <b>by </b><i><a class="ref" href="nat_d.html#T6" title="NAT_D:th.6">NAT_D:6</a></i>;<br/>


</div><b>end;</b></div><b>consider </b><font color="Maroon" title="c5">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="E6:52_1_1_1_2"/><i><font color="Green" title="E29">A9</font></i>: 
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">k</font>] &amp; ( for <font color="Olive" title="b1">n</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">n</font>] holds <br/><font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">k</font> ) )
 <b>from </b><i><a class="ref" href="nat_1.html#S6" title="NAT_1:sch.6">NAT_1:sch 6</a>(<a class="txt" href="nat_3.html#E3:52_1_1_1_2"><i><font color="Green" title="E27">A6</font></i></a>, <a class="txt" href="nat_3.html#E4:52_1_1_1_2"><i><font color="Green" title="E28">A8</font></i></a>);<br/></i><b>take </b>
<font color="Maroon" title="c5">k</font>
;<br/><a NAME="E7:52_1_1_1_2"/>
<font color="Maroon" title="c5">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c5">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><b>hence </b><a NAME="E8:52_1_1_1_2"/>
( <font color="Maroon" title="c2">d</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c5">k</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c1">n</font> &amp; not <font color="Maroon" title="c2">d</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c1">n</font> )
 <b>by </b><i><a class="txt" href="nat_3.html#E6:52_1_1_1_2"><i><font color="Green" title="E29">A9</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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