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

<b>let </b><font color="Maroon" title="c1">i</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:101"/><i><font color="Green" title="E55">A1</font></i>: 
<font color="Maroon" title="c1">i</font> is <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>
 ;<br/>

<b>given </b><font color="Maroon" title="c2">m</font>, <font color="Maroon" title="c3">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><a NAME="E3:101"/><i><font color="Green" title="E56">A2</font></i>: 
( <font color="Maroon" title="c1">i</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">n</font> &amp; not <font color="Maroon" title="c1">i</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">m</font> &amp; not <font color="Maroon" title="c1">i</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c3">n</font> )
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <b>means </b>( $1 is <a href="int_2.html#V1" title="INT_2:attr.1">prime</a> &amp;  ex <font color="Olive" title="b1">m</font>, <font color="Olive" title="b2">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( $1 <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Olive" title="b1">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b2">n</font> &amp; not $1 <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Olive" title="b1">m</font> &amp; not $1 <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Olive" title="b2">n</font> ) );<br/>
<a NAME="E4:101"/><i><font color="Green" title="E57">A3</font></i>: 
 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 <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">i</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="101_1"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon" title="c4">i</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E2:101_1"/><i><font color="Green" title="E58">A4</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">i</font>]
 <b>by </b><i><a class="txt" href="newton.html#E1:101"><i><font color="Green" title="E55">A1</font></i></a>, <a class="txt" href="newton.html#E3:101"><i><font color="Green" title="E56">A2</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c5">i</font> = <font color="Maroon" title="c4">i</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="c5">i</font>
;<br/>

<b>thus </b><a NAME="E4:101_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">i</font>]
 <b>by </b><i><a class="txt" href="newton.html#E2:101_1"><i><font color="Green" title="E58">A4</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c4">p'</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="E6:101"/><i><font color="Green" title="E58">A5</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">p'</font>]
 <b>and </b><br/><a NAME="E7:101"/><i><font color="Green" title="E59">A6</font></i>: 
for <font color="Olive" title="b1">p1</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">p1</font>] holds <br/><font color="Maroon" title="c4">p'</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p1</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="newton.html#E4:101"><i><font color="Green" title="E57">A3</font></i></a>);<br/></i>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <b>means </b> ex <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <font color="Maroon" title="c4">p'</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> $1 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b1">n</font> &amp; not <font color="Maroon" title="c4">p'</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> $1 &amp; not <font color="Maroon" title="c4">p'</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Olive" title="b1">n</font> );<br/>
<a NAME="E8:101"/><i><font color="Green" title="E60">A7</font></i>: 
 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 <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">i</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="101_2"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon" title="c5">i</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E2:101_2"/><i><font color="Green" title="E61">A8</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c5">i</font>]
 <b>by </b><i><a class="txt" href="newton.html#E6:101"><i><font color="Green" title="E58">A5</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c6">i</font> = <font color="Maroon" title="c5">i</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="c6">i</font>
;<br/>

<b>thus </b><a NAME="E4:101_2"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c6">i</font>]
 <b>by </b><i><a class="txt" href="newton.html#E2:101_2"><i><font color="Green" title="E61">A8</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c5">m</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="E10:101"/><i><font color="Green" title="E61">A9</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c5">m</font>]
 <b>and </b><br/><a NAME="E11:101"/><i><font color="Green" title="E62">A10</font></i>: 
for <font color="Olive" title="b1">m1</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>2</sub></font>[<font color="Olive" title="b1">m1</font>] holds <br/><font color="Maroon" title="c5">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">m1</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="newton.html#E8:101"><i><font color="Green" title="E60">A7</font></i></a>);<br/></i>
<a NAME="E12:101"/>
<font color="Maroon" title="c5">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0
 
<b>by </b><i><a class="txt" href="newton.html#E10:101"><i><font color="Green" title="E61">A9</font></i></a></i>;<br/>
<a NAME="E13:101"/><b>then </b><i><font color="Green" title="E63">A11</font></i>: 
<font color="Maroon" title="c5">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 
;<br/>
<b>consider </b><font color="Maroon" title="c6">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E15:101"/><i><font color="Green" title="E64">A12</font></i>: 
( <font color="Maroon" title="c4">p'</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c5">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c6">n</font> &amp; not <font color="Maroon" title="c4">p'</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c5">m</font> &amp; not <font color="Maroon" title="c4">p'</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c6">n</font> )
 <b>by </b><i><a class="txt" href="newton.html#E10:101"><i><font color="Green" title="E61">A9</font></i></a></i>;<br/>
<a NAME="E16:101"/><i><font color="Green" title="E65">A13</font></i>: 
<font color="Maroon" title="c5">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">p'</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="101_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:101_3"/><i><font color="Green" title="E66">A14</font></i>: 
<font color="Maroon" title="c5">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c4">p'</font>
 ;<br/>

<b>set </b><font color="Maroon" title="c7">m1</font> = <font color="Maroon" title="c5">m</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c4">p'</font>;<br/>
<a NAME="E2:101_3"/><i><font color="Green" title="E67">A15</font></i>: 
<font color="Maroon" title="c4">p'</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 
<b>by </b><i><a class="txt" href="newton.html#E6:101"><i><font color="Green" title="E58">A5</font></i></a>, <a class="ref" href="int_2.html#D5" title="INT_2:def.5">INT_2:def 5</a></i>;<br/>
<a NAME="E3:101_3"/><b>then </b>
<font color="Maroon" title="c5">m</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c4">p'</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">p'</font>
 
<b>by </b><i><a class="ref" href="nat_d.html#T1" title="NAT_D:th.1">NAT_D:1</a></i>;<br/>
<a NAME="E4:101_3"/><b>then </b><i><font color="Green" title="E68">A16</font></i>: 
<font color="Maroon" title="c5">m</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c4">p'</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">m</font>
 
<b>by </b><i><a class="txt" href="newton.html#E1:101_3"><i><font color="Green" title="E66">A14</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E5:101_3"/><i><font color="Green" title="E69">A17</font></i>: 
<font color="Maroon" title="c4">p'</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c4">p'</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">m</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c4">p'</font></span>)</span>
 
<b>by </b><i><a class="ref" href="nat_d.html#D3" title="NAT_D:def.3">NAT_D:def 3</a></i>;<br/>
<a NAME="E6:101_3"/><i><font color="Green" title="E70">A18</font></i>: 
<font color="Maroon" title="c5">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">p'</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">m</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c4">p'</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">m</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c4">p'</font></span>)</span>
 
<b>by </b><i><a class="txt" href="newton.html#E2:101_3"><i><font color="Green" title="E67">A15</font></i></a>, <a class="ref" href="nat_d.html#T2" title="NAT_D:th.2">NAT_D:2</a></i>;<br/>
<a NAME="E7:101_3"/><i><font color="Green" title="E71">A19</font></i>: 
<font color="Maroon" title="c4">p'</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">m</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c4">p'</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c6">n</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="101_3_1"><b>proof </b></a><div class="add">

<a NAME="E1:101_3_1"/><i><font color="Green" title="E72">A20</font></i>: 
<font color="Maroon" title="c4">p'</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c4">p'</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">m</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c4">p'</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c6">n</font></span>)</span>
 
<b>by </b><i><a class="ref" href="nat_d.html#D3" title="NAT_D:def.3">NAT_D:def 3</a></i>;<br/>
<a NAME="E2:101_3_1"/>
<font color="Maroon" title="c5">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">p'</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">m</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c4">p'</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">m</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c4">p'</font></span>)</span>
 
<b>by </b><i><a class="txt" href="newton.html#E2:101_3"><i><font color="Green" title="E67">A15</font></i></a>, <a class="ref" href="nat_d.html#T2" title="NAT_D:th.2">NAT_D:2</a></i>;<br/>
<a NAME="E3:101_3_1"/><b>then </b>
<font color="Maroon" title="c5">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c6">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">p'</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">m</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c4">p'</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c6">n</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">m</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c4">p'</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c6">n</font></span>)</span>
 
;<br/>
<b>hence </b><a NAME="E4:101_3_1"/>
<font color="Maroon" title="c4">p'</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">m</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c4">p'</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c6">n</font>
 <b>by </b><i><a class="txt" href="newton.html#E15:101"><i><font color="Green" title="E64">A12</font></i></a>, <a class="txt" href="newton.html#E1:101_3_1"><i><font color="Green" title="E72">A20</font></i></a>, <a class="ref" href="nat_d.html#T10" title="NAT_D:th.10">NAT_D:10</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:101_3"/>
not <font color="Maroon" title="c4">p'</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c5">m</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c4">p'</font>
 
<b>by </b><i><a class="txt" href="newton.html#E15:101"><i><font color="Green" title="E64">A12</font></i></a>, <a class="txt" href="newton.html#E5:101_3"><i><font color="Green" title="E69">A17</font></i></a>, <a class="txt" href="newton.html#E6:101_3"><i><font color="Green" title="E70">A18</font></i></a>, <a class="ref" href="nat_d.html#T8" title="NAT_D:th.8">NAT_D:8</a></i>;<br/>
<b>hence </b><a NAME="E9:101_3"/>
contradiction
 <b>by </b><i><a class="txt" href="newton.html#E11:101"><i><font color="Green" title="E62">A10</font></i></a>, <a class="txt" href="newton.html#E15:101"><i><font color="Green" title="E64">A12</font></i></a>, <a class="txt" href="newton.html#E4:101_3"><i><font color="Green" title="E68">A16</font></i></a>, <a class="txt" href="newton.html#E7:101_3"><i><font color="Green" title="E71">A19</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E17:101"/><i><font color="Green" title="E66">A21</font></i>: 
<font color="Maroon" title="c5">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 2
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="101_4"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:101_4"/>
<font color="Maroon" title="c5">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2
 ;<br/>

<b>then consider </b><font color="Maroon" title="c7">p1</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="E3:101_4"/><i><font color="Green" title="E67">A22</font></i>: 
( <font color="Maroon" title="c7">p1</font> is <a href="int_2.html#V1" title="INT_2:attr.1">prime</a> &amp; <font color="Maroon" title="c7">p1</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c5">m</font> )
 <b>by </b><i><a class="ref" href="int_2.html#T48" title="INT_2:th.48">INT_2:48</a></i>;<br/>
<a NAME="E4:101_4"/><i><font color="Green" title="E68">A23</font></i>: 
<font color="Maroon" title="c7">p1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 
<b>by </b><i><a class="txt" href="newton.html#E3:101_4"><i><font color="Green" title="E67">A22</font></i></a>, <a class="ref" href="int_2.html#D5" title="INT_2:def.5">INT_2:def 5</a></i>;<br/>
<a NAME="E5:101_4"/><i><font color="Green" title="E69">A24</font></i>: 
<font color="Maroon" title="c7">p1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0
 
<b>by </b><i><a class="txt" href="newton.html#E3:101_4"><i><font color="Green" title="E67">A22</font></i></a>, <a class="ref" href="int_2.html#D5" title="INT_2:def.5">INT_2:def 5</a></i>;<br/>
<a NAME="E6:101_4"/>
<font color="Maroon" title="c7">p1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">m</font>
 
<b>by </b><i><a class="txt" href="newton.html#E13:101"><i><font color="Green" title="E63">A11</font></i></a>, <a class="txt" href="newton.html#E3:101_4"><i><font color="Green" title="E67">A22</font></i></a>, <a class="ref" href="nat_d.html#T7" title="NAT_D:th.7">NAT_D:7</a></i>;<br/>
<a NAME="E7:101_4"/><b>then </b><i><font color="Green" title="E70">A25</font></i>: 
not <font color="Maroon" title="c4">p'</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">p1</font>
 
<b>by </b><i><a class="txt" href="newton.html#E16:101"><i><font color="Green" title="E65">A13</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E8:101_4"/><i><font color="Green" title="E71">A26</font></i>: 
not <font color="Maroon" title="c7">p1</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c4">p'</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="101_4_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:101_4_1"/>
<font color="Maroon" title="c7">p1</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c4">p'</font>
 ;<br/>

<a NAME="E2:101_4_1"/><b>then </b>
( <font color="Maroon" title="c7">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c7">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p'</font> )
 
<b>by </b><i><a class="txt" href="newton.html#E6:101"><i><font color="Green" title="E58">A5</font></i></a>, <a class="ref" href="int_2.html#D5" title="INT_2:def.5">INT_2:def 5</a></i>;<br/>
<b>hence </b><a NAME="E3:101_4_1"/>
contradiction
 <b>by </b><i><a class="txt" href="newton.html#E13:101"><i><font color="Green" title="E63">A11</font></i></a>, <a class="txt" href="newton.html#E16:101"><i><font color="Green" title="E65">A13</font></i></a>, <a class="txt" href="newton.html#E3:101_4"><i><font color="Green" title="E67">A22</font></i></a>, <a class="ref" href="int_2.html#D5" title="INT_2:def.5">INT_2:def 5</a>, <a class="ref" href="nat_d.html#T7" title="NAT_D:th.7">NAT_D:7</a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c8">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E10:101_4"/><i><font color="Green" title="E72">A27</font></i>: 
<font color="Maroon" title="c5">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c6">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p'</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c8">k</font>
 <b>by </b><i><a class="txt" href="newton.html#E15:101"><i><font color="Green" title="E64">A12</font></i></a>, <a class="ref" href="nat_d.html#D3" title="NAT_D:def.3">NAT_D:def 3</a></i>;<br/>
<a NAME="E11:101_4"/>
<font color="Maroon" title="c7">p1</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c4">p'</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c8">k</font>
 
<b>by </b><i><a class="txt" href="newton.html#E3:101_4"><i><font color="Green" title="E67">A22</font></i></a>, <a class="txt" href="newton.html#E10:101_4"><i><font color="Green" title="E72">A27</font></i></a>, <a class="ref" href="nat_d.html#T9" title="NAT_D:th.9">NAT_D:9</a></i>;<br/>
<a NAME="E12:101_4"/><b>then </b>
<font color="Maroon" title="c7">p1</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c8">k</font>
 
<b>by </b><i><a class="txt" href="newton.html#E7:101"><i><font color="Green" title="E59">A6</font></i></a>, <a class="txt" href="newton.html#E3:101_4"><i><font color="Green" title="E67">A22</font></i></a>, <a class="txt" href="newton.html#E7:101_4"><i><font color="Green" title="E70">A25</font></i></a>, <a class="txt" href="newton.html#E8:101_4"><i><font color="Green" title="E71">A26</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c9">k1</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E14:101_4"/><i><font color="Green" title="E73">A28</font></i>: 
<font color="Maroon" title="c8">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">p1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c9">k1</font>
 <b>by </b><i><a class="ref" href="nat_d.html#D3" title="NAT_D:def.3">NAT_D:def 3</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c10">m1</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E16:101_4"/><i><font color="Green" title="E74">A29</font></i>: 
<font color="Maroon" title="c5">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">p1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c10">m1</font>
 <b>by </b><i><a class="txt" href="newton.html#E3:101_4"><i><font color="Green" title="E67">A22</font></i></a>, <a class="ref" href="nat_d.html#D3" title="NAT_D:def.3">NAT_D:def 3</a></i>;<br/>
<a NAME="E17:101_4"/><i><font color="Green" title="E75">A30</font></i>: 
<font color="Maroon" title="c10">m1</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="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E18:101_4"/><i><font color="Green" title="E76">A31</font></i>: 
<font color="Maroon" title="c10">m1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="101_4_2"><b>proof </b></a><div class="add">

<a NAME="E1:101_4_2"/>
<font color="Maroon" title="c10">m1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0
 
<b>by </b><i><a class="txt" href="newton.html#E10:101"><i><font color="Green" title="E61">A9</font></i></a>, <a class="txt" href="newton.html#E16:101_4"><i><font color="Green" title="E74">A29</font></i></a></i>;<br/>
<b>hence </b><a NAME="E2:101_4_2"/>
<font color="Maroon" title="c10">m1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 ;<br/>


</div><b>end;</b></div>
<a NAME="E19:101_4"/><i><font color="Green" title="E77">A32</font></i>: 
<font color="Maroon" title="c10">m1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">m</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="101_4_3"><b>proof </b></a><div class="add">

<a NAME="E1:101_4_3"/>
<font color="Maroon" title="c10">m1</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c5">m</font>
 
<b>by </b><i><a class="txt" href="newton.html#E16:101_4"><i><font color="Green" title="E74">A29</font></i></a>, <a class="ref" href="nat_d.html#D3" title="NAT_D:def.3">NAT_D:def 3</a></i>;<br/>
<a NAME="E2:101_4_3"/><b>then </b><i><font color="Green" title="E78">A33</font></i>: 
<font color="Maroon" title="c10">m1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">m</font>
 
<b>by </b><i><a class="txt" href="newton.html#E13:101"><i><font color="Green" title="E63">A11</font></i></a>, <a class="ref" href="nat_d.html#T7" title="NAT_D:th.7">NAT_D:7</a></i>;<br/>
<a NAME="E3:101_4_3"/>
<font color="Maroon" title="c10">m1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">m</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="101_4_3_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:101_4_3_1"/>
<font color="Maroon" title="c10">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">m</font>
 ;<br/>

<a NAME="E2:101_4_3_1"/><b>then </b>
<font color="Maroon" title="c5">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c10">m1</font>
 
;<br/>
<b>hence </b><a NAME="E3:101_4_3_1"/>
contradiction
 <b>by </b><i><a class="txt" href="newton.html#E4:101_4"><i><font color="Green" title="E68">A23</font></i></a>, <a class="txt" href="newton.html#E16:101_4"><i><font color="Green" title="E74">A29</font></i></a>, <a class="txt" href="newton.html#E18:101_4"><i><font color="Green" title="E76">A31</font></i></a>, <a class="ref" href="xreal_1.html#T70" title="XREAL_1:th.70">XREAL_1:70</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:101_4_3"/>
<font color="Maroon" title="c10">m1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">m</font>
 <b>by </b><i><a class="txt" href="newton.html#E2:101_4_3"><i><font color="Green" title="E78">A33</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E20:101_4"/><i><font color="Green" title="E78">A34</font></i>: 
<font color="Maroon" title="c4">p'</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c10">m1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c6">n</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="101_4_4"><b>proof </b></a><div class="add">

<a NAME="E1:101_4_4"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">p'</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c9">k1</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c7">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">p1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">m1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c6">n</font></span>)</span>
 
<b>by </b><i><a class="txt" href="newton.html#E10:101_4"><i><font color="Green" title="E72">A27</font></i></a>, <a class="txt" href="newton.html#E14:101_4"><i><font color="Green" title="E73">A28</font></i></a>, <a class="txt" href="newton.html#E16:101_4"><i><font color="Green" title="E74">A29</font></i></a></i>;<br/>
<a NAME="E2:101_4_4"/><b>then </b>
<font color="Maroon" title="c10">m1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c6">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p'</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c9">k1</font>
 
<b>by </b><i><a class="txt" href="newton.html#E5:101_4"><i><font color="Green" title="E69">A24</font></i></a>, <a class="ref" href="xcmplx_1.html#T5" title="XCMPLX_1:th.5">XCMPLX_1:5</a></i>;<br/>
<b>hence </b><a NAME="E3:101_4_4"/>
<font color="Maroon" title="c4">p'</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c10">m1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c6">n</font>
 <b>by </b><i><a class="ref" href="nat_d.html#D3" title="NAT_D:def.3">NAT_D:def 3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E21:101_4"/>
not <font color="Maroon" title="c4">p'</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c10">m1</font>
 
<b>by </b><i><a class="txt" href="newton.html#E15:101"><i><font color="Green" title="E64">A12</font></i></a>, <a class="txt" href="newton.html#E16:101_4"><i><font color="Green" title="E74">A29</font></i></a>, <a class="ref" href="nat_d.html#T9" title="NAT_D:th.9">NAT_D:9</a></i>;<br/>
<b>hence </b><a NAME="E22:101_4"/>
contradiction
 <b>by </b><i><a class="txt" href="newton.html#E17:101_4"><i><font color="Green" title="E75">A30</font></i></a>, <a class="txt" href="newton.html#E11:101"><i><font color="Green" title="E62">A10</font></i></a>, <a class="txt" href="newton.html#E15:101"><i><font color="Green" title="E64">A12</font></i></a>, <a class="txt" href="newton.html#E19:101_4"><i><font color="Green" title="E77">A32</font></i></a>, <a class="txt" href="newton.html#E20:101_4"><i><font color="Green" title="E78">A34</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E18:101"/>
<font color="Maroon" title="c5">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="101_5"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:101_5"/>
<font color="Maroon" title="c5">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 2
 ;<br/>

<a NAME="E2:101_5"/><b>then </b><i><font color="Green" title="E67">A35</font></i>: 
( <font color="Maroon" title="c5">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c5">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 2 )
 
;<br/>
<a NAME="E3:101_5"/>
<font color="Maroon" title="c5">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<b>by </b><i><a class="txt" href="newton.html#E13:101"><i><font color="Green" title="E63">A11</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E4:101_5"/><b>then </b>
<font color="Maroon" title="c5">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="txt" href="newton.html#E2:101_5"><i><font color="Green" title="E67">A35</font></i></a>, <a class="ref" href="nat_1.html#T9" title="NAT_1:th.9">NAT_1:9</a></i>;<br/>
<b>hence </b><a NAME="E5:101_5"/>
contradiction
 <b>by </b><i><a class="txt" href="newton.html#E15:101"><i><font color="Green" title="E64">A12</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E19:101"/>
contradiction
 <b>by </b><i><a class="txt" href="newton.html#E17:101"><i><font color="Green" title="E66">A21</font></i></a></i>;<br/>


</div>
