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

<b>let </b><font color="Maroon">i</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:104"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">i</font> is <a href="int_2.html#V1">prime</a>
 ;<br/>

<b>given </b><font color="Maroon">m</font> being   <a href="nat_1.html#NM1">Nat</a>, <font color="Maroon">n</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><a NAME="E2:104"/><i><font color="Green">E42</font></i>: 
( <font color="Maroon">i</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">m</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">n</font> &amp; not <font color="Maroon">i</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">m</font> &amp; not <font color="Maroon">i</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">n</font> )
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] <b>means </b>( <font color="Maroon">a<sub>1</sub></font> is <a href="int_2.html#V1">prime</a> &amp;  ex <font color="Olive">m</font>, <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( <font color="Maroon">a<sub>1</sub></font> <a href="nat_1.html#R1">divides</a> <font color="Olive">m</font> <a href="xcmplx_0.html#K3">*</a> <font color="Olive">n</font> &amp; not <font color="Maroon">a<sub>1</sub></font> <a href="nat_1.html#R1">divides</a> <font color="Olive">m</font> &amp; not <font color="Maroon">a<sub>1</sub></font> <a href="nat_1.html#R1">divides</a> <font color="Olive">n</font> ) );<br/>
<a NAME="E3:104"/><i><font color="Green">E45</font></i>: 
 ex <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">i</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="104_1"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon">i</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E2:104_1"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">i</font>]
 <b>by </b><i><a class="ref" href="newton.html#T10" target="_self">Th10</a>, <a class="ref" href="newton.html#T11" target="_self">Th11</a></i>;<br/>
<b>reconsider </b><font color="Maroon">i</font> = <font color="Maroon">i</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/>
<b>take </b>
<font color="Maroon">i</font>
;<br/>

<b>thus </b><a NAME="E4:104_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">i</font>]
 <b>by </b><i><a class="ref" href="newton.html#T15" target="_self">Th15</a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">p'</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E5:104"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">p'</font>]
 <b>and </b><br/><a NAME="E6:104"/><i><font color="Green">E48</font></i>: 
for <font color="Olive">p1</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">p1</font>] holds <br/><font color="Maroon">p'</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">p1</font>
 <b>from </b><i><a class="ref" href="nat_1.html#S5">NAT_1:sch 5</a>();<br/></i>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] <b>means </b> ex <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( <font color="Maroon">p'</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">a<sub>1</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Olive">n</font> &amp; not <font color="Maroon">p'</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">a<sub>1</sub></font> &amp; not <font color="Maroon">p'</font> <a href="nat_1.html#R1">divides</a> <font color="Olive">n</font> );<br/>
<a NAME="E7:104"/><i><font color="Green">E49</font></i>: 
 ex <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">i</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="104_2"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon">i</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E2:104_2"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">i</font>]
 <b>by </b><i><a class="ref" href="newton.html#T15" target="_self">Th15</a></i>;<br/>
<b>reconsider </b><font color="Maroon">i</font> = <font color="Maroon">i</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/>
<b>take </b>
<font color="Maroon">i</font>
;<br/>

<b>thus </b><a NAME="E4:104_2"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">i</font>]
 <b>by </b><i><a class="ref" href="newton.html#T19" target="_self">Th19</a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">m</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E9:104"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">m</font>]
 <b>and </b><br/><a NAME="E10:104"/><i><font color="Green">E80</font></i>: 
for <font color="Olive">m1</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">m1</font>] holds <br/><font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m1</font>
 <b>from </b><i><a class="ref" href="nat_1.html#S5">NAT_1:sch 5</a>(<a class="ref" href="newton.html#T18" target="_self">Th18</a>);<br/></i>
<a NAME="E11:104"/>
<font color="Maroon">m</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i><a class="ref" href="newton.html#T19" target="_self">Th19</a></i>;<br/>
<a NAME="E12:104"/><b>then </b><i><font color="Green">E83</font></i>: 
<font color="Maroon">m</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
;<br/>
<b>consider </b><font color="Maroon">n</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E14:104"/><i><font color="Green">E84</font></i>: 
( <font color="Maroon">p'</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">m</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">n</font> &amp; not <font color="Maroon">p'</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">m</font> &amp; not <font color="Maroon">p'</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">n</font> )
 <b>by </b><i><a class="ref" href="newton.html#T19" target="_self">Th19</a></i>;<br/>
<a NAME="E15:104"/><i><font color="Green">E85</font></i>: 
<font color="Maroon">m</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">p'</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="104_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:104_3"/><i><font color="Green">E86</font></i>: 
<font color="Maroon">m</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">p'</font>
 ;<br/>

<b>set </b><font color="Maroon">m1</font> = <font color="Maroon">m</font> <a href="nat_1.html#K6">mod</a> <font color="Maroon">p'</font>;<br/>
<a NAME="E2:104_3"/><i><font color="Green">E87</font></i>: 
<font color="Maroon">p'</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="ref" href="newton.html#T15" target="_self">Th15</a>, <a class="ref" href="newton.html#T9" target="_self">Th9</a></i>;<br/>
<a NAME="E3:104_3"/><b>then </b>
<font color="Maroon">m</font> <a href="nat_1.html#K6">mod</a> <font color="Maroon">p'</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">p'</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T46">NAT_1:46</a></i>;<br/>
<a NAME="E4:104_3"/><b>then </b><i><font color="Green">E88</font></i>: 
<font color="Maroon">m</font> <a href="nat_1.html#K6">mod</a> <font color="Maroon">p'</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m</font>
 
<b>by </b><i><a class="ref" href="newton.html#D3" target="_self">Def3</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E5:104_3"/><i><font color="Green">E103</font></i>: 
<font color="Maroon">p'</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">p'</font> <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K5">div</a> <font color="Maroon">p'</font></span>)</span>
 
<b>by </b><i><a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/>
<a NAME="E6:104_3"/><i><font color="Green">E104</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">p'</font> <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K5">div</a> <font color="Maroon">p'</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K6">mod</a> <font color="Maroon">p'</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="nat_1.html#T47">NAT_1:47</a></i>;<br/>
<a NAME="E7:104_3"/><i><font color="Green">E105</font></i>: 
<font color="Maroon">p'</font> <a href="nat_1.html#R1">divides</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K6">mod</a> <font color="Maroon">p'</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">n</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="104_3_1"><b>proof </b></a><div class="add">

<a NAME="E1:104_3_1"/><i><font color="Green">E106</font></i>: 
<font color="Maroon">p'</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">p'</font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K5">div</a> <font color="Maroon">p'</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">n</font></span>)</span>
 
<b>by </b><i><a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/>
<a NAME="E2:104_3_1"/>
<font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">p'</font> <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K5">div</a> <font color="Maroon">p'</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K6">mod</a> <font color="Maroon">p'</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="nat_1.html#T47">NAT_1:47</a></i>;<br/>
<a NAME="E3:104_3_1"/><b>then </b>
<font color="Maroon">m</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p'</font> <a href="nat_1.html#K2">*</a> <span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K5">div</a> <font color="Maroon">p'</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">n</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K6">mod</a> <font color="Maroon">p'</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">n</font></span>)</span>
 
;<br/>
<b>hence </b><a NAME="E4:104_3_1"/>
<font color="Maroon">p'</font> <a href="nat_1.html#R1">divides</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K6">mod</a> <font color="Maroon">p'</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="newton.html#T23" target="_self">Th23</a>, <a class="ref" href="newton.html#T32" target="_self">Th32</a>, <a class="ref" href="nat_1.html#T57">NAT_1:57</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:104_3"/>
not <font color="Maroon">p'</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">m</font> <a href="nat_1.html#K6">mod</a> <font color="Maroon">p'</font>
 
<b>by </b><i><a class="ref" href="newton.html#T23" target="_self">Th23</a>, <a class="ref" href="newton.html#T29" target="_self">Th29</a>, <a class="ref" href="newton.html#T30" target="_self">Th30</a>, <a class="ref" href="nat_1.html#T55">NAT_1:55</a></i>;<br/>
<b>hence </b><a NAME="E9:104_3"/>
contradiction
 <b>by </b><i><a class="ref" href="newton.html#T21" target="_self">Th21</a>, <a class="ref" href="newton.html#T23" target="_self">Th23</a>, <a class="ref" href="newton.html#T27" target="_self">Th27</a>, <a class="ref" href="newton.html#T31" target="_self">Th31</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E16:104"/><i><font color="Green">E107</font></i>: 
<font color="Maroon">m</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 2
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="104_4"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:104_4"/>
<font color="Maroon">m</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2
 ;<br/>

<b>then consider </b><font color="Maroon">p1</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:104_4"/><i><font color="Green">E108</font></i>: 
( <font color="Maroon">p1</font> is <a href="int_2.html#V1">prime</a> &amp; <font color="Maroon">p1</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">m</font> )
 <b>by </b><i><a class="ref" href="int_2.html#T48">INT_2:48</a></i>;<br/>
<a NAME="E4:104_4"/><i><font color="Green">E109</font></i>: 
<font color="Maroon">p1</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 
<b>by </b><i>, <a class="ref" href="int_2.html#D5">INT_2:def 5</a></i>;<br/>
<a NAME="E5:104_4"/><i><font color="Green">E110</font></i>: 
<font color="Maroon">p1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i>, <a class="ref" href="int_2.html#D5">INT_2:def 5</a></i>;<br/>
<a NAME="E6:104_4"/>
<font color="Maroon">p1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 
<b>by </b><i><a class="ref" href="newton.html#T22" target="_self">Th22</a>, , <a class="ref" href="nat_1.html#T54">NAT_1:54</a></i>;<br/>
<a NAME="E7:104_4"/><b>then </b><i><font color="Green">E111</font></i>: 
not <font color="Maroon">p'</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p1</font>
 
<b>by </b><i><a class="ref" href="newton.html#T25" target="_self">Th25</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E8:104_4"/><i><font color="Green">E112</font></i>: 
not <font color="Maroon">p1</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">p'</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="104_4_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:104_4_1"/>
<font color="Maroon">p1</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">p'</font>
 ;<br/>

<a NAME="E2:104_4_1"/><b>then </b>
( <font color="Maroon">p1</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">p1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p'</font> )
 
<b>by </b><i><a class="ref" href="newton.html#T15" target="_self">Th15</a>, <a class="ref" href="int_2.html#D5">INT_2:def 5</a></i>;<br/>
<b>hence </b><a NAME="E3:104_4_1"/>
contradiction
 <b>by </b><i><a class="ref" href="newton.html#T22" target="_self">Th22</a>, <a class="ref" href="newton.html#T25" target="_self">Th25</a>, , <a class="ref" href="int_2.html#D5">INT_2:def 5</a>, <a class="ref" href="nat_1.html#T54">NAT_1:54</a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">k</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E10:104_4"/><i><font color="Green">E113</font></i>: 
<font color="Maroon">m</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p'</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="newton.html#T23" target="_self">Th23</a>, <a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/>
<a NAME="E11:104_4"/>
<font color="Maroon">p1</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">p'</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">k</font>
 
<b>by </b><i>, <a class="ref" href="newton.html#T32" target="_self">Th32</a>, <a class="ref" href="nat_1.html#T56">NAT_1:56</a></i>;<br/>
<a NAME="E12:104_4"/><b>then </b>
<font color="Maroon">p1</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">k</font>
 
<b>by </b><i><a class="ref" href="newton.html#T16" target="_self">Th16</a>, , <a class="ref" href="newton.html#T30" target="_self">Th30</a>, <a class="ref" href="newton.html#T31" target="_self">Th31</a></i>;<br/>
<b>then consider </b><font color="Maroon">k1</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E14:104_4"/><i><font color="Green">E114</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p1</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">k1</font>
 <b>by </b><i><a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/>
<b>consider </b><font color="Maroon">m1</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E16:104_4"/><i><font color="Green">E115</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p1</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">m1</font>
 <b>by </b><i>, <a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/>
<a NAME="E17:104_4"/><i><font color="Green">E116</font></i>: 
<font color="Maroon">m1</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E18:104_4"/><i><font color="Green">E117</font></i>: 
<font color="Maroon">m1</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="104_4_2"><b>proof </b></a><div class="add">

<a NAME="E1:104_4_2"/>
<font color="Maroon">m1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i><a class="ref" href="newton.html#T19" target="_self">Th19</a>, </i>;<br/>
<b>hence </b><a NAME="E2:104_4_2"/>
<font color="Maroon">m1</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/>


</div><b>end;</b></div>
<a NAME="E19:104_4"/><i><font color="Green">E118</font></i>: 
<font color="Maroon">m1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="104_4_3"><b>proof </b></a><div class="add">

<a NAME="E1:104_4_3"/>
<font color="Maroon">m1</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">m</font>
 
<b>by </b><i>, <a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/>
<a NAME="E2:104_4_3"/><b>then </b><i><font color="Green">E119</font></i>: 
<font color="Maroon">m1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 
<b>by </b><i><a class="ref" href="newton.html#T22" target="_self">Th22</a>, <a class="ref" href="nat_1.html#T54">NAT_1:54</a></i>;<br/>
<a NAME="E3:104_4_3"/>
<font color="Maroon">m1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">m</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="104_4_3_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:104_4_3_1"/>
<font color="Maroon">m1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font>
 ;<br/>

<a NAME="E2:104_4_3_1"/><b>then </b>
<font color="Maroon">m</font> <a href="hidden.html#R1">=</a> 1 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">m1</font>
 
;<br/>
<b>hence </b><a NAME="E3:104_4_3_1"/>
contradiction
 <b>by </b><i><a class="ref" href="newton.html#T27" target="_self">Th27</a>, , <a class="ref" href="newton.html#T33" target="_self">Th33</a>, <a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:104_4_3"/>
<font color="Maroon">m1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="txt" href="newton.html#E12:104"><i><font color="Green">E83</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E20:104_4"/><i><font color="Green">E120</font></i>: 
<font color="Maroon">p'</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">m1</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">n</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="104_4_4"><b>proof </b></a><div class="add">

<a NAME="E1:104_4_4"/>
<font color="Maroon">m</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p1</font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">m1</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">n</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E2:104_4_4"/><b>then </b>
<font color="Maroon">p'</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p1</font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">m1</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">n</font></span>)</span>
 
<b>by </b><i><a class="ref" href="newton.html#T32" target="_self">Th32</a></i>;<br/>
<a NAME="E3:104_4_4"/><b>then </b>
<font color="Maroon">p'</font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">k1</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">p1</font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">m1</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">n</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E4:104_4_4"/><b>then </b>
<font color="Maroon">p'</font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">k1</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">p1</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">p1</font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">m1</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">n</font></span>)</span>
 
;<br/>
<a NAME="E5:104_4_4"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">p'</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">k1</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">p1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p1</font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">m1</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">n</font></span>)</span>
 
;<br/>
<a NAME="E6:104_4_4"/><b>then </b>
<font color="Maroon">m1</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p'</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">k1</font>
 
<b>by </b><i><a class="ref" href="newton.html#T29" target="_self">Th29</a>, <a class="ref" href="xcmplx_1.html#T5">XCMPLX_1:5</a></i>;<br/>
<b>hence </b><a NAME="E7:104_4_4"/>
<font color="Maroon">p'</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">m1</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E21:104_4"/>
not <font color="Maroon">p'</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">m1</font>
 
<b>by </b><i><a class="ref" href="newton.html#T23" target="_self">Th23</a>, , <a class="ref" href="nat_1.html#T56">NAT_1:56</a></i>;<br/>
<b>hence </b><a NAME="E22:104_4"/>
contradiction
 <b>by </b><i><a class="txt" href="newton.html#E10:104"><i><font color="Green">E80</font></i></a>, <a class="ref" href="newton.html#T21" target="_self">Th21</a>, <a class="ref" href="newton.html#T23" target="_self">Th23</a>, <a class="ref" href="newton.html#T35" target="_self">Th35</a>, <a class="txt" href="newton.html#E12:104"><i><font color="Green">E83</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E17:104"/>
<font color="Maroon">m</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="104_5"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:104_5"/>
<font color="Maroon">m</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 2
 ;<br/>

<a NAME="E2:104_5"/><b>then </b><i><font color="Green">E121</font></i>: 
( <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">m</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 2 )
 
;<br/>
<a NAME="E3:104_5"/>
<font color="Maroon">m</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="newton.html#T22" target="_self">Th22</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E4:104_5"/><b>then </b>
<font color="Maroon">m</font> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i>, <a class="ref" href="nat_1.html#T27">NAT_1:27</a></i>;<br/>
<b>hence </b><a NAME="E5:104_5"/>
contradiction
 <b>by </b><i><a class="ref" href="newton.html#T23" target="_self">Th23</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E18:104"/>
contradiction
 <b>by </b><i><a class="ref" href="newton.html#D3" target="_self">Def3</a></i>;<br/>


</div>
