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

<b>let </b><font color="Maroon">m</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>



<b>assume </b><a NAME="E1:85"/><i><font color="Green">E41</font></i>: 
( <font color="Maroon">m</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 or <font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>( <font color="Maroon">a<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp;  ex <font color="Olive">i</font>, <font color="Olive">i1</font> being  <a href="int_1.html#NM1">Integer</a> st <span class="p1">(<span class="default"><font color="Olive">i</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">m</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Olive">i1</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> );<br/>
<a NAME="E2:85"/><i><font color="Green">E42</font></i>: 
 ex <font color="Olive">p</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">p</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="85_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="85_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:85_1_1"/>
( <font color="Maroon">m</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 or <font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 </span><b>by </b><i><a class="txt" href="newton.html#E6:85"><i><font color="Green">E48</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="85_1_1_1"><b>suppose </b></a><a NAME="E1:85_1_1_1"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">m</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:85_1_1_1"/>
 ex <font color="Olive">i</font>, <font color="Olive">i1</font> being  <a href="int_1.html#NM1">Integer</a> st <span class="p1">(<span class="default"><font color="Olive">i</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">m</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Olive">i1</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="85_1_1_1_1"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon">i1</font> = 0;<br/>
<a NAME="E1:85_1_1_1_1"/>
<span class="p1">(<span class="default">1 <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default">0 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font>
 
;<br/>
<b>hence </b><a NAME="E2:85_1_1_1_1"/>
 ex <font color="Olive">i</font>, <font color="Olive">i1</font> being  <a href="int_1.html#NM1">Integer</a> st <span class="p1">(<span class="default"><font color="Olive">i</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">m</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Olive">i1</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font>
 ;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E3:85_1_1_1"/>
 ex <font color="Olive">p</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">p</font>]
 <b>by </b><i><a class="txt" href="newton.html#E9:85"><i><font color="Green">E50</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="85_1_1_2"><b>suppose </b></a><a NAME="E1:85_1_1_2"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:85_1_1_2"/>
 ex <font color="Olive">i</font>, <font color="Olive">i1</font> being  <a href="int_1.html#NM1">Integer</a> st <span class="p1">(<span class="default"><font color="Olive">i</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">m</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Olive">i1</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="85_1_1_2_1"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon">i</font> = 0;<br/>
<a NAME="E1:85_1_1_2_1"/>
<span class="p1">(<span class="default">0 <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 
;<br/>
<b>hence </b><a NAME="E2:85_1_1_2_1"/>
 ex <font color="Olive">i</font>, <font color="Olive">i1</font> being  <a href="int_1.html#NM1">Integer</a> st <span class="p1">(<span class="default"><font color="Olive">i</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">m</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Olive">i1</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 ;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E3:85_1_1_2"/>
 ex <font color="Olive">p</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">p</font>]
 <b>by </b><i><a class="txt" href="newton.html#E9:85"><i><font color="Green">E50</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E3:85"/>
 ex <font color="Olive">p</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">p</font>] &amp; ( for <font color="Olive">n</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">n</font>] holds <br/><font color="Olive">p</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">n</font> ) )
 
<b>from </b><i><a class="ref" href="nat_1.html#S5">NAT_1:sch 5</a>(<a class="txt" href="newton.html#E7:85"><i><font color="Green">E49</font></i></a>);<br/></i>
<b>then 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="E5:85"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">p</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>and </b><br/><a NAME="E6:85"/><i><font color="Green">E48</font></i>: 
 ex <font color="Olive">i</font>, <font color="Olive">i0</font> being  <a href="int_1.html#NM1">Integer</a> st <span class="p1">(<span class="default"><font color="Olive">i</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">m</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Olive">i0</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font>
 <b>and </b><br/><a NAME="E7:85"/><i><font color="Green">E49</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp;  ex <font color="Olive">i1</font>, <font color="Olive">i2</font> being  <a href="int_1.html#NM1">Integer</a> st <span class="p1">(<span class="default"><font color="Olive">i1</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">m</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Olive">i2</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">k</font> holds <br/><font color="Maroon">p</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font>
 ;<br/>
<b>consider </b><font color="Maroon">i</font> being   <a href="int_1.html#NM1">Integer</a>, <font color="Maroon">i0</font> being   <a href="int_1.html#NM1">Integer</a><b> such that </b><br/><a NAME="E9:85"/><i><font color="Green">E50</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">m</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">i0</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font>
 <b>by </b><i><a class="txt" href="newton.html#E10:85"><i><font color="Green">E51</font></i></a></i>;<br/>
<a NAME="E10:85"/><i><font color="Green">E51</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st  ex <font color="Olive">i1</font>, <font color="Olive">i2</font> being  <a href="int_1.html#NM1">Integer</a> st <span class="p1">(<span class="default"><font color="Olive">i1</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">m</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Olive">i2</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">k</font> holds <br/><font color="Maroon">p</font> <a href="nat_1.html#R1">divides</a> <font color="Olive">k</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="85_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>given </b><font color="Maroon">i1</font> being   <a href="int_1.html#NM1">Integer</a>, <font color="Maroon">i2</font> being   <a href="int_1.html#NM1">Integer</a><b> such that </b><a NAME="E1:85_2"/><i><font color="Green">E80</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">m</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">i2</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font>
 ;<br/>

<b>consider </b><font color="Maroon">l</font> being   <a href="nat_1.html#NM1">Nat</a>, <font color="Maroon">s</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:85_2"/><i><font color="Green">E83</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">l</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">s</font>
 <b>and </b><br/><a NAME="E4:85_2"/><i><font color="Green">E84</font></i>: 
<font color="Maroon">s</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">p</font>
 <b>by </b><i><a class="txt" href="newton.html#E9:85"><i><font color="Green">E50</font></i></a>, <a class="ref" href="nat_1.html#T42">NAT_1:42</a></i>;<br/>
<b>reconsider </b><font color="Maroon">l</font> = <font color="Maroon">l</font>, <font color="Maroon">s</font> = <font color="Maroon">s</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/>
<a NAME="E6:85_2"/>
<font color="Maroon">s</font> <a href="hidden.html#R1">=</a> 0
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="85_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:85_2_1"/>
<font color="Maroon">s</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/>

<a NAME="E2:85_2_1"/><b>then </b><i><font color="Green">E85</font></i>: 
<font color="Maroon">s</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
;<br/>
<a NAME="E3:85_2_1"/>
 ex <font color="Olive">i3</font>, <font color="Olive">i4</font> being  <a href="int_1.html#NM1">Integer</a> st <span class="p1">(<span class="default"><font color="Olive">i3</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">m</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Olive">i4</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="85_2_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:85_2_1_1"/>
<font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">l</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E2:85_2_1_1"/><b>then </b>
<font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i1</font> <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">l</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">m</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">i2</font> <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">i0</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">l</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">n</font></span>)</span>
 
<b>by </b><i><a class="ref" href="newton.html#T7" target="_self">Th7</a>, </i>;<br/>
<b>hence </b><a NAME="E3:85_2_1_1"/>
 ex <font color="Olive">i3</font>, <font color="Olive">i4</font> being  <a href="int_1.html#NM1">Integer</a> st <span class="p1">(<span class="default"><font color="Olive">i3</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">m</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Olive">i4</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:85_2_1"/>
contradiction
 <b>by </b><i><a class="ref" href="newton.html#T6" target="_self">Th6</a>, , </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:85_2"/>
<font color="Maroon">p</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">k</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="E11:85"/>
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">n</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="85_3"><b>proof </b></a><div class="add">

<a NAME="E1:85_3"/><i><font color="Green">E86</font></i>: 
 ex <font color="Olive">i</font>, <font color="Olive">i1</font> being  <a href="int_1.html#NM1">Integer</a> st <span class="p1">(<span class="default"><font color="Olive">i</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">m</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Olive">i1</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="85_3_1"><b>proof </b></a><div class="add">

<a NAME="E1:85_3_1"/>
<span class="p1">(<span class="default">1 <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default">0 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font>
 
;<br/>
<b>hence </b><a NAME="E2:85_3_1"/>
 ex <font color="Olive">i</font>, <font color="Olive">i1</font> being  <a href="int_1.html#NM1">Integer</a> st <span class="p1">(<span class="default"><font color="Olive">i</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">m</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Olive">i1</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E2:85_3"/><i><font color="Green">E87</font></i>: 
 ex <font color="Olive">i</font>, <font color="Olive">i1</font> being  <a href="int_1.html#NM1">Integer</a> st <span class="p1">(<span class="default"><font color="Olive">i</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">m</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Olive">i1</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="85_3_2"><b>proof </b></a><div class="add">

<a NAME="E1:85_3_2"/>
<span class="p1">(<span class="default">0 <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 
;<br/>
<b>hence </b><a NAME="E2:85_3_2"/>
 ex <font color="Olive">i</font>, <font color="Olive">i1</font> being  <a href="int_1.html#NM1">Integer</a> st <span class="p1">(<span class="default"><font color="Olive">i</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">m</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Olive">i1</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E3:85_3"/><i><font color="Green">E88</font></i>: 
<font color="Maroon">p</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">m</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E4:85_3"/><i><font color="Green">E103</font></i>: 
<font color="Maroon">p</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">n</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E5:85_3"/>
for <font color="Olive">s</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">s</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">m</font> &amp; <font color="Olive">s</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">n</font> holds <br/><font color="Olive">s</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">p</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="85_3_3"><b>proof </b></a><div class="add">

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

<b>assume </b><b>that </b><br/><a NAME="E1:85_3_3"/><i><font color="Green">E104</font></i>: 
<font color="Maroon">s</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">m</font>
 <b>and </b><br/><a NAME="E2:85_3_3"/><i><font color="Green">E105</font></i>: 
<font color="Maroon">s</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">n</font>
 ;<br/>

<b>consider </b><font color="Maroon">l</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E4:85_3_3"/><i><font color="Green">E106</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">l</font>
 <b>by </b><i><a class="ref" href="newton.html#T9" target="_self">Th9</a>, <a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/>
<b>consider </b><font color="Maroon">r</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E6:85_3_3"/><i><font color="Green">E107</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="newton.html#T10" target="_self">Th10</a>, <a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">p'</font> = <font color="Maroon">p</font>, <font color="Maroon">s'</font> = <font color="Maroon">s</font> as   <a href="int_1.html#NM1">Integer</a> ;<br/>
<a NAME="E8:85_3_3"/><i><font color="Green">E108</font></i>: 
<font color="Maroon">s</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="E9:85_3_3"/>
 ex <font color="Olive">i4</font> being  <a href="int_1.html#NM1">Integer</a> st <font color="Maroon">p'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s'</font> <a href="xcmplx_0.html#K3">*</a> <font color="Olive">i4</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="85_3_3_1"><b>proof </b></a><div class="add">

<b>take </b>
<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">l</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">i0</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r</font></span>)</span>
;<br/>

<b>thus </b><a NAME="E1:85_3_3_1"/>
<font color="Maroon">p'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s'</font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">l</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">i0</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="newton.html#T7" target="_self">Th7</a>, <a class="ref" href="newton.html#T11" target="_self">Th11</a>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E10:85_3_3"/><b>then </b><i><font color="Green">E109</font></i>: 
<font color="Maroon">s'</font> <a href="int_1.html#R2">divides</a> <font color="Maroon">p'</font>
 
<b>by </b><i><a class="ref" href="int_1.html#D9">INT_1:def 9</a></i>;<br/>
<a NAME="E11:85_3_3"/><i><font color="Green">E110</font></i>: 
for <font color="Olive">z</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds   <a href="int_2.html#K1">abs</a> <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <font color="Olive">z</font>
 
<b>by </b><i><a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/>
<a NAME="E12:85_3_3"/><b>then </b><i><font color="Green">E111</font></i>: 
 <a href="int_2.html#K1">abs</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font>
 
<b>by </b><i><a class="ref" href="newton.html#T15" target="_self">Th15</a></i>;<br/>
<a NAME="E13:85_3_3"/>
 <a href="int_2.html#K1">abs</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font>
 
<b>by </b><i><a class="ref" href="newton.html#T18" target="_self">Th18</a></i>;<br/>
<b>hence </b><a NAME="E14:85_3_3"/>
<font color="Maroon">s</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">p</font>
 <b>by </b><i><a class="ref" href="newton.html#T16" target="_self">Th16</a>, <a class="ref" href="newton.html#T19" target="_self">Th19</a>, <a class="ref" href="int_2.html#T21">INT_2:21</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:85_3"/>
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">n</font>
 <b>by </b><i>, , <a class="ref" href="nat_1.html#D5">NAT_1:def 5</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E12:85"/>
 ex <font color="Olive">i</font>, <font color="Olive">i1</font> being  <a href="int_1.html#NM1">Integer</a> st <span class="p1">(<span class="default"><font color="Olive">i</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">m</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Olive">i1</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="txt" href="newton.html#E10:85"><i><font color="Green">E51</font></i></a></i>;<br/>


</div>
