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

<b>let </b><font color="Maroon" title="c1">a</font>, <font color="Maroon" title="c2">b</font>, <font color="Maroon" title="c3">c</font>, <font color="Maroon" title="c4">d</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="E1:71"/><i><font color="Green" title="E51">A1</font></i>: 
( <font color="Maroon" title="c1">a</font> <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> <font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c1">a</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">c</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c4">d</font> )
 ;<br/>

<b>set </b><font color="Maroon" title="c5">e</font> = <font color="Maroon" title="c1">a</font> <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> <font color="Maroon" title="c3">c</font>;<br/>
<a NAME="E2:71"/><i><font color="Green" title="E52">A2</font></i>: 
<font color="Maroon" title="c1">a</font> <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> <font color="Maroon" title="c3">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">a</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c3">c</font>
 
<b>by </b><i><a class="txt" href="wsierp_1.html#E18"><i><font color="Green" title="E13">Lm8</font></i></a></i>;<br/>
<a NAME="E3:71"/><i><font color="Green" title="E53">A3</font></i>: 
( <font color="Maroon" title="c1">a</font> <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> <font color="Maroon" title="c3">c</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c3">c</font> &amp; <font color="Maroon" title="c1">a</font> <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> <font color="Maroon" title="c3">c</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c1">a</font> )
 
<b>by </b><i><a class="ref" href="nat_d.html#D5" title="NAT_D:def.5">NAT_D:def 5</a></i>;<br/>
<a NAME="E4:71"/><b>then </b><i><font color="Green" title="E54">A4</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c4">d</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c3">c</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c4">d</font>
 
<b>by </b><i><a class="ref" href="wsierp_1.html#T19" target="_self" title="WSIERP_1:th.19">Th19</a></i>;<br/>
<a NAME="E5:71"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> <font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="txt" href="wsierp_1.html#E1:71"><i><font color="Green" title="E51">A1</font></i></a>, <a class="txt" href="wsierp_1.html#E3:71"><i><font color="Green" title="E53">A3</font></i></a>, <a class="ref" href="wsierp_1.html#T21" target="_self" title="WSIERP_1:th.21">Th21</a></i>;<br/>
<a NAME="E6:71"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c4">d</font></span>)</span> <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> <font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="txt" href="wsierp_1.html#E27"><i><font color="Green" title="E22">Lm10</font></i></a></i>;<br/>
<a NAME="E7:71"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c4">d</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c1">a</font>
 
<b>by </b><i><a class="txt" href="wsierp_1.html#E1:71"><i><font color="Green" title="E51">A1</font></i></a>, <a class="txt" href="wsierp_1.html#E4:71"><i><font color="Green" title="E54">A4</font></i></a>, <a class="ref" href="wsierp_1.html#T37" target="_self" title="WSIERP_1:th.37">Th37</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c6">g</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E9:71"/><i><font color="Green" title="E55">A5</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c4">d</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c6">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">a</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>reconsider </b><font color="Maroon" title="c7">g</font> = <font color="Maroon" title="c6">g</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><i><font color="Green" title="E56">A6</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="71_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:71_1"/><i><font color="Green" title="E57">A7</font></i>: 
<font color="Maroon" title="c4">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 ;<br/><a NAME="E2:71_1"/><b>then </b>
<font color="Maroon" title="c1">a</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="txt" href="wsierp_1.html#E1:71"><i><font color="Green" title="E51">A1</font></i></a>, <a class="ref" href="newton.html#T9" title="NEWTON:th.9">NEWTON:9</a></i>;<br/><a NAME="E3:71_1"/><b>then </b>
( <font color="Maroon" title="c1">a</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> 1 &amp; <font color="Maroon" title="c2">b</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> 1 )
 <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="E4:71_1"/><b>then </b>
( <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 <b>by </b><i><a class="ref" href="wsierp_1.html#T20" target="_self" title="WSIERP_1:th.20">Th20</a></i>;<br/><a NAME="E5:71_1"/><b>then </b>
( <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> 0 &amp; <font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> 0 )
 <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="E6:71_1"/>
 ex <font color="Olive" title="b1">e</font>, <font color="Olive" title="b2">f</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">e</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c4">d</font> &amp; <font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">f</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c4">d</font> )
 <b>by </b><i><a class="txt" href="wsierp_1.html#E1:71_1"><i><font color="Green" title="E57">A7</font></i></a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:71_2"/>
<font color="Maroon" title="c4">d</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0
 ;<br/><a NAME="E2:71_2"/><b>then </b><i><font color="Green" title="E57">A8</font></i>: 
<font color="Maroon" title="c4">d</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T14" title="NAT_1:th.14">NAT_1:14</a></i>;<br/><b>then consider </b><font color="Maroon" title="c8">d1</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E4:71_2"/><i><font color="Green" title="E58">A9</font></i>: 
<font color="Maroon" title="c4">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c8">d1</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T10" title="NAT_1:th.10">NAT_1:10</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c9">d1</font> = <font color="Maroon" title="c8">d1</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><i><font color="Green" title="E59">A10</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="71_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:71_2_1"/>
<font color="Maroon" title="c1">a</font> <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> <font color="Maroon" title="c3">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 ;<br/><a NAME="E2:71_2_1"/><b>then </b><i><font color="Green" title="E60">A11</font></i>: 
( <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 &amp; 0 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">c</font> )
 <b>by </b><i><a class="ref" href="int_2.html#T5" title="INT_2:th.5">INT_2:5</a></i>;<br/><a NAME="E3:71_2_1"/><b>then </b>
<font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="txt" href="wsierp_1.html#E1:71"><i><font color="Green" title="E51">A1</font></i></a>, <a class="ref" href="newton.html#T65" title="NEWTON:th.65">NEWTON:65</a></i>;<br/><a NAME="E4:71_2_1"/><b>then </b><i><font color="Green" title="E61">A12</font></i>: 
<font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c4">d</font>
 <b>by </b><i><a class="ref" href="newton.html#T15" title="NEWTON:th.15">NEWTON:15</a></i>;<br/><a NAME="E5:71_2_1"/>
<font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c4">d</font>
 <b>by </b><i><a class="txt" href="wsierp_1.html#E2:71_2"><i><font color="Green" title="E57">A8</font></i></a>, <a class="txt" href="wsierp_1.html#E2:71_2_1"><i><font color="Green" title="E60">A11</font></i></a>, <a class="ref" href="newton.html#T16" title="NEWTON:th.16">NEWTON:16</a></i>;<br/><b>hence </b><a NAME="E6:71_2_1"/>
 ex <font color="Olive" title="b1">e</font>, <font color="Olive" title="b2">f</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">e</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c4">d</font> &amp; <font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">f</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c4">d</font> )
 <b>by </b><i><a class="txt" href="wsierp_1.html#E4:71_2_1"><i><font color="Green" title="E61">A12</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:71_2_2"/><i><font color="Green" title="E60">A13</font></i>: 
<font color="Maroon" title="c1">a</font> <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> <font color="Maroon" title="c3">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0
 ;<br/><a NAME="E2:71_2_2"/><b>then </b><i><font color="Green" title="E61">A14</font></i>: 
( <font color="Maroon" title="c1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0 or <font color="Maroon" title="c3">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0 )
 ;<br/><a NAME="E3:71_2_2"/><b>then </b><i><font color="Green" title="E62">A15</font></i>: 
<font color="Maroon" title="c1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0
 <b>by </b><i><a class="txt" href="wsierp_1.html#E1:71"><i><font color="Green" title="E51">A1</font></i></a>, <a class="ref" href="card_4.html#T51" title="CARD_4:th.51">CARD_4:51</a></i>;<br/><div><i><font color="Green" title="E63">A16</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="71_2_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:71_2_2_1"/><i><font color="Green" title="E64">A17</font></i>: 
0 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">c</font>
 ;<br/><a NAME="E2:71_2_2_1"/><b>then </b>
<font color="Maroon" title="c1">a</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 <b>by </b><i><a class="txt" href="wsierp_1.html#E1:71"><i><font color="Green" title="E51">A1</font></i></a>, <a class="txt" href="wsierp_1.html#E2:71_2"><i><font color="Green" title="E57">A8</font></i></a>, <a class="ref" href="newton.html#T16" title="NEWTON:th.16">NEWTON:16</a></i>;<br/><a NAME="E3:71_2_2_1"/><b>then </b><i><font color="Green" title="E65">A18</font></i>: 
<font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 <b>by </b><i><a class="txt" href="wsierp_1.html#E2:71_2_2"><i><font color="Green" title="E61">A14</font></i></a>, <a class="txt" href="wsierp_1.html#E1:71_2_2_1"><i><font color="Green" title="E64">A17</font></i></a>, <a class="ref" href="xcmplx_1.html#T6" title="XCMPLX_1:th.6">XCMPLX_1:6</a></i>;<br/><a NAME="E4:71_2_2_1"/><b>then </b>
<font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="txt" href="wsierp_1.html#E1:71"><i><font color="Green" title="E51">A1</font></i></a>, <a class="ref" href="newton.html#T65" title="NEWTON:th.65">NEWTON:65</a></i>;<br/><a NAME="E5:71_2_2_1"/><b>then </b><i><font color="Green" title="E66">A19</font></i>: 
<font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c4">d</font>
 <b>by </b><i><a class="ref" href="newton.html#T15" title="NEWTON:th.15">NEWTON:15</a></i>;<br/><a NAME="E6:71_2_2_1"/>
<font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c4">d</font>
 <b>by </b><i><a class="txt" href="wsierp_1.html#E2:71_2"><i><font color="Green" title="E57">A8</font></i></a>, <a class="txt" href="wsierp_1.html#E3:71_2_2_1"><i><font color="Green" title="E65">A18</font></i></a>, <a class="ref" href="newton.html#T16" title="NEWTON:th.16">NEWTON:16</a></i>;<br/><b>hence </b><a NAME="E7:71_2_2_1"/>
 ex <font color="Olive" title="b1">e</font>, <font color="Olive" title="b2">f</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">e</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c4">d</font> &amp; <font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">f</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c4">d</font> )
 <b>by </b><i><a class="txt" href="wsierp_1.html#E5:71_2_2_1"><i><font color="Green" title="E66">A19</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_2_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:71_2_2_2"/><i><font color="Green" title="E64">A20</font></i>: 
<font color="Maroon" title="c3">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0
 ;<br/><a NAME="E2:71_2_2_2"/><i><font color="Green" title="E65">A21</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c4">d</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0
 <b>by </b><i><a class="txt" href="wsierp_1.html#E1:71_2_2"><i><font color="Green" title="E60">A13</font></i></a>, <a class="ref" href="card_4.html#T51" title="CARD_4:th.51">CARD_4:51</a></i>;<br/><b>consider </b><font color="Maroon" title="c10">a1</font>, <font color="Maroon" title="c11">c1</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a><b> such that </b><br/><a NAME="E4:71_2_2_2"/><i><font color="Green" title="E66">A22</font></i>: 
( <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c10">a1</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c11">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">c</font> &amp; <font color="Maroon" title="c10">a1</font>,<font color="Maroon" title="c11">c1</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_relative_prime</a>  )
 <b>by </b><i><a class="txt" href="wsierp_1.html#E2:71"><i><font color="Green" title="E52">A2</font></i></a>, <a class="txt" href="wsierp_1.html#E1:71_2_2_2"><i><font color="Green" title="E64">A20</font></i></a>, <a class="ref" href="int_2.html#T38" title="INT_2:th.38">INT_2:38</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c12">a1</font> = <font color="Maroon" title="c10">a1</font>, <font color="Maroon" title="c13">c1</font> = <font color="Maroon" title="c11">c1</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="txt" href="wsierp_1.html#E3:71_2_2"><i><font color="Green" title="E62">A15</font></i></a>, <a class="txt" href="wsierp_1.html#E1:71_2_2_2"><i><font color="Green" title="E64">A20</font></i></a>, <a class="txt" href="wsierp_1.html#E4:71_2_2_2"><i><font color="Green" title="E66">A22</font></i></a>, <a class="txt" href="wsierp_1.html#E34"><i><font color="Green" title="E27">Lm12</font></i></a></i>;<br/><a NAME="E6:71_2_2_2"/><i><font color="Green" title="E67">A23</font></i>: 
( <font color="Maroon" title="c12">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">a</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> <font color="Maroon" title="c3">c</font></span>)</span> &amp; <font color="Maroon" title="c13">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">c</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> <font color="Maroon" title="c3">c</font></span>)</span> )
 <b>by </b><i><a class="txt" href="wsierp_1.html#E1:71_2_2"><i><font color="Green" title="E60">A13</font></i></a>, <a class="txt" href="wsierp_1.html#E4:71_2_2_2"><i><font color="Green" title="E66">A22</font></i></a>, <a class="ref" href="xcmplx_1.html#T90" title="XCMPLX_1:th.90">XCMPLX_1:90</a></i>;<br/><a NAME="E7:71_2_2_2"/>
<font color="Maroon" title="c12">a1</font>,<font color="Maroon" title="c13">c1</font> <a href="int_2.html#R2" title="INT_2:pred.2">are_relative_prime</a> 
 <b>by </b><i><a class="txt" href="wsierp_1.html#E4:71_2_2_2"><i><font color="Green" title="E66">A22</font></i></a>, <a class="txt" href="wsierp_1.html#E19"><i><font color="Green" title="E14">Lm9</font></i></a></i>;<br/><a NAME="E8:71_2_2_2"/><b>then </b><i><font color="Green" title="E68">A24</font></i>: 
<font color="Maroon" title="c12">a1</font> <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> <font color="Maroon" title="c13">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="ref" href="int_2.html#D6" title="INT_2:def.6">INT_2:def 6</a></i>;<br/><font color="Maroon" title="c12">a1</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">d1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c7">g</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> <font color="Maroon" title="c3">c</font></span>)</span>
<b>by </b><i><a class="txt" href="wsierp_1.html#E9:71"><i><font color="Green" title="E55">A5</font></i></a>, <a class="txt" href="wsierp_1.html#E4:71_2"><i><font color="Green" title="E58">A9</font></i></a>, <a class="txt" href="wsierp_1.html#E1:71_2_2"><i><font color="Green" title="E60">A13</font></i></a>, <a class="txt" href="wsierp_1.html#E4:71_2_2_2"><i><font color="Green" title="E66">A22</font></i></a>, <a class="ref" href="xcmplx_1.html#T90" title="XCMPLX_1:th.90">XCMPLX_1:90</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c9">d1</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c7">g</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> <font color="Maroon" title="c3">c</font></span>)</span>
<b>by </b><i><a class="ref" href="newton.html#T11" title="NEWTON:th.11">NEWTON:11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c9">d1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c7">g</font></span>)</span></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> <font color="Maroon" title="c3">c</font></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c9">d1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c7">g</font>
<b>by </b><i><a class="txt" href="wsierp_1.html#E1:71_2_2"><i><font color="Green" title="E60">A13</font></i></a>, <a class="ref" href="xcmplx_1.html#T90" title="XCMPLX_1:th.90">XCMPLX_1:90</a></i>
;<br/><a NAME="E10:71_2_2_2"/><b>then </b>
<font color="Maroon" title="c7">g</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c12">a1</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/><a NAME="E11:71_2_2_2"/><b>then </b>
<font color="Maroon" title="c7">g</font> <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> <font color="Maroon" title="c13">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="txt" href="wsierp_1.html#E8:71_2_2_2"><i><font color="Green" title="E68">A24</font></i></a>, <a class="ref" href="wsierp_1.html#T21" target="_self" title="WSIERP_1:th.21">Th21</a></i>;<br/><a NAME="E12:71_2_2_2"/><b>then </b><i><font color="Green" title="E69">A25</font></i>: 
<font color="Maroon" title="c7">g</font> <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">c1</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c4">d</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="txt" href="wsierp_1.html#E27"><i><font color="Green" title="E22">Lm10</font></i></a></i>;<br/><i><font color="Green" title="E70">A26</font></i>: <font color="Maroon" title="c13">c1</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c4">d</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c4">d</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">g</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c4">d</font></span>)</span>
<b>by </b><i><a class="txt" href="wsierp_1.html#E6:71_2_2_2"><i><font color="Green" title="E67">A23</font></i></a>, <a class="ref" href="prepower.html#T15" title="PREPOWER:th.15">PREPOWER:15</a>, <a class="txt" href="wsierp_1.html#E1:71"><i><font color="Green" title="E51">A1</font></i></a>, <a class="txt" href="wsierp_1.html#E9:71"><i><font color="Green" title="E55">A5</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c7">g</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c2">b</font>
<b>by </b><i><a class="txt" href="wsierp_1.html#E2:71_2_2_2"><i><font color="Green" title="E65">A21</font></i></a>, <a class="ref" href="xcmplx_1.html#T90" title="XCMPLX_1:th.90">XCMPLX_1:90</a></i>
;<br/><a NAME="E14:71_2_2_2"/><b>then </b>
<font color="Maroon" title="c7">g</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c13">c1</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c4">d</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/><a NAME="E15:71_2_2_2"/><b>then </b>
<font color="Maroon" title="c7">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="txt" href="wsierp_1.html#E12:71_2_2_2"><i><font color="Green" title="E69">A25</font></i></a>, <a class="ref" href="newton.html#T62" title="NEWTON:th.62">NEWTON:62</a></i>;<br/><b>hence </b><a NAME="E16:71_2_2_2"/>
 ex <font color="Olive" title="b1">e</font>, <font color="Olive" title="b2">f</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">e</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c4">d</font> &amp; <font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">f</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c4">d</font> )
 <b>by </b><i><a class="txt" href="wsierp_1.html#E9:71"><i><font color="Green" title="E55">A5</font></i></a>, <a class="txt" href="wsierp_1.html#E13:71_2_2_2"><i><font color="Green" title="E70">A26</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E6:71_2_2"/>
 ex <font color="Olive" title="b1">e</font>, <font color="Olive" title="b2">f</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">e</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c4">d</font> &amp; <font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">f</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c4">d</font> )
 <b>by </b><i><a class="txt" href="wsierp_1.html#E4:71_2_2"><i><font color="Green" title="E63">A16</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E8:71_2"/>
 ex <font color="Olive" title="b1">e</font>, <font color="Olive" title="b2">f</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">e</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c4">d</font> &amp; <font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">f</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c4">d</font> )
 <b>by </b><i><a class="txt" href="wsierp_1.html#E6:71_2"><i><font color="Green" title="E59">A10</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E13:71"/>
 ex <font color="Olive" title="b1">e</font>, <font color="Olive" title="b2">f</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">e</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c4">d</font> &amp; <font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">f</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c4">d</font> )
 <b>by </b><i><a class="txt" href="wsierp_1.html#E11:71"><i><font color="Green" title="E56">A6</font></i></a></i>;<br/>


</div>
