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<b>let </b><font color="Maroon" title="c1">m</font>, <font color="Maroon" title="c2">b</font>, <font color="Maroon" title="c3">n</font>, <font color="Maroon" title="c4">a</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>







<b>assume </b><a NAME="E1:9"/><i><font color="Green" title="E10">A1</font></i>: 
( <font color="Maroon" title="c1">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 &amp; <font color="Maroon" title="c2">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  &amp; <font color="Maroon" title="c1">m</font>,<font color="Maroon" title="c3">n</font> <a href="int_2.html#R2" title="INT_2:pred.2">are_relative_prime</a>  &amp; <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c1">m</font> <a href="int_2.html#R2" title="INT_2:pred.2">are_relative_prime</a>  &amp; <font color="Maroon" title="c3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c1">m</font> )
 ;<br/>

<b>assume </b><a NAME="E2:9"/>
not <font color="Maroon" title="c1">m</font>,<font color="Maroon" title="c2">b</font> <a href="int_2.html#R2" title="INT_2:pred.2">are_relative_prime</a> 
 ;<br/>

<a NAME="E3:9"/><b>then </b><i><font color="Green" title="E11">A2</font></i>: 
<font color="Maroon" title="c1">m</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#NR2" title="HIDDEN:NR.2">&lt;&gt;</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/>
<b>set </b><font color="Maroon" title="c5">k</font> = <font color="Maroon" title="c1">m</font> <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> <font color="Maroon" title="c2">b</font>;<br/>
<a NAME="E4:9"/><i><font color="Green" title="E12">A3</font></i>: 
<font color="Maroon" title="c1">m</font> <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> <font color="Maroon" title="c2">b</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> 
 
<b>by </b><i><a class="txt" href="euler_2.html#E1:9"><i><font color="Green" title="E10">A1</font></i></a>, <a class="ref" href="newton.html#T71" title="NEWTON:th.71">NEWTON:71</a></i>;<br/>
<a NAME="E5:9"/><i><font color="Green" title="E13">A4</font></i>: 
( <font color="Maroon" title="c1">m</font> <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> <font color="Maroon" title="c2">b</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c1">m</font> &amp; <font color="Maroon" title="c1">m</font> <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> <font color="Maroon" title="c2">b</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">b</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/>
<b>then consider </b><font color="Maroon" title="c6">km</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E7:9"/><i><font color="Green" title="E14">A5</font></i>: 
<font color="Maroon" title="c1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c6">km</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="E8:9"/><i><font color="Green" title="E15">A6</font></i>: 
( <font color="Maroon" title="c1">m</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#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  &amp; <font color="Maroon" title="c6">km</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  )
 
<b>by </b><i><a class="txt" href="euler_2.html#E1:9"><i><font color="Green" title="E10">A1</font></i></a>, <a class="txt" href="euler_2.html#E7:9"><i><font color="Green" title="E14">A5</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">km</font> = <font color="Maroon" title="c6">km</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>consider </b><font color="Maroon" title="c8">kb</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E11:9"/><i><font color="Green" title="E16">A7</font></i>: 
<font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c8">kb</font>
 <b>by </b><i><a class="txt" href="euler_2.html#E5:9"><i><font color="Green" title="E13">A4</font></i></a>, <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="c9">p</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E13:9"/><i><font color="Green" title="E17">A8</font></i>: 
( <font color="Maroon" title="c4">a</font> <a href="nat_1.html#K3" title="NAT_1:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c8">kb</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c7">km</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c9">p</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="nat_1.html#K3" title="NAT_1:func.3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c8">kb</font></span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c7">km</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="nat_1.html#K3" title="NAT_1:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c8">kb</font></span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c7">km</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c7">km</font> )
 <b>by </b><i><a class="txt" href="euler_2.html#E1:9"><i><font color="Green" title="E10">A1</font></i></a>, <a class="txt" href="euler_2.html#E7:9"><i><font color="Green" title="E14">A5</font></i></a>, <a class="ref" href="nat_d.html#D2" title="NAT_D:def.2">NAT_D:def 2</a></i>;<br/>
<a NAME="E14:9"/><i><font color="Green" title="E18">A9</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="nat_1.html#K3" title="NAT_1:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c8">kb</font></span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c7">km</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> <font color="Maroon" title="c2">b</font></span>)</span> <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="c4">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c8">kb</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">km</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c9">p</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="euler_2.html#E13:9"><i><font color="Green" title="E17">A8</font></i></a></i>;<br/>
<b>set </b><font color="Maroon" title="c10">tm</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c8">kb</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">km</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c9">p</font></span>)</span>;<br/>
<a NAME="E15:9"/><i><font color="Green" title="E19">A10</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c8">kb</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">km</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c9">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  implies <font color="Maroon" title="c1">m</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c3">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:9_1"/><i><font color="Green" title="E20">A11</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c8">kb</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">km</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c9">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> 
 ;<br/>

<a NAME="E2:9_1"/><i><font color="Green" title="E21">A12</font></i>: 
 <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="int_2.html#K1" title="INT_2:func.1">abs</a> <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> 
 
<b>by </b><i><a class="ref" href="absvalue.html#T7" title="ABSVALUE:th.7">ABSVALUE:7</a></i>;<br/>
<a NAME="E3:9_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c7">km</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="int_2.html#K1" title="INT_2:func.1">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c7">km</font></span>)</span>
 
<b>by </b><i><a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></i>;<br/>
<b>then </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c7">km</font></span>)</span> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c7">km</font></span>)</span> <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> 
<b>by </b><i><a class="txt" href="euler_2.html#E2:9_1"><i><font color="Green" title="E21">A12</font></i></a>, <a class="ref" href="int_2.html#D3" title="INT_2:def.3">INT_2:def 3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c7">km</font>
<b>by </b><i><a class="ref" href="newton.html#T65" title="NEWTON:th.65">NEWTON:65</a></i>
;<br/>
<b>hence </b><a NAME="E5:9_1"/>
<font color="Maroon" title="c1">m</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c3">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 <b>by </b><i><a class="txt" href="euler_2.html#E1:9"><i><font color="Green" title="E10">A1</font></i></a>, <a class="txt" href="euler_2.html#E7:9"><i><font color="Green" title="E14">A5</font></i></a>, <a class="txt" href="euler_2.html#E11:9"><i><font color="Green" title="E16">A7</font></i></a>, <a class="txt" href="euler_2.html#E14:9"><i><font color="Green" title="E18">A9</font></i></a>, <a class="txt" href="euler_2.html#E1:9_1"><i><font color="Green" title="E20">A11</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E16:9"/><i><font color="Green" title="E20">A13</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c8">kb</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">km</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c9">p</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  implies <font color="Maroon" title="c1">m</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c3">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:9_2"/><i><font color="Green" title="E21">A14</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c8">kb</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">km</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c9">p</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> 
 ;<br/>

<b>assume </b><a NAME="E2:9_2"/><i><font color="Green" title="E22">A15</font></i>: 
<font color="Maroon" title="c1">m</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/>

<a NAME="E3:9_2"/><i><font color="Green" title="E23">A16</font></i>: 
<font color="Maroon" title="c1">m</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#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T2" title="NAT_1:th.2">NAT_1:2</a></i>;<br/>
<a NAME="E4:9_2"/>
<font color="Maroon" title="c1">m</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">km</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c8">kb</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">km</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c9">p</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="euler_2.html#E1:9"><i><font color="Green" title="E10">A1</font></i></a>, <a class="txt" href="euler_2.html#E4:9"><i><font color="Green" title="E12">A3</font></i></a>, <a class="txt" href="euler_2.html#E7:9"><i><font color="Green" title="E14">A5</font></i></a>, <a class="txt" href="euler_2.html#E8:9"><i><font color="Green" title="E15">A6</font></i></a>, <a class="txt" href="euler_2.html#E11:9"><i><font color="Green" title="E16">A7</font></i></a>, <a class="txt" href="euler_2.html#E14:9"><i><font color="Green" title="E18">A9</font></i></a>, <a class="txt" href="euler_2.html#E1:9_2"><i><font color="Green" title="E21">A14</font></i></a>, <a class="ref" href="euler_1.html#T16" title="EULER_1:th.16">EULER_1:16</a></i>;<br/>
<b>hence </b><a NAME="E5:9_2"/>
contradiction
 <b>by </b><i><a class="txt" href="euler_2.html#E3:9"><i><font color="Green" title="E11">A2</font></i></a>, <a class="txt" href="euler_2.html#E2:9_2"><i><font color="Green" title="E22">A15</font></i></a>, <a class="txt" href="euler_2.html#E3:9_2"><i><font color="Green" title="E23">A16</font></i></a>, <a class="ref" href="int_1.html#T22" title="INT_1:th.22">INT_1:22</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E17:9"/><i><font color="Green" title="E21">A17</font></i>: 
<font color="Maroon" title="c1">m</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="int_2.html#K1" title="INT_2:func.1">abs</a> <font color="Maroon" title="c1">m</font></span>)</span> <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> <span class="p1">(<span class="default"><a href="int_2.html#K1" title="INT_2:func.1">abs</a> <font color="Maroon" title="c3">n</font></span>)</span>
 
<b>by </b><i><a class="ref" href="int_2.html#D3" title="INT_2:def.3">INT_2:def 3</a></i>;<br/>
<a NAME="E18:9"/>
( <font color="Maroon" title="c1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="int_2.html#K1" title="INT_2:func.1">abs</a> <font color="Maroon" title="c1">m</font> &amp; <font color="Maroon" title="c3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="int_2.html#K1" title="INT_2:func.1">abs</a> <font color="Maroon" title="c3">n</font> )
 
<b>by </b><i><a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></i>;<br/>
<b>hence </b><a NAME="E19:9"/>
contradiction
 <b>by </b><i><a class="txt" href="euler_2.html#E1:9"><i><font color="Green" title="E10">A1</font></i></a>, <a class="txt" href="euler_2.html#E15:9"><i><font color="Green" title="E19">A10</font></i></a>, <a class="txt" href="euler_2.html#E16:9"><i><font color="Green" title="E20">A13</font></i></a>, <a class="txt" href="euler_2.html#E17:9"><i><font color="Green" title="E21">A17</font></i></a>, <a class="ref" href="int_2.html#D6" title="INT_2:def.6">INT_2:def 6</a></i>;<br/>


</div>
