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<span class="kw">let </span><font color="Maroon" title="c1">p</font> be   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">m</font>, <font color="Olive" title="b2">n</font> being   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">zero</a>   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">m</font>,<font color="Olive" title="b2">n</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_relative_prime</a>  &amp; <font color="Maroon" title="c1">p</font> <a href="newton.html#K15" title="NEWTON:func.15">|^</a> 2 <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Olive" title="b1">m</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Olive" title="b2">n</font> &amp;  not <font color="Maroon" title="c1">p</font> <a href="newton.html#K15" title="NEWTON:func.15">|^</a> 2 <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Olive" title="b1">m</font> holds <br/><font color="Maroon" title="c1">p</font> <a href="newton.html#K15" title="NEWTON:func.15">|^</a> 2 <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Olive" title="b2">n</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">b</font> be   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">zero</a>   <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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_relative_prime</a>  &amp; <font color="Maroon" title="c1">p</font> <a href="newton.html#K15" title="NEWTON:func.15">|^</a> 2 <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">a</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c3">b</font> &amp;  not <font color="Maroon" title="c1">p</font> <a href="newton.html#K15" title="NEWTON:func.15">|^</a> 2 <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">a</font> implies <font color="Maroon" title="c1">p</font> <a href="newton.html#K15" title="NEWTON:func.15">|^</a> 2 <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c3">b</font> )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:15"/><span class="lab"><font color="Green" title="E12">A1</font></span>: 
<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_relative_prime</a> 
 <span class="kw">and </span><br/><a NAME="E2:15"/><span class="lab"><font color="Green" title="E13">A2</font></span>: 
<font color="Maroon" title="c1">p</font> <a href="newton.html#K15" title="NEWTON:func.15">|^</a> 2 <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">a</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c3">b</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">p</font> <a href="newton.html#K15" title="NEWTON:func.15">|^</a> 2 <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">a</font> or <font color="Maroon" title="c1">p</font> <a href="newton.html#K15" title="NEWTON:func.15">|^</a> 2 <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c3">b</font> )</span><br/>

<a NAME="E3:15"/>
<font color="Maroon" title="c1">p</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c1">p</font> <a href="newton.html#K15" title="NEWTON:func.15">|^</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_3.html#T3" title="NAT_3:th.3">NAT_3:3</a></span>;<br/>
<a NAME="E4:15"/><span class="kw">then </span><span class="lab"><font color="Green" title="E14">A3</font></span>: 
<font color="Maroon" title="c1">p</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">a</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c3">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:15"><span class="lab"><font color="Green" title="E13">A2</font></span></a>, <a class="ref" href="nat_d.html#T4" title="NAT_D:th.4">NAT_D:4</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:15_1"/>
( <font color="Maroon" title="c1">p</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">a</font> or <font color="Maroon" title="c1">p</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c3">b</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:15"><span class="lab"><font color="Green" title="E14">A3</font></span></a>, <a class="ref" href="newton.html#T80" title="NEWTON:th.80">NEWTON:80</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_1_1"><span class="kw">suppose </span></a><a NAME="E1:15_1_1"/>
<font color="Maroon" title="c1">p</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">a</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">p</font> <a href="newton.html#K15" title="NEWTON:func.15">|^</a> 2 <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">a</font> or <font color="Maroon" title="c1">p</font> <a href="newton.html#K15" title="NEWTON:func.15">|^</a> 2 <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c3">b</font> )</span><br/><div class="add"><a NAME="E2:15_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E15">A4</font></span>: 
 not <font color="Maroon" title="c1">p</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c3">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:15"><span class="lab"><font color="Green" title="E12">A1</font></span></a>, <a class="ref" href="pythtrip.html#D2" title="PYTHTRIP:def.2">PYTHTRIP:def 2</a></span>;<br/><a NAME="E3:15_1_1"/>
<font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c3">b</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_relative_prime</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">reconsider </span><font color="Maroon" title="c4">k</font> = <font color="Maroon" title="c1">p</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c3">b</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>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" title="ORDINAL1:def.12">ORDINAL1:def 12</a></span>;<br/>
<span class="kw">assume </span><a NAME="E2:15_1_1_1"/>
 not <font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c3">b</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_relative_prime</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E3:15_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E16">A5</font></span>: 
<font color="Maroon" title="c4">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="int_2.html#D3" title="INT_2:def.3">INT_2:def 3</a></span>;<br/>
<a NAME="E4:15_1_1_1"/>
( <font color="Maroon" title="c4">k</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c1">p</font> &amp; <font color="Maroon" title="c4">k</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c3">b</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#D5" title="NAT_D:def.5">NAT_D:def 5</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:15_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:15_1_1"><span class="lab"><font color="Green" title="E15">A4</font></span></a>, <a class="txt" href="#E3:15_1_1_1"><span class="lab"><font color="Green" title="E16">A5</font></span></a>, <a class="ref" href="int_2.html#D4" title="INT_2:def.4">INT_2:def 4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E4:15_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c1">p</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c3">b</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_relative_prime</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="euler_1.html#T14" title="EULER_1:th.14">EULER_1:14</a></span>;<br/><a NAME="E5:15_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c1">p</font> <a href="newton.html#K15" title="NEWTON:func.15">|^</a> 2,<font color="Maroon" title="c3">b</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_relative_prime</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="wsierp_1.html#T1" title="WSIERP_1:th.1">WSIERP_1:1</a></span>;<br/><span class="kw">hence </span><a NAME="E6:15_1_1"/>
( <font color="Maroon" title="c1">p</font> <a href="newton.html#K15" title="NEWTON:func.15">|^</a> 2 <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">a</font> or <font color="Maroon" title="c1">p</font> <a href="newton.html#K15" title="NEWTON:func.15">|^</a> 2 <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c3">b</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:15"><span class="lab"><font color="Green" title="E13">A2</font></span></a>, <a class="ref" href="euler_1.html#T13" title="EULER_1:th.13">EULER_1:13</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_1_2"><span class="kw">suppose </span></a><a NAME="E1:15_1_2"/>
<font color="Maroon" title="c1">p</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c3">b</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">p</font> <a href="newton.html#K15" title="NEWTON:func.15">|^</a> 2 <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">a</font> or <font color="Maroon" title="c1">p</font> <a href="newton.html#K15" title="NEWTON:func.15">|^</a> 2 <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c3">b</font> )</span><br/><div class="add"><a NAME="E2:15_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E15">A6</font></span>: 
 not <font color="Maroon" title="c1">p</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:15"><span class="lab"><font color="Green" title="E12">A1</font></span></a>, <a class="ref" href="pythtrip.html#D2" title="PYTHTRIP:def.2">PYTHTRIP:def 2</a></span>;<br/><a NAME="E3:15_1_2"/>
<font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">a</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_relative_prime</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">reconsider </span><font color="Maroon" title="c4">k</font> = <font color="Maroon" title="c1">p</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c2">a</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>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" title="ORDINAL1:def.12">ORDINAL1:def 12</a></span>;<br/>
<span class="kw">assume </span><a NAME="E2:15_1_2_1"/>
 not <font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">a</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_relative_prime</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E3:15_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E16">A7</font></span>: 
<font color="Maroon" title="c4">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="int_2.html#D3" title="INT_2:def.3">INT_2:def 3</a></span>;<br/>
<a NAME="E4:15_1_2_1"/>
( <font color="Maroon" title="c4">k</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c1">p</font> &amp; <font color="Maroon" title="c4">k</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">a</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#D5" title="NAT_D:def.5">NAT_D:def 5</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:15_1_2_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:15_1_2"><span class="lab"><font color="Green" title="E15">A6</font></span></a>, <a class="txt" href="#E3:15_1_2_1"><span class="lab"><font color="Green" title="E16">A7</font></span></a>, <a class="ref" href="int_2.html#D4" title="INT_2:def.4">INT_2:def 4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E4:15_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c1">p</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">a</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_relative_prime</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="euler_1.html#T14" title="EULER_1:th.14">EULER_1:14</a></span>;<br/><a NAME="E5:15_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c1">p</font> <a href="newton.html#K15" title="NEWTON:func.15">|^</a> 2,<font color="Maroon" title="c2">a</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_relative_prime</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="wsierp_1.html#T1" title="WSIERP_1:th.1">WSIERP_1:1</a></span>;<br/><span class="kw">hence </span><a NAME="E6:15_1_2"/>
( <font color="Maroon" title="c1">p</font> <a href="newton.html#K15" title="NEWTON:func.15">|^</a> 2 <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">a</font> or <font color="Maroon" title="c1">p</font> <a href="newton.html#K15" title="NEWTON:func.15">|^</a> 2 <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c3">b</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:15"><span class="lab"><font color="Green" title="E13">A2</font></span></a>, <a class="ref" href="euler_1.html#T13" title="EULER_1:th.13">EULER_1:13</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div>
