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

<span class="kw">let </span><font color="Maroon" title="c1">n</font>, <font color="Maroon" title="c2">m</font>, <font color="Maroon" title="c3">n1</font>, <font color="Maroon" title="c4">m1</font> be   <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</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="c3">n1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c4">m1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c2">m</font> &amp; <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">m</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_relative_prime</a>  implies <font color="Maroon" title="c3">n1</font>,<font color="Maroon" title="c4">m1</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_relative_prime</a>  )</span><br/>







<a NAME="E1:17"/><span class="lab"><font color="Green" title="E18">A1</font></span>: 
<font color="Maroon" title="c3">n1</font> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <font color="Maroon" title="c4">m1</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c4">m1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="int_2.html#D2" title="INT_2:def.2">INT_2:def 2</a></span>;<br/>
<span class="kw">assume </span><a NAME="E2:17"/>
<font color="Maroon" title="c3">n1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c1">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c4">m1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c2">m</font> or  not <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">m</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_relative_prime</a>  or <font color="Maroon" title="c3">n1</font>,<font color="Maroon" title="c4">m1</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_relative_prime</a>  )</span><br/>

<a NAME="E3:17"/><span class="kw">then </span><span class="lab"><font color="Green" title="E19">A2</font></span>: 
<font color="Maroon" title="c3">n1</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="moebius1.html#T39" title="MOEBIUS1:th.39">MOEBIUS1:39</a></span>;<br/>
<a NAME="E4:17"/>
<font color="Maroon" title="c3">n1</font> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <font color="Maroon" title="c4">m1</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c3">n1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="int_2.html#D2" title="INT_2:def.2">INT_2:def 2</a></span>;<br/>
<a NAME="E5:17"/><span class="kw">then </span><span class="lab"><font color="Green" title="E20">A3</font></span>: 
<font color="Maroon" title="c3">n1</font> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <font color="Maroon" title="c4">m1</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:17"><span class="lab"><font color="Green" title="E19">A2</font></span></a>, <a class="ref" href="int_2.html#T9" title="INT_2:th.9">INT_2:9</a></span>;<br/>
<span class="kw">assume </span><a NAME="E6:17"/><span class="lab"><font color="Green" title="E21">A4</font></span>: 
<font color="Maroon" title="c4">m1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c2">m</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">m</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_relative_prime</a>  or <font color="Maroon" title="c3">n1</font>,<font color="Maroon" title="c4">m1</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_relative_prime</a>  )</span><br/>

<a NAME="E7:17"/><span class="kw">then </span>
<font color="Maroon" title="c4">m1</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="moebius1.html#T39" title="MOEBIUS1:th.39">MOEBIUS1:39</a></span>;<br/>
<a NAME="E8:17"/><span class="kw">then </span><span class="lab"><font color="Green" title="E22">A5</font></span>: 
<font color="Maroon" title="c3">n1</font> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <font color="Maroon" title="c4">m1</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:17"><span class="lab"><font color="Green" title="E18">A1</font></span></a>, <a class="ref" href="int_2.html#T9" title="INT_2:th.9">INT_2:9</a></span>;<br/>
<a NAME="E9:17"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">m1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:17"><span class="lab"><font color="Green" title="E21">A4</font></span></a>, <a class="ref" href="moebius1.html#T39" title="MOEBIUS1:th.39">MOEBIUS1:39</a></span>;<br/>
<a NAME="E10:17"/><span class="kw">then </span>
<font color="Maroon" title="c3">n1</font> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <font color="Maroon" title="c4">m1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="int_2.html#T5" title="INT_2:th.5">INT_2:5</a></span>;<br/>
<a NAME="E11:17"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">n1</font> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <font color="Maroon" title="c4">m1</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/>
<a NAME="E12:17"/><span class="kw">then </span><span class="lab"><font color="Green" title="E23">A6</font></span>: 
<font color="Maroon" title="c3">n1</font> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <font color="Maroon" title="c4">m1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<span class="kw">assume </span><a NAME="E13:17"/><span class="lab"><font color="Green" title="E24">A7</font></span>: 
<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">m</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"> <font color="Maroon" title="c3">n1</font>,<font color="Maroon" title="c4">m1</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_relative_prime</a> </span><br/>

<span class="kw">assume </span><a NAME="E14:17"/>
 not <font color="Maroon" title="c3">n1</font>,<font color="Maroon" title="c4">m1</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="E15:17"/><span class="kw">then </span><span class="lab"><font color="Green" title="E25">A8</font></span>: 
<font color="Maroon" title="c3">n1</font> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <font color="Maroon" title="c4">m1</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="E16:17"/>
<font color="Maroon" title="c1">n</font> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <font color="Maroon" title="c2">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:17"><span class="lab"><font color="Green" title="E24">A7</font></span></a>, <a class="ref" href="int_2.html#D3" title="INT_2:def.3">INT_2:def 3</a></span>;<br/>
<a NAME="E17:17"/><span class="kw">then </span>
<font color="Maroon" title="c3">n1</font> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <font color="Maroon" title="c4">m1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <font color="Maroon" title="c2">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:17"><span class="lab"><font color="Green" title="E20">A3</font></span></a>, <a class="txt" href="#E8:17"><span class="lab"><font color="Green" title="E22">A5</font></span></a>, <a class="ref" href="int_2.html#T22" title="INT_2:th.22">INT_2:22</a>, <a class="ref" href="int_2.html#T27" title="INT_2:th.27">INT_2:27</a></span>;<br/>
<a NAME="E18:17"/><span class="kw">then </span>
<font color="Maroon" title="c1">n</font> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <font color="Maroon" title="c2">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:17"><span class="lab"><font color="Green" title="E25">A8</font></span></a>, <a class="txt" href="#E12:17"><span class="lab"><font color="Green" title="E23">A6</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E19:17"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:17"><span class="lab"><font color="Green" title="E24">A7</font></span></a>, <a class="ref" href="int_2.html#D3" title="INT_2:def.3">INT_2:def 3</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>
