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<span class="kw">let </span><font color="Maroon" title="c1">k</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">  for <font color="Olive" title="b1">n0</font>, <font color="Olive" title="b2">m0</font> being   <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">zero</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Olive" title="b1">n0</font>,<font color="Olive" title="b2">m0</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_relative_prime</a>  &amp; <font color="Maroon" title="c1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n0</font> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Olive" title="b2">m0</font></span>)</span> holds <br/> ex <font color="Olive" title="b3">n1</font>, <font color="Olive" title="b4">m1</font> being   <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st <br/>( <font color="Olive" title="b3">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="Olive" title="b1">n0</font> &amp; <font color="Olive" title="b4">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="Olive" title="b2">m0</font> &amp; <font color="Maroon" title="c1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">n1</font> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Olive" title="b4">m1</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">n0</font>, <font color="Maroon" title="c3">m0</font> be   <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">zero</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="c2">n0</font>,<font color="Maroon" title="c3">m0</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_relative_prime</a>  &amp; <font color="Maroon" title="c1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n0</font> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c3">m0</font></span>)</span> implies  ex <font color="Olive" title="b1">n1</font>, <font color="Olive" title="b2">m1</font> being   <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st <br/>( <font color="Olive" title="b1">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="c2">n0</font> &amp; <font color="Olive" title="b2">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="c3">m0</font> &amp; <font color="Maroon" title="c1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">n1</font> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Olive" title="b2">m1</font> ) )</span><br/>





<span class="kw">assume </span><a NAME="E1:21"/><span class="lab"><font color="Green" title="E22">A1</font></span>: 
<font color="Maroon" title="c2">n0</font>,<font color="Maroon" title="c3">m0</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"> (  not <font color="Maroon" title="c1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n0</font> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c3">m0</font></span>)</span> or  ex <font color="Olive" title="b1">n1</font>, <font color="Olive" title="b2">m1</font> being   <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st <br/>( <font color="Olive" title="b1">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="c2">n0</font> &amp; <font color="Olive" title="b2">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="c3">m0</font> &amp; <font color="Maroon" title="c1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">n1</font> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Olive" title="b2">m1</font> ) )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c4">m1</font> = <font color="Maroon" title="c1">k</font> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <font color="Maroon" title="c3">m0</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c5">n1</font> = <font color="Maroon" title="c1">k</font> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <font color="Maroon" title="c2">n0</font>;<br/>
<span class="kw">assume </span><a NAME="E2:21"/><span class="lab"><font color="Green" title="E23">A2</font></span>: 
<font color="Maroon" title="c1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n0</font> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c3">m0</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">n1</font>, <font color="Olive" title="b2">m1</font> being   <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st <br/>( <font color="Olive" title="b1">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="c2">n0</font> &amp; <font color="Olive" title="b2">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="c3">m0</font> &amp; <font color="Maroon" title="c1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">n1</font> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Olive" title="b2">m1</font> )</span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c1">k</font> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <font color="Maroon" title="c2">n0</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">m1</font> being   <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st <br/>( <font color="Maroon" title="c1">k</font> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <font color="Maroon" title="c2">n0</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">n0</font> &amp; <font color="Olive" title="b1">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="c3">m0</font> &amp; <font color="Maroon" title="c1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <font color="Maroon" title="c2">n0</font></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Olive" title="b1">m1</font> )</span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c1">k</font> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <font color="Maroon" title="c3">m0</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">k</font> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <font color="Maroon" title="c2">n0</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">n0</font> &amp; <font color="Maroon" title="c1">k</font> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <font color="Maroon" title="c3">m0</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="c3">m0</font> &amp; <font color="Maroon" title="c1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <font color="Maroon" title="c2">n0</font></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <font color="Maroon" title="c3">m0</font></span>)</span> )</span><br/>

<a NAME="E3:21"/>
( <font color="Maroon" title="c1">k</font> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <font color="Maroon" title="c2">n0</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">n0</font> &amp; <font color="Maroon" title="c1">k</font> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <font color="Maroon" title="c2">n0</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="ref" href="nat_d.html#D5" title="NAT_D:def.5">NAT_D:def 5</a>, <a class="ref" href="newton.html#T58" title="NEWTON:th.58">NEWTON:58</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:21"/>
<font color="Maroon" title="c1">k</font> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <font color="Maroon" title="c2">n0</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">n0</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">k</font> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <font color="Maroon" title="c3">m0</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="c3">m0</font> &amp; <font color="Maroon" title="c1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <font color="Maroon" title="c2">n0</font></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <font color="Maroon" title="c3">m0</font></span>)</span> )</span><br/>

<a NAME="E5:21"/>
( <font color="Maroon" title="c1">k</font> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <font color="Maroon" title="c3">m0</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c3">m0</font> &amp; <font color="Maroon" title="c1">k</font> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <font color="Maroon" title="c3">m0</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="ref" href="nat_d.html#D5" title="NAT_D:def.5">NAT_D:def 5</a>, <a class="ref" href="newton.html#T58" title="NEWTON:th.58">NEWTON:58</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:21"/>
<font color="Maroon" title="c1">k</font> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <font color="Maroon" title="c3">m0</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="c3">m0</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">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <font color="Maroon" title="c2">n0</font></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <font color="Maroon" title="c3">m0</font></span>)</span></span><br/>

<a NAME="E7:21"/>
<font color="Maroon" title="c1">k</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">n0</font> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c3">m0</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:21"><span class="lab"><font color="Green" title="E23">A2</font></span></a>, <a class="ref" href="moebius1.html#T39" title="MOEBIUS1:th.39">MOEBIUS1:39</a></span>;<br/>
<span class="kw">hence </span><font color="Maroon" title="c1">k</font> = 
<font color="Maroon" title="c1">k</font> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n0</font> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c3">m0</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="newton.html#T49" title="NEWTON:th.49">NEWTON:49</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <font color="Maroon" title="c2">n0</font></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <font color="Maroon" title="c3">m0</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:21"><span class="lab"><font color="Green" title="E22">A1</font></span></a>, <a class="ref" href="nat_5.html#T17" target="_self" title="NAT_5:th.17">Th17</a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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