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



<b>set </b><font color="Maroon" title="c3">X</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">kk</font> where <font color="Olive" title="b1">kk</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( <font color="Maroon" title="c1">n</font>,<font color="Olive" title="b1">kk</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_relative_prime</a>  &amp; <font color="Olive" title="b1">kk</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp; <font color="Olive" title="b1">kk</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> ) </span>}</span> ;<br/>
<b>thus </b><a NAME="E1:5"/>
( <font color="Maroon" title="c1">n</font> is <a href="int_2.html#V1" title="INT_2:attr.1">prime</a> &amp; <font color="Maroon" title="c2">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">kk</font> where <font color="Olive" title="b1">kk</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( <font color="Maroon" title="c1">n</font>,<font color="Olive" title="b1">kk</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_relative_prime</a>  &amp; <font color="Olive" title="b1">kk</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp; <font color="Olive" title="b1">kk</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> ) </span>}</span>  implies ( <font color="Maroon" title="c1">n</font> is <a href="int_2.html#V1" title="INT_2:attr.1">prime</a> &amp; <font color="Maroon" title="c2">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">n</font> &amp; not <font color="Maroon" title="c2">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:5_1"/><i><font color="Green" title="E7">A1</font></i>: 
( <font color="Maroon" title="c1">n</font> is <a href="int_2.html#V1" title="INT_2:attr.1">prime</a> &amp; <font color="Maroon" title="c2">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">kk</font> where <font color="Olive" title="b1">kk</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( <font color="Maroon" title="c1">n</font>,<font color="Olive" title="b1">kk</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_relative_prime</a>  &amp; <font color="Olive" title="b1">kk</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp; <font color="Olive" title="b1">kk</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> ) </span>}</span>  )
 ;<br/>

<a NAME="E2:5_1"/><b>then </b><i><font color="Green" title="E8">A2</font></i>: 
 ex <font color="Olive" title="b1">kk</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Olive" title="b1">kk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">k</font> &amp; <font color="Maroon" title="c1">n</font>,<font color="Olive" title="b1">kk</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_relative_prime</a>  &amp; <font color="Olive" title="b1">kk</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp; <font color="Olive" title="b1">kk</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> )
 
;<br/>
<a NAME="E3:5_1"/><b>then </b>
<font color="Maroon" title="c2">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">n</font>
 
<b>by </b><i><a class="txt" href="euler_1.html#E1:5_1"><i><font color="Green" title="E7">A1</font></i></a>, <a class="txt" href="euler_1.html#E3"><i><font color="Green" title="E3">Lm2</font></i></a>, <a class="ref" href="euler_1.html#T2" target="_self" title="EULER_1:th.2">Th2</a></i>;<br/>
<a NAME="E4:5_1"/><b>then </b>
<font color="Maroon" title="c2">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">n</font>
 
<b>by </b><i><a class="txt" href="euler_1.html#E2:5_1"><i><font color="Green" title="E8">A2</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E5:5_1"/>
( <font color="Maroon" title="c1">n</font> is <a href="int_2.html#V1" title="INT_2:attr.1">prime</a> &amp; <font color="Maroon" title="c2">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">n</font> &amp; not <font color="Maroon" title="c2">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 <b>by </b><i><a class="txt" href="euler_1.html#E1:5_1"><i><font color="Green" title="E7">A1</font></i></a>, <a class="txt" href="euler_1.html#E2:5_1"><i><font color="Green" title="E8">A2</font></i></a>, <a class="ref" href="euler_1.html#T1" target="_self" title="EULER_1:th.1">Th1</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E2:5"/><i><font color="Green" title="E7">A3</font></i>: 
( <font color="Maroon" title="c1">n</font> is <a href="int_2.html#V1" title="INT_2:attr.1">prime</a> &amp; <font color="Maroon" title="c2">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">n</font> &amp; not <font color="Maroon" title="c2">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 ;<br/>

<a NAME="E3:5"/><b>then </b><i><font color="Green" title="E8">A4</font></i>: 
<font color="Maroon" title="c2">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">n</font>
 
<b>by </b><i><a class="ref" href="euler_1.html#T1" target="_self" title="EULER_1:th.1">Th1</a></i>;<br/>
<a NAME="E4:5"/><i><font color="Green" title="E9">A5</font></i>: 
<font color="Maroon" title="c2">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="euler_1.html#E2:5"><i><font color="Green" title="E7">A3</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<a NAME="E5:5"/><b>then </b><i><font color="Green" title="E10">A6</font></i>: 
<font color="Maroon" title="c2">k</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/>
<a NAME="E6:5"/><i><font color="Green" title="E11">A7</font></i>: 
<font color="Maroon" title="c2">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <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/>
<a NAME="E7:5"/>
<font color="Maroon" title="c2">k</font>,<font color="Maroon" title="c1">n</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_relative_prime</a> 
 
<b>by </b><i><a class="txt" href="euler_1.html#E2:5"><i><font color="Green" title="E7">A3</font></i></a>, <a class="txt" href="euler_1.html#E3:5"><i><font color="Green" title="E8">A4</font></i></a>, <a class="txt" href="euler_1.html#E4:5"><i><font color="Green" title="E9">A5</font></i></a>, <a class="ref" href="euler_1.html#T3" target="_self" title="EULER_1:th.3">Th3</a></i>;<br/>
<b>hence </b><a NAME="E8:5"/>
( <font color="Maroon" title="c1">n</font> is <a href="int_2.html#V1" title="INT_2:attr.1">prime</a> &amp; <font color="Maroon" title="c2">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">kk</font> where <font color="Olive" title="b1">kk</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( <font color="Maroon" title="c1">n</font>,<font color="Olive" title="b1">kk</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_relative_prime</a>  &amp; <font color="Olive" title="b1">kk</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp; <font color="Olive" title="b1">kk</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> ) </span>}</span>  )
 <b>by </b><i><a class="txt" href="euler_1.html#E2:5"><i><font color="Green" title="E7">A3</font></i></a>, <a class="txt" href="euler_1.html#E3:5"><i><font color="Green" title="E8">A4</font></i></a>, <a class="txt" href="euler_1.html#E5:5"><i><font color="Green" title="E10">A6</font></i></a>, <a class="txt" href="euler_1.html#E6:5"><i><font color="Green" title="E11">A7</font></i></a></i>;<br/>


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