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<span class="kw">let </span><font color="Maroon" title="c1">k</font>, <font color="Maroon" title="c2">n</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp; <font color="Maroon" title="c1">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">n</font> &amp; <font color="Maroon" title="c2">n</font> is  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>  implies <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">n</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_coprime</a>  )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:3"/><span class="lab"><font color="Green" title="E5">A1</font></span>: 
( <font color="Maroon" title="c1">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp; <font color="Maroon" title="c1">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">n</font> )
 <span class="kw">and </span><br/><a NAME="E2:3"/><span class="lab"><font color="Green" title="E6">A2</font></span>: 
<font color="Maroon" title="c2">n</font> is  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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>,<font color="Maroon" title="c2">n</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_coprime</a> </span><br/>

<a NAME="E3:3"/><span class="lab"><font color="Green" title="E7">A3</font></span>: 
<font color="Maroon" title="c1">k</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c2">n</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#D5" onmouseover="rs('nat_d/D5')" onmouseout="rh()">NAT_D:def 5</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="3_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:3_1"/>
( <font color="Maroon" title="c1">k</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c1">k</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">n</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:3"><span class="lab"><font color="Green" title="E6">A2</font></span></a>, <a class="txt" href="#E3:3"><span class="lab"><font color="Green" title="E7">A3</font></span></a>, <a class="ref" href="int_2.html#D4" onmouseover="rs('int_2/D4')" onmouseout="rh()">INT_2:def 4</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="3_1_1"><span class="kw">suppose </span></a><a NAME="E1:3_1_1"/>
<font color="Maroon" title="c1">k</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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>,<font color="Maroon" title="c2">n</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_coprime</a> </span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:3_1_1"/>
<font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">n</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_coprime</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="int_2.html#D3" onmouseover="rs('int_2/D3')" onmouseout="rh()">INT_2:def 3</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="3_1_2"><span class="kw">suppose </span></a><a NAME="E1:3_1_2"/>
<font color="Maroon" title="c1">k</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">n</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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>,<font color="Maroon" title="c2">n</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_coprime</a> </span><br/><div class="add"><a NAME="E2:3_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c2">n</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c1">k</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#D5" onmouseover="rs('nat_d/D5')" onmouseout="rh()">NAT_D:def 5</a></span>;<br/><span class="kw">hence </span><a NAME="E3:3_1_2"/>
<font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">n</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_coprime</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:3"><span class="lab"><font color="Green" title="E5">A1</font></span></a>, <a class="ref" href="nat_d.html#T7" onmouseover="rs('nat_d/T7')" onmouseout="rh()">NAT_D:7</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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>

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