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<span class="kw">let </span><font color="Maroon" title="c1">b</font>, <font color="Maroon" title="c2">a</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">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">k</font>, <font color="Olive" title="b2">l</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>  st <font color="Olive" title="b1">k</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Olive" title="b2">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 holds <br/>( <font color="Olive" title="b1">k</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">l</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c1">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <span class="p1">(<span class="default"><font color="Olive" title="b1">k</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">l</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c1">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">k</font>, <font color="Maroon" title="c4">l</font> be   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</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">k</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c4">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 implies ( <font color="Maroon" title="c3">k</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">l</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c1">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">l</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c1">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) )</span><br/>







<span class="kw">assume </span><a NAME="E1:16"/>
<font color="Maroon" title="c3">k</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c4">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <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">k</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">l</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c1">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">l</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c1">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )</span><br/>

<a NAME="E2:16"/><span class="kw">then </span>
<font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c4">l</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="int_2.html#D3" title="INT_2:def.3">INT_2:def 3</a></span>;<br/>
<a NAME="E3:16"/><span class="kw">then </span>
( <font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c4">l</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c1">b</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_relative_prime</a>  &amp; <font color="Maroon" title="c3">k</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">l</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c1">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#T10" target="_self" title="WSIERP_1:th.10">Th15</a>, <a class="ref" href="wsierp_1.html#T11" target="_self" title="WSIERP_1:th.11">Th16</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:16"/>
( <font color="Maroon" title="c3">k</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">l</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c1">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">l</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c1">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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>; <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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