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<span class="kw">let </span><font color="Maroon" title="c1">x</font>, <font color="Maroon" title="c2">y</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="c1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c2">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) implies <font color="Maroon" title="c1">x</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c2">y</font></span>)</span>,<font color="Maroon" title="c2">y</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="int_2.html#R1" title="INT_2:pred.1">are_relative_prime</a>  )</span><br/>



<span class="kw">assume </span><a NAME="E1:56"/><span class="lab"><font color="Green" title="E38">A1</font></span>: 
( <font color="Maroon" title="c1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c2">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</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="c1">x</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c2">y</font></span>)</span>,<font color="Maroon" title="c2">y</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="int_2.html#R1" title="INT_2:pred.1">are_relative_prime</a> </span><br/>

<a NAME="E2:56"/><span class="kw">then </span><span class="lab"><font color="Green" title="E39">A2</font></span>: 
<font color="Maroon" title="c1">x</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c2">y</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="E3:56"/><span class="lab"><font color="Green" title="E40">A3</font></span>: 
<font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c2">y</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c2">y</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="int_4.html#E51"><span class="lab"><font color="Green" title="E36">Lm11</font></span></a>, <a class="ref" href="int_2.html#T21" title="INT_2:th.21">INT_2:21</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c3">a</font>, <font color="Maroon" title="c4">b</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a><span class="kw"> such that </span><br/><a NAME="E5:56"/><span class="lab"><font color="Green" title="E41">A4</font></span>: 
<font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">a</font>
 <span class="kw">and </span><br/><a NAME="E6:56"/><span class="lab"><font color="Green" title="E42">A5</font></span>: 
( <font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">b</font> &amp; <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">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="txt" href="#E1:56"><span class="lab"><font color="Green" title="E38">A1</font></span></a>, <a class="ref" href="int_2.html#T23" title="INT_2:th.23">INT_2:23</a></span>;<br/>
<a NAME="E7:56"/>
<font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c2">y</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c2">y</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="int_4.html#E51"><span class="lab"><font color="Green" title="E36">Lm11</font></span></a>, <a class="ref" href="int_2.html#T21" title="INT_2:th.21">INT_2:21</a></span>;<br/>
<a NAME="E8:56"/><span class="kw">then </span>
<font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c2">y</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:56"><span class="lab"><font color="Green" title="E39">A2</font></span></a>, <a class="txt" href="#E5:56"><span class="lab"><font color="Green" title="E41">A4</font></span></a>, <a class="ref" href="xcmplx_1.html#T5" title="XCMPLX_1:th.5">XCMPLX_1:5</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:56"/>
<font color="Maroon" title="c1">x</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c2">y</font></span>)</span>,<font color="Maroon" title="c2">y</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c2">y</font></span>)</span> <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="txt" href="#E2:56"><span class="lab"><font color="Green" title="E39">A2</font></span></a>, <a class="txt" href="#E3:56"><span class="lab"><font color="Green" title="E40">A3</font></span></a>, <a class="txt" href="#E6:56"><span class="lab"><font color="Green" title="E42">A5</font></span></a>, <a class="ref" href="xcmplx_1.html#T5" title="XCMPLX_1:th.5">XCMPLX_1:5</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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