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<span class="kw">let </span><font color="Maroon" title="c1">i</font>, <font color="Maroon" title="c2">x</font>, <font color="Maroon" title="c3">y</font> be   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</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="c2">x</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">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  implies <span class="p1">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">x</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="c1">i</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">i</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c3">y</font></span>)</span> )</span><br/>





<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:15"/><span class="lab"><font color="Green" title="E17">A1</font></span>: 
<font color="Maroon" title="c2">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 <span class="kw">and </span><br/><a NAME="E2:15"/><span class="lab"><font color="Green" title="E18">A2</font></span>: 
<font color="Maroon" title="c1">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</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"> <span class="p1">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">x</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="c1">i</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">i</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c3">y</font></span>)</span></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c4">a2</font>, <font color="Maroon" title="c5">b2</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a><span class="kw"> such that </span><br/><a NAME="E4:15"/><span class="lab"><font color="Green" title="E19">A3</font></span>: 
( <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">a2</font> &amp; <font color="Maroon" title="c3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c5">b2</font> )
 <span class="kw">and </span><br/><a NAME="E5:15"/><span class="lab"><font color="Green" title="E20">A4</font></span>: 
<font color="Maroon" title="c4">a2</font>,<font color="Maroon" title="c5">b2</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="#E1:15"><span class="lab"><font color="Green" title="E17">A1</font></span></a>, <a class="ref" href="int_2.html#T23" onmouseover="rs('int_2/T23')" onmouseout="rh()">INT_2:23</a></span>;<br/>
<a NAME="E6:15"/>
( <font color="Maroon" title="c1">i</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c3">y</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">a2</font> &amp; <font color="Maroon" title="c1">i</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c3">y</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c5">b2</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:15"><span class="lab"><font color="Green" title="E19">A3</font></span></a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E21">A5</font></span>: <span class="p1">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">x</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="c1">i</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">y</font></span>)</span> = 
<span class="p1"><a href="int_2.html#K1" title="INT_2:func.1">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c3">y</font></span>)</span></span>)</span></span><a href="int_2.html#K1" title="INT_2:func.1">.|</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:15"><span class="lab"><font color="Green" title="E20">A4</font></span></a>, <a class="ref" href="int_2.html#T24" onmouseover="rs('int_2/T24')" onmouseout="rh()">INT_2:24</a></span>
<br/>.= 
<span class="p1"><a href="int_2.html#K1" title="INT_2:func.1">|.</a><span class="default"><font color="Maroon" title="c1">i</font></span><a href="int_2.html#K1" title="INT_2:func.1">.|</a></span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p1"><a href="int_2.html#K1" title="INT_2:func.1">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c3">y</font></span>)</span></span><a href="int_2.html#K1" title="INT_2:func.1">.|</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="complex1.html#T65" onmouseover="rs('complex1/T65')" onmouseout="rh()">COMPLEX1:65</a></span>
;<br/>
<a NAME="E8:15"/>
<font color="Maroon" title="c1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="int_2.html#K1" title="INT_2:func.1">|.</a><span class="default"><font color="Maroon" title="c1">i</font></span><a href="int_2.html#K1" title="INT_2:func.1">.|</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:15"><span class="lab"><font color="Green" title="E18">A2</font></span></a>, <a class="ref" href="absvalue.html#D1" onmouseover="rs('absvalue/D1')" onmouseout="rh()">ABSVALUE:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:15"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">x</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="c1">i</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">i</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c3">y</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:15"><span class="lab"><font color="Green" title="E21">A5</font></span></a>, <a class="ref" href="absvalue.html#D1" onmouseover="rs('absvalue/D1')" onmouseout="rh()">ABSVALUE:def 1</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/>


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