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<b>let </b><font color="Maroon">m</font> be   <a href="int_1.html#NM1">Integer</a>;<br/><b>let </b><font color="Maroon">n</font> be   <a href="int_1.html#NM1">Integer</a>;<br/>



<a NAME="E1:65"/>
<font color="Maroon">m</font> <a href="int_2.html#K3">gcd</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="prepower.html#K1">abs</a> <font color="Maroon">m</font></span>)</span> <a href="nat_1.html#K10">hcf</a> <span class="p1">(<span class="default"><a href="prepower.html#K1">abs</a> <font color="Maroon">n</font></span>)</span>
 
<b>by </b><i><a class="ref" href="int_2.html#D3">INT_2:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon">m1</font> being   <a href="int_1.html#NM1">Integer</a>, <font color="Maroon">n1</font> being   <a href="int_1.html#NM1">Integer</a><b> such that </b><br/><a NAME="E3:65"/><i><font color="Green">E40</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="prepower.html#K1">abs</a> <font color="Maroon">m</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">m1</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="prepower.html#K1">abs</a> <font color="Maroon">n</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">n1</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> <a href="int_2.html#K3">gcd</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="txt" href="wsierp_1.html#E8"><i><font color="Green">Lemma45</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="65_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="65_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:65_1_1"/>
( ( <font color="Maroon">m</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 ) or ( <font color="Maroon">m</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 ) or ( <font color="Maroon">m</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 ) or ( <font color="Maroon">m</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="65_1_1_1"><b>suppose </b></a><a NAME="E1:65_1_1_1"/>
( <font color="Maroon">m</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 )
 ;<br/><div class="add"><a NAME="E2:65_1_1_1"/><b>then </b>
(  <a href="prepower.html#K1">abs</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> &amp;  <a href="prepower.html#K1">abs</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> )
 <b>by </b><i><a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/><b>hence </b><a NAME="E3:65_1_1_1"/>
 ex <font color="Olive">m1</font>, <font color="Olive">n1</font> being  <a href="int_1.html#NM1">Integer</a> st <font color="Maroon">m</font> <a href="int_2.html#K3">gcd</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K3">*</a> <font color="Olive">m1</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</a> <font color="Olive">n1</font></span>)</span>
 <b>by </b><i><a class="txt" href="wsierp_1.html#E12"><i><font color="Green">Lemma47</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="65_1_1_2"><b>suppose </b></a><a NAME="E1:65_1_1_2"/>
( <font color="Maroon">m</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 )
 ;<br/><div class="add"><a NAME="E2:65_1_1_2"/><b>then </b>
(  <a href="prepower.html#K1">abs</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> &amp;  <a href="prepower.html#K1">abs</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a>  <a href="xcmplx_0.html#K4">-</a> <font color="Maroon">n</font> )
 <b>by </b><i><a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/><a NAME="E3:65_1_1_2"/><b>then </b>
<font color="Maroon">m</font> <a href="int_2.html#K3">gcd</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">m1</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">n1</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="wsierp_1.html#E12"><i><font color="Green">Lemma47</font></i></a></i>;<br/><b>hence </b><a NAME="E4:65_1_1_2"/>
 ex <font color="Olive">m1</font>, <font color="Olive">n1</font> being  <a href="int_1.html#NM1">Integer</a> st <font color="Maroon">m</font> <a href="int_2.html#K3">gcd</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K3">*</a> <font color="Olive">m1</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</a> <font color="Olive">n1</font></span>)</span>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="65_1_1_3"><b>suppose </b></a><a NAME="E1:65_1_1_3"/>
( <font color="Maroon">m</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 )
 ;<br/><div class="add"><a NAME="E2:65_1_1_3"/><b>then </b>
(  <a href="prepower.html#K1">abs</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a>  <a href="xcmplx_0.html#K4">-</a> <font color="Maroon">m</font> &amp;  <a href="prepower.html#K1">abs</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> )
 <b>by </b><i><a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/><a NAME="E3:65_1_1_3"/><b>then </b>
<font color="Maroon">m</font> <a href="int_2.html#K3">gcd</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">m1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">n1</font></span>)</span>
 <b>by </b><i><a class="txt" href="wsierp_1.html#E12"><i><font color="Green">Lemma47</font></i></a></i>;<br/><b>hence </b><a NAME="E4:65_1_1_3"/>
 ex <font color="Olive">m1</font>, <font color="Olive">n1</font> being  <a href="int_1.html#NM1">Integer</a> st <font color="Maroon">m</font> <a href="int_2.html#K3">gcd</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K3">*</a> <font color="Olive">m1</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</a> <font color="Olive">n1</font></span>)</span>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="65_1_1_4"><b>suppose </b></a><a NAME="E1:65_1_1_4"/>
( <font color="Maroon">m</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 )
 ;<br/><div class="add"><a NAME="E2:65_1_1_4"/><b>then </b>
(  <a href="prepower.html#K1">abs</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a>  <a href="xcmplx_0.html#K4">-</a> <font color="Maroon">m</font> &amp;  <a href="prepower.html#K1">abs</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a>  <a href="xcmplx_0.html#K4">-</a> <font color="Maroon">n</font> )
 <b>by </b><i><a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/><a NAME="E3:65_1_1_4"/><b>then </b>
<font color="Maroon">m</font> <a href="int_2.html#K3">gcd</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">m1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">n1</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="wsierp_1.html#E12"><i><font color="Green">Lemma47</font></i></a></i>;<br/><b>hence </b><a NAME="E4:65_1_1_4"/>
 ex <font color="Olive">m1</font>, <font color="Olive">n1</font> being  <a href="int_1.html#NM1">Integer</a> st <font color="Maroon">m</font> <a href="int_2.html#K3">gcd</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K3">*</a> <font color="Olive">m1</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</a> <font color="Olive">n1</font></span>)</span>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E5:65"/>
 ex <font color="Olive">m1</font>, <font color="Olive">n1</font> being  <a href="int_1.html#NM1">Integer</a> st <font color="Maroon">m</font> <a href="int_2.html#K3">gcd</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K3">*</a> <font color="Olive">m1</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</a> <font color="Olive">n1</font></span>)</span>
 ;<br/>


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