<?xml version="1.0"?>
<div class="add">

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





<div><i><font color="Green">E40</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="70_1"><b>now </b></a><div class="add"><b>given </b><font color="Maroon">x</font> being   <a href="int_1.html#NM1">Integer</a>, <font color="Maroon">y</font> being   <a href="int_1.html#NM1">Integer</a><b> such that </b><a NAME="E1:70_1"/><i><font color="Green">E41</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">x</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="Maroon">y</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font>
 ;<br/><a NAME="E2:70_1"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">m</font> <a href="int_2.html#K3">gcd</a> <font color="Maroon">n</font> <a href="int_1.html#R2">divides</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="ref" href="int_2.html#T31">INT_2:31</a></i>;<br/><a NAME="E3:70_1"/>
<font color="Maroon">m</font> <a href="int_2.html#K3">gcd</a> <font color="Maroon">n</font> <a href="int_1.html#R2">divides</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="int_2.html#T32">INT_2:32</a></i>;<br/><b>hence </b><a NAME="E4:70_1"/>
<font color="Maroon">m</font> <a href="int_2.html#K3">gcd</a> <font color="Maroon">n</font> <a href="int_1.html#R2">divides</a> <font color="Maroon">k</font>
 <b>by </b><i>, <a class="txt" href="wsierp_1.html#E14"><i><font color="Green">Lemma53</font></i></a>, </i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:70_2"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">m</font> <a href="int_2.html#K3">gcd</a> <font color="Maroon">n</font> <a href="int_1.html#R2">divides</a> <font color="Maroon">k</font>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_2_1"><b>now </b></a><div class="add"><b>consider </b><font color="Maroon">l</font> being   <a href="int_1.html#NM1">Integer</a><b> such that </b><br/><a NAME="E2:70_2_1"/><i><font color="Green">E50</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="int_2.html#K3">gcd</a> <font color="Maroon">n</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">l</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font>
 <b>by </b><i>, <a class="ref" href="int_1.html#D9">INT_1:def 9</a></i>;<br/><b>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="E4:70_2_1"/><i><font color="Green">E102</font></i>: 
<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> <font color="Maroon">n1</font></span>)</span>
 <b>by </b><i><a class="txt" href="wsierp_1.html#E9"><i><font color="Green">Lemma46</font></i></a></i>;<br/><a NAME="E5:70_2_1"/>
<font color="Maroon">k</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"><font color="Maroon">m1</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">l</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"><font color="Maroon">n1</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">l</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="wsierp_1.html#E14"><i><font color="Green">Lemma53</font></i></a>, <a class="txt" href="wsierp_1.html#E15"><i><font color="Green">Lemma54</font></i></a></i>;<br/><b>hence </b><a NAME="E6:70_2_1"/>
 ex <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="int_1.html#NM1">Integer</a> st <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K3">*</a> <font color="Olive">x</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">y</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font>
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:70_2"/>
 ex <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="int_1.html#NM1">Integer</a> st <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K3">*</a> <font color="Olive">x</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">y</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:70"/>
(  ex <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="int_1.html#NM1">Integer</a> st <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K3">*</a> <font color="Olive">x</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">y</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> iff <font color="Maroon">m</font> <a href="int_2.html#K3">gcd</a> <font color="Maroon">n</font> <a href="int_1.html#R2">divides</a> <font color="Maroon">k</font> )
 <b>by </b><i><a class="txt" href="wsierp_1.html#E13"><i><font color="Green">Lemma51</font></i></a></i>;<br/>


</div>
