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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/><b>let </b><font color="Maroon">m1</font> be   <a href="int_1.html#NM1">Integer</a>;<br/><b>let </b><font color="Maroon">n1</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/>









<b>assume </b><a NAME="E1:71"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">m</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <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> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> )
 ;<br/>

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



<b>assume </b><a NAME="E2:71"/><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="E3:71"/><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="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i><a class="txt" href="wsierp_1.html#E13"><i><font color="Green">Lemma51</font></i></a>, <a class="ref" href="int_2.html#T35">INT_2:35</a></i>;<br/>
<b>consider </b><font color="Maroon">m2</font> being   <a href="int_1.html#NM1">Integer</a>, <font color="Maroon">n2</font> being   <a href="int_1.html#NM1">Integer</a><b> such that </b><br/><a NAME="E5:71"/><i><font color="Green">E49</font></i>: 
( <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <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">m2</font> &amp; <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <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">n2</font> &amp; <font color="Maroon">m2</font>,<font color="Maroon">n2</font> <a href="int_2.html#R1">are_relative_prime</a>  )
 <b>by </b><i><a class="txt" href="wsierp_1.html#E13"><i><font color="Green">Lemma51</font></i></a>, <a class="ref" href="int_2.html#T38">INT_2:38</a></i>;<br/>
<a NAME="E6:71"/><i><font color="Green">E50</font></i>: 
( <font color="Maroon">m2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> <a href="xcmplx_0.html#K7">/</a> <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> &amp; <font color="Maroon">n2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="xcmplx_0.html#K7">/</a> <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> )
 
<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>, <a class="ref" href="xcmplx_1.html#T90">XCMPLX_1:90</a></i>;<br/>
<a NAME="E7:71"/><i><font color="Green">E102</font></i>: 
<font color="Maroon">n2</font> <a href="int_2.html#K3">gcd</a> <font color="Maroon">m2</font> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="txt" href="wsierp_1.html#E15"><i><font color="Green">Lemma54</font></i></a>, <a class="ref" href="int_2.html#D4">INT_2:def 4</a></i>;<br/>
<a NAME="E8:71"/><i><font color="Green">E105</font></i>: 
( <font color="Maroon">m2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">n2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 
<b>by </b><i><a class="txt" href="wsierp_1.html#E13"><i><font color="Green">Lemma51</font></i></a>, <a class="txt" href="wsierp_1.html#E15"><i><font color="Green">Lemma54</font></i></a></i>;<br/>
<a NAME="E9:71"/>
<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#K6">-</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="hidden.html#R1">=</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> <a href="xcmplx_0.html#K6">-</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>
 
<b>by </b><i><a class="txt" href="wsierp_1.html#E13"><i><font color="Green">Lemma51</font></i></a>, </i>;<br/>
<a NAME="E10:71"/><b>then </b>
<font color="Maroon">m</font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">m1</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">n1</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">y</font></span>)</span>
 
;<br/>
<b>then </b><i><font color="Green">E106</font></i>: <font color="Maroon">m2</font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">m1</font></span>)</span> = 
<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#K6">-</a> <font color="Maroon">y</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <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>
<b>by </b><i><a class="ref" href="wsierp_1.html#T9" target="_self">Th9</a>, <a class="ref" href="xcmplx_1.html#T75">XCMPLX_1:75</a></i>
<br/>.= 
<font color="Maroon">n2</font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">n1</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">y</font></span>)</span>
<b>by </b><i><a class="ref" href="wsierp_1.html#T9" target="_self">Th9</a>, <a class="ref" href="xcmplx_1.html#T75">XCMPLX_1:75</a></i>
;<br/>
<a NAME="E12:71"/><b>then </b><i><font color="Green">E107</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">m1</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">n2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">n1</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">y</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">m2</font>
 
<b>by </b><i><a class="txt" href="wsierp_1.html#E18"><i><font color="Green">Lemma57</font></i></a>, <a class="ref" href="xcmplx_1.html#T95">XCMPLX_1:95</a></i>;<br/>
<a NAME="E13:71"/>
<font color="Maroon">n2</font> <a href="int_1.html#R2">divides</a> <font color="Maroon">m2</font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">m1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="wsierp_1.html#E19"><i><font color="Green">Lemma58</font></i></a>, <a class="ref" href="int_1.html#D9">INT_1:def 9</a></i>;<br/>
<a NAME="E14:71"/><b>then </b>
<font color="Maroon">n2</font> <a href="int_1.html#R2">divides</a> <font color="Maroon">x</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">m1</font>
 
<b>by </b><i><a class="ref" href="wsierp_1.html#T10" target="_self">Th10</a>, <a class="txt" href="wsierp_1.html#E12"><i><font color="Green">Lemma47</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon">t</font> being   <a href="int_1.html#NM1">Integer</a><b> such that </b><br/><a NAME="E16:71"/><i><font color="Green">E108</font></i>: 
<font color="Maroon">n2</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">m1</font>
 <b>by </b><i><a class="ref" href="int_1.html#D9">INT_1:def 9</a></i>;<br/>
<a NAME="E17:71"/>
<font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">m1</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">n2</font>
 
<b>by </b><i><a class="txt" href="wsierp_1.html#E18"><i><font color="Green">Lemma57</font></i></a>, <a class="ref" href="wsierp_1.html#T12" target="_self">Th12</a>, <a class="ref" href="xcmplx_1.html#T90">XCMPLX_1:90</a></i>;<br/>
<a NAME="E18:71"/><b>then </b>
<font color="Maroon">t</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">m2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n1</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">y</font>
 
<b>by </b><i><a class="txt" href="wsierp_1.html#E18"><i><font color="Green">Lemma57</font></i></a>, <a class="ref" href="wsierp_1.html#T11" target="_self">Th11</a>, <a class="ref" href="xcmplx_1.html#T88">XCMPLX_1:88</a></i>;<br/>
<a NAME="E19:71"/><b>then </b><i><font color="Green">E109</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n1</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">m2</font></span>)</span>
 
;<br/>
<a NAME="E20:71"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m1</font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">n2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="wsierp_1.html#T12" target="_self">Th12</a></i>;<br/>
<b>hence </b><a NAME="E21:71"/>
 ex <font color="Olive">t</font> being  <a href="int_1.html#NM1">Integer</a> st <br/>( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m1</font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Olive">t</font> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K7">/</a> <span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="int_2.html#K3">gcd</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n1</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Olive">t</font> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K7">/</a> <span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="int_2.html#K3">gcd</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="wsierp_1.html#T9" target="_self">Th9</a>, <a class="ref" href="wsierp_1.html#T13" target="_self">Th13</a></i>;<br/>


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