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<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font> be   <a href="int_1.html#NM1">Integer</a>;<br/>









<b>assume </b><a NAME="E1:72"/><i><font color="Green">E51</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> )
 ;<br/>

<b>let </b><font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>7</sub></font> be   <a href="int_1.html#NM1">Integer</a>;<br/>



<b>assume </b><a NAME="E2:72"/><i><font color="Green">E52</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/>

<a NAME="E3:72"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="int_2.html#K3">gcd</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i><a class="txt" href="wsierp_1.html#E1:72"><i><font color="Green">E51</font></i></a>, <a class="ref" href="int_2.html#T35">INT_2:35</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>9</sub></font> being   <a href="int_1.html#NM1">Integer</a><b> such that </b><br/><a NAME="E5:72"/><i><font color="Green">E54</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="int_2.html#K3">gcd</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="int_2.html#K3">gcd</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font> <a href="int_2.html#R1">are_relative_prime</a>  )
 <b>by </b><i><a class="txt" href="wsierp_1.html#E1:72"><i><font color="Green">E51</font></i></a>, <a class="ref" href="int_2.html#T38">INT_2:38</a></i>;<br/>
<a NAME="E6:72"/><i><font color="Green">E55</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="int_2.html#K3">gcd</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="int_2.html#K3">gcd</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> )
 
<b>by </b><i><a class="txt" href="wsierp_1.html#E3:72"><i><font color="Green">E53</font></i></a>, <a class="txt" href="wsierp_1.html#E5:72"><i><font color="Green">E54</font></i></a>, <a class="ref" href="xcmplx_1.html#T90">XCMPLX_1:90</a></i>;<br/>
<a NAME="E7:72"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="int_2.html#K3">gcd</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="txt" href="wsierp_1.html#E5:72"><i><font color="Green">E54</font></i></a>, <a class="ref" href="int_2.html#D4">INT_2:def 4</a></i>;<br/>
<a NAME="E8:72"/><i><font color="Green">E57</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 
<b>by </b><i><a class="txt" href="wsierp_1.html#E1:72"><i><font color="Green">E51</font></i></a>, <a class="txt" href="wsierp_1.html#E5:72"><i><font color="Green">E54</font></i></a></i>;<br/>
<a NAME="E9:72"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="wsierp_1.html#E1:72"><i><font color="Green">E51</font></i></a>, <a class="txt" href="wsierp_1.html#E2:72"><i><font color="Green">E52</font></i></a></i>;<br/>
<a NAME="E10:72"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
 
;<br/>
<b>then </b><i><font color="Green">E58</font></i>: <font color="Maroon">c<sub>8</sub></font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="int_2.html#K3">gcd</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="wsierp_1.html#E6:72"><i><font color="Green">E55</font></i></a>, <a class="ref" href="xcmplx_1.html#T75">XCMPLX_1:75</a></i>
<br/>.= 
<font color="Maroon">c<sub>9</sub></font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="wsierp_1.html#E6:72"><i><font color="Green">E55</font></i></a>, <a class="ref" href="xcmplx_1.html#T75">XCMPLX_1:75</a></i>
;<br/>
<a NAME="E12:72"/><b>then </b><i><font color="Green">E59</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="txt" href="wsierp_1.html#E8:72"><i><font color="Green">E57</font></i></a>, <a class="ref" href="xcmplx_1.html#T95">XCMPLX_1:95</a></i>;<br/>
<a NAME="E13:72"/>
<font color="Maroon">c<sub>9</sub></font> <a href="int_1.html#R2">divides</a> <font color="Maroon">c<sub>8</sub></font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="wsierp_1.html#E11:72"><i><font color="Green">E58</font></i></a>, <a class="ref" href="int_1.html#D9">INT_1:def 9</a></i>;<br/>
<a NAME="E14:72"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font> <a href="int_1.html#R2">divides</a> <font color="Maroon">c<sub>6</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="wsierp_1.html#E7:72"><i><font color="Green">E56</font></i></a>, <a class="ref" href="wsierp_1.html#T36" target="_self">Th36</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>10</sub></font> being   <a href="int_1.html#NM1">Integer</a><b> such that </b><br/><a NAME="E16:72"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="int_1.html#D9">INT_1:def 9</a></i>;<br/>
<a NAME="E17:72"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="txt" href="wsierp_1.html#E8:72"><i><font color="Green">E57</font></i></a>, <a class="txt" href="wsierp_1.html#E16:72"><i><font color="Green">E60</font></i></a>, <a class="ref" href="xcmplx_1.html#T90">XCMPLX_1:90</a></i>;<br/>
<a NAME="E18:72"/><b>then </b>
<font color="Maroon">c<sub>10</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="wsierp_1.html#E8:72"><i><font color="Green">E57</font></i></a>, <a class="txt" href="wsierp_1.html#E12:72"><i><font color="Green">E59</font></i></a>, <a class="ref" href="xcmplx_1.html#T88">XCMPLX_1:88</a></i>;<br/>
<a NAME="E19:72"/><b>then </b><i><font color="Green">E61</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
 
;<br/>
<a NAME="E20:72"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="wsierp_1.html#E16:72"><i><font color="Green">E60</font></i></a></i>;<br/>
<b>hence </b><a NAME="E21:72"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="int_1.html#NM1">Integer</a> st <br/>( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="xcmplx_0.html#K7">/</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="int_2.html#K3">gcd</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K7">/</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="int_2.html#K3">gcd</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="wsierp_1.html#E6:72"><i><font color="Green">E55</font></i></a>, <a class="txt" href="wsierp_1.html#E19:72"><i><font color="Green">E61</font></i></a></i>;<br/>


</div>
