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



<b>assume </b><a NAME="E1:26"/><i><font color="Green">E18</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 or <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 ;<br/>

<a NAME="E2:26"/>
<font color="Maroon">c<sub>1</sub></font> <a href="int_2.html#K3" target="_self">gcd</a> <font color="Maroon">c<sub>2</sub></font> <a href="int_1.html#R2">divides</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="int_2.html#T31" target="_self">Th31</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>3</sub></font> being   <a href="int_1.html#NM1">Integer</a><b> such that </b><br/><a NAME="E4:26"/><i><font color="Green">E19</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" target="_self">gcd</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</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="E5:26"/>
<font color="Maroon">c<sub>1</sub></font> <a href="int_2.html#K3" target="_self">gcd</a> <font color="Maroon">c<sub>2</sub></font> <a href="int_1.html#R2">divides</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="int_2.html#T31" target="_self">Th31</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>4</sub></font> being   <a href="int_1.html#NM1">Integer</a><b> such that </b><br/><a NAME="E7:26"/><i><font color="Green">E20</font></i>: 
<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" target="_self">gcd</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="int_1.html#D9">INT_1:def 9</a></i>;<br/>
<a NAME="E8:26"/>
<font color="Maroon">c<sub>3</sub></font> <a href="int_2.html#K3" target="_self">gcd</a> <font color="Maroon">c<sub>4</sub></font> <a href="int_1.html#R2">divides</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="int_2.html#T31" target="_self">Th31</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="int_1.html#NM1">Integer</a><b> such that </b><br/><a NAME="E10:26"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="int_2.html#K3" target="_self">gcd</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="int_1.html#D9">INT_1:def 9</a></i>;<br/>
<a NAME="E11:26"/>
<font color="Maroon">c<sub>3</sub></font> <a href="int_2.html#K3" target="_self">gcd</a> <font color="Maroon">c<sub>4</sub></font> <a href="int_1.html#R2">divides</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="ref" href="int_2.html#T31" target="_self">Th31</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>6</sub></font> being   <a href="int_1.html#NM1">Integer</a><b> such that </b><br/><a NAME="E13:26"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="int_2.html#K3" target="_self">gcd</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="int_1.html#D9">INT_1:def 9</a></i>;<br/>
<a NAME="E14:26"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="int_2.html#K3" target="_self">gcd</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="int_2.html#K3" target="_self">gcd</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="int_2.html#E4:26"><i><font color="Green">E19</font></i></a>, <a class="txt" href="int_2.html#E10:26"><i><font color="Green">E21</font></i></a></i>;<br/>
<a NAME="E15:26"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="int_2.html#K3" target="_self">gcd</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="int_2.html#K3" target="_self">gcd</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="int_2.html#E7:26"><i><font color="Green">E20</font></i></a>, <a class="txt" href="int_2.html#E13:26"><i><font color="Green">E22</font></i></a></i>;<br/>
<a NAME="E16:26"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="int_2.html#K3" target="_self">gcd</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="int_2.html#K3" target="_self">gcd</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="int_1.html#R2">divides</a> <font color="Maroon">c<sub>1</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="int_2.html#K3" target="_self">gcd</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="int_2.html#K3" target="_self">gcd</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="int_1.html#R2">divides</a> <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="txt" href="int_2.html#E14:26"><i><font color="Green">E23</font></i></a>, <a class="ref" href="int_1.html#D9">INT_1:def 9</a></i>;<br/>
<a NAME="E17:26"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="int_2.html#K3" target="_self">gcd</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="int_2.html#K3" target="_self">gcd</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="int_1.html#R2">divides</a> <font color="Maroon">c<sub>1</sub></font> <a href="int_2.html#K3" target="_self">gcd</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="int_2.html#T33" target="_self">Th33</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="int_1.html#NM1">Integer</a><b> such that </b><br/><a NAME="E19:26"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="int_2.html#K3" target="_self">gcd</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="int_2.html#K3" target="_self">gcd</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="int_2.html#K3" target="_self">gcd</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="ref" href="int_1.html#D9">INT_1:def 9</a></i>;<br/>
<a NAME="E20:26"/><i><font color="Green">E25</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="int_2.html#K3" target="_self">gcd</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> 1 <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" target="_self">gcd</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="int_2.html#K3" target="_self">gcd</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <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="int_2.html#E19:26"><i><font color="Green">E24</font></i></a></i>;<br/>
<a NAME="E21:26"/>
<font color="Maroon">c<sub>1</sub></font> <a href="int_2.html#K3" target="_self">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="int_2.html#E1:26"><i><font color="Green">E18</font></i></a>, <a class="ref" href="int_2.html#T35" target="_self">Th35</a></i>;<br/>
<a NAME="E22:26"/><b>then </b>
1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="int_2.html#K3" target="_self">gcd</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="int_2.html#E20:26"><i><font color="Green">E25</font></i></a>, <a class="ref" href="xcmplx_1.html#T5">XCMPLX_1:5</a></i>;<br/>
<a NAME="E23:26"/><b>then </b>
( <font color="Maroon">c<sub>3</sub></font> <a href="int_2.html#K3" target="_self">gcd</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">c<sub>3</sub></font> <a href="int_2.html#K3" target="_self">gcd</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1 )
 
<b>by </b><i><a class="ref" href="int_1.html#T22">INT_1:22</a></i>;<br/>
<a NAME="E24:26"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="int_2.html#R1" target="_self">are_relative_prime</a> 
 
<b>by </b><i><a class="ref" href="int_2.html#D4" target="_self">Def4</a>, <a class="ref" href="int_2.html#T8" target="_self">Th8</a></i>;<br/>
<b>hence </b><a NAME="E25:26"/>
ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="int_1.html#NM1">Integer</a> st <br/>( <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" target="_self">gcd</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Olive">b<sub>1</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" target="_self">gcd</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="int_2.html#R1" target="_self">are_relative_prime</a>  )
 <b>by </b><i><a class="txt" href="int_2.html#E4:26"><i><font color="Green">E19</font></i></a>, <a class="txt" href="int_2.html#E7:26"><i><font color="Green">E20</font></i></a></i>;<br/>


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