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<b>let </b><font color="Maroon" title="c1">a</font>, <font color="Maroon" title="c2">b</font> be   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( <font color="Maroon" title="c1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c2">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) implies  ex <font color="Olive" title="b1">a1</font>, <font color="Olive" title="b2">b1</font> being  <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> st <br/>( <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b1">a1</font> &amp; <font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b2">b1</font> &amp; <font color="Olive" title="b1">a1</font>,<font color="Olive" title="b2">b1</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_relative_prime</a>  ) )</span><br/>



<b>assume </b><a NAME="E1:29"/><i><font color="Green" title="E24">A1</font></i>: 
( <font color="Maroon" title="c1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c2">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">a1</font>, <font color="Olive" title="b2">b1</font> being  <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> st <br/>( <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b1">a1</font> &amp; <font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b2">b1</font> &amp; <font color="Olive" title="b1">a1</font>,<font color="Olive" title="b2">b1</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_relative_prime</a>  )</span><br/>

<a NAME="E2:29"/>
<font color="Maroon" title="c1">a</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c2">b</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c1">a</font>
 
<b>by </b><i><a class="ref" href="int_2.html#D3" target="_self" title="INT_2:def.3">Def3</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c3">a1</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a><b> such that </b><br/><a NAME="E4:29"/><i><font color="Green" title="E25">A2</font></i>: 
<font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">a1</font>
 <b>by </b><i><a class="ref" href="int_1.html#D9" title="INT_1:def.9">INT_1:def 9</a></i>;<br/>
<a NAME="E5:29"/>
<font color="Maroon" title="c1">a</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c2">b</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c2">b</font>
 
<b>by </b><i><a class="ref" href="int_2.html#D3" target="_self" title="INT_2:def.3">Def3</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c4">b1</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a><b> such that </b><br/><a NAME="E7:29"/><i><font color="Green" title="E26">A3</font></i>: 
<font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">b1</font>
 <b>by </b><i><a class="ref" href="int_1.html#D9" title="INT_1:def.9">INT_1:def 9</a></i>;<br/>
<a NAME="E8:29"/>
<font color="Maroon" title="c3">a1</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c4">b1</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c3">a1</font>
 
<b>by </b><i><a class="ref" href="int_2.html#D3" target="_self" title="INT_2:def.3">Def3</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c5">a2</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a><b> such that </b><br/><a NAME="E10:29"/><i><font color="Green" title="E27">A4</font></i>: 
<font color="Maroon" title="c3">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a1</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c4">b1</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c5">a2</font>
 <b>by </b><i><a class="ref" href="int_1.html#D9" title="INT_1:def.9">INT_1:def 9</a></i>;<br/>
<a NAME="E11:29"/>
<font color="Maroon" title="c3">a1</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c4">b1</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c4">b1</font>
 
<b>by </b><i><a class="ref" href="int_2.html#D3" target="_self" title="INT_2:def.3">Def3</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c6">b2</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a><b> such that </b><br/><a NAME="E13:29"/><i><font color="Green" title="E28">A5</font></i>: 
<font color="Maroon" title="c4">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a1</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c4">b1</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c6">b2</font>
 <b>by </b><i><a class="ref" href="int_1.html#D9" title="INT_1:def.9">INT_1:def 9</a></i>;<br/>
<a NAME="E14:29"/><i><font color="Green" title="E29">A6</font></i>: 
<font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">a1</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c4">b1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c5">a2</font>
 
<b>by </b><i><a class="txt" href="#E4:29"><i><font color="Green" title="E25">A2</font></i></a>, <a class="txt" href="#E10:29"><i><font color="Green" title="E27">A4</font></i></a></i>;<br/>
<a NAME="E15:29"/>
<font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">a1</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c4">b1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c6">b2</font>
 
<b>by </b><i><a class="txt" href="#E7:29"><i><font color="Green" title="E26">A3</font></i></a>, <a class="txt" href="#E13:29"><i><font color="Green" title="E28">A5</font></i></a></i>;<br/>
<a NAME="E16:29"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a1</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c4">b1</font></span>)</span> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c1">a</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a1</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c4">b1</font></span>)</span> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c2">b</font> )
 
<b>by </b><i><a class="txt" href="#E14:29"><i><font color="Green" title="E29">A6</font></i></a>, <a class="ref" href="int_1.html#D9" title="INT_1:def.9">INT_1:def 9</a></i>;<br/>
<a NAME="E17:29"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a1</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c4">b1</font></span>)</span> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c1">a</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c2">b</font>
 
<b>by </b><i><a class="ref" href="int_2.html#D3" target="_self" title="INT_2:def.3">Def3</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c7">c</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a><b> such that </b><br/><a NAME="E19:29"/><i><font color="Green" title="E30">A7</font></i>: 
<font color="Maroon" title="c1">a</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">a1</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c4">b1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c7">c</font>
 <b>by </b><i><a class="ref" href="int_1.html#D9" title="INT_1:def.9">INT_1:def 9</a></i>;<br/>
<a NAME="E20:29"/><i><font color="Green" title="E31">A8</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="nat_1.html#K3" title="NAT_1:func.3">*</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">a1</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c4">b1</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c7">c</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E19:29"><i><font color="Green" title="E30">A7</font></i></a></i>;<br/>
<a NAME="E21:29"/>
<font color="Maroon" title="c1">a</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c2">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="#E1:29"><i><font color="Green" title="E24">A1</font></i></a>, <a class="ref" href="int_2.html#T35" target="_self" title="INT_2:th.35">Th35</a></i>;<br/>
<a NAME="E22:29"/><b>then </b>
1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a1</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c4">b1</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c7">c</font>
 
<b>by </b><i><a class="txt" href="#E20:29"><i><font color="Green" title="E31">A8</font></i></a>, <a class="ref" href="xcmplx_1.html#T5" title="XCMPLX_1:th.5">XCMPLX_1:5</a></i>;<br/>
<a NAME="E23:29"/><b>then </b>
( <font color="Maroon" title="c3">a1</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c4">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c3">a1</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c4">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1 )
 
<b>by </b><i><a class="ref" href="int_1.html#T22" title="INT_1:th.22">INT_1:22</a></i>;<br/>
<a NAME="E24:29"/><b>then </b>
<font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c4">b1</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_relative_prime</a> 
 
<b>by </b><i><a class="ref" href="int_2.html#D4" target="_self" title="INT_2:def.4">Def4</a>, <a class="ref" href="nat_1.html#T2" title="NAT_1:th.2">NAT_1:2</a></i>;<br/>
<b>hence </b><a NAME="E25:29"/>
 ex <font color="Olive" title="b1">a1</font>, <font color="Olive" title="b2">b1</font> being  <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> st <br/>( <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b1">a1</font> &amp; <font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b2">b1</font> &amp; <font color="Olive" title="b1">a1</font>,<font color="Olive" title="b2">b1</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_relative_prime</a>  )
 <b>by </b><i><a class="txt" href="#E4:29"><i><font color="Green" title="E25">A2</font></i></a>, <a class="txt" href="#E7:29"><i><font color="Green" title="E26">A3</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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