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<span class="kw">given </span><font color="Maroon" title="c1">a</font>, <font color="Maroon" title="c2">b</font>, <font color="Maroon" title="c3">c</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><a NAME="E2:46"/><span class="lab"><font color="Green" title="E46">A1</font></span>: 
17 <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="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">c</font>
 <span class="kw">and </span><br/><a NAME="E3:46"/><span class="lab"><font color="Green" title="E47">A2</font></span>: 
<font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 <span class="kw">and </span><br/><a NAME="E4:46"/><span class="lab"><font color="Green" title="E48">A3</font></span>: 
<font color="Maroon" title="c2">b</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 <span class="kw">and </span><br/><a NAME="E5:46"/><span class="lab"><font color="Green" title="E49">A4</font></span>: 
<font color="Maroon" title="c3">c</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 <span class="kw">and </span><br/><a NAME="E6:46"/><span class="lab"><font color="Green" title="E50">A5</font></span>: 
<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font> <a href="number03.html#R1" title="NUMBER03:pred.1">are_mutually_coprime</a> 
 ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="number07.html#D1" target="_self" onmouseover="rs('number07/D1')" onmouseout="rh()">NUMBER07:def 1</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E7:46"/><span class="lab"><font color="Green" title="E51">A6</font></span>: 
17 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> 8</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 
;<br/>
<a NAME="E8:46"/><span class="kw">then </span>
<font color="Maroon" title="c1">a</font> is  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:46"><span class="lab"><font color="Green" title="E46">A1</font></span></a>, <a class="txt" href="#E6:46"><span class="lab"><font color="Green" title="E50">A5</font></span></a>, <a class="ref" href="number07.html#T7" target="_self" onmouseover="rs('number07/T7')" onmouseout="rh()">Th7</a></span>;<br/>
<a NAME="E9:46"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A7</font></span>: 
( <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 or <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 5 or <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 7 or <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 9 or <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 11 or <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 13 )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:46"><span class="lab"><font color="Green" title="E46">A1</font></span></a>, <a class="txt" href="#E3:46"><span class="lab"><font color="Green" title="E47">A2</font></span></a>, <a class="txt" href="#E4:46"><span class="lab"><font color="Green" title="E48">A3</font></span></a>, <a class="txt" href="#E5:46"><span class="lab"><font color="Green" title="E49">A4</font></span></a>, <a class="txt" href="number07.html#E49"><span class="lab"><font color="Green" title="E45">Lm4</font></span></a>, <a class="txt" href="number07.html#E48"><span class="lab"><font color="Green" title="E44">Lm3</font></span></a></span>;<br/>
<a NAME="E10:46"/>
<font color="Maroon" title="c2">b</font> is  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:46"><span class="lab"><font color="Green" title="E46">A1</font></span></a>, <a class="txt" href="#E6:46"><span class="lab"><font color="Green" title="E50">A5</font></span></a>, <a class="txt" href="#E7:46"><span class="lab"><font color="Green" title="E51">A6</font></span></a>, <a class="ref" href="number07.html#T7" target="_self" onmouseover="rs('number07/T7')" onmouseout="rh()">Th7</a></span>;<br/>
<a NAME="E11:46"/><span class="kw">then </span><span class="lab"><font color="Green" title="E53">A8</font></span>: 
( <font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 or <font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 5 or <font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 7 or <font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 9 or <font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 11 or <font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 13 )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:46"><span class="lab"><font color="Green" title="E46">A1</font></span></a>, <a class="txt" href="#E3:46"><span class="lab"><font color="Green" title="E47">A2</font></span></a>, <a class="txt" href="#E4:46"><span class="lab"><font color="Green" title="E48">A3</font></span></a>, <a class="txt" href="#E5:46"><span class="lab"><font color="Green" title="E49">A4</font></span></a>, <a class="txt" href="number07.html#E49"><span class="lab"><font color="Green" title="E45">Lm4</font></span></a>, <a class="txt" href="number07.html#E48"><span class="lab"><font color="Green" title="E44">Lm3</font></span></a></span>;<br/>
<a NAME="E12:46"/><span class="lab"><font color="Green" title="E54">A9</font></span>: 
<font color="Maroon" title="c3">c</font> is  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:46"><span class="lab"><font color="Green" title="E46">A1</font></span></a>, <a class="txt" href="#E6:46"><span class="lab"><font color="Green" title="E50">A5</font></span></a>, <a class="txt" href="#E7:46"><span class="lab"><font color="Green" title="E51">A6</font></span></a>, <a class="ref" href="number07.html#T7" target="_self" onmouseover="rs('number07/T7')" onmouseout="rh()">Th7</a></span>;<br/>
<a NAME="E13:46"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">c</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">b</font>
 
;<br/>
<a NAME="E14:46"/><span class="kw">then </span><span class="lab"><font color="Green" title="E55">A10</font></span>: 
( <font color="Maroon" title="c3">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 or <font color="Maroon" title="c3">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 5 or <font color="Maroon" title="c3">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 7 or <font color="Maroon" title="c3">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 9 or <font color="Maroon" title="c3">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 11 or <font color="Maroon" title="c3">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 13 )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:46"><span class="lab"><font color="Green" title="E46">A1</font></span></a>, <a class="txt" href="#E3:46"><span class="lab"><font color="Green" title="E47">A2</font></span></a>, <a class="txt" href="#E4:46"><span class="lab"><font color="Green" title="E48">A3</font></span></a>, <a class="txt" href="#E5:46"><span class="lab"><font color="Green" title="E49">A4</font></span></a>, <a class="txt" href="#E12:46"><span class="lab"><font color="Green" title="E54">A9</font></span></a>, <a class="txt" href="number07.html#E49"><span class="lab"><font color="Green" title="E45">Lm4</font></span></a>, <a class="txt" href="number07.html#E48"><span class="lab"><font color="Green" title="E44">Lm3</font></span></a></span>;<br/>
<a NAME="E15:46"/>
 not 3,3 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> 3 <a href="int_2.html#R1" title="INT_2:pred.1">are_coprime</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="number07.html#T1" target="_self" onmouseover="rs('number07/T1')" onmouseout="rh()">Th1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E16:46"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:46"><span class="lab"><font color="Green" title="E46">A1</font></span></a>, <a class="txt" href="#E6:46"><span class="lab"><font color="Green" title="E50">A5</font></span></a>, <a class="txt" href="#E9:46"><span class="lab"><font color="Green" title="E52">A7</font></span></a>, <a class="txt" href="#E11:46"><span class="lab"><font color="Green" title="E53">A8</font></span></a>, <a class="txt" href="#E14:46"><span class="lab"><font color="Green" title="E55">A10</font></span></a>, <a class="ref" href="euler_1.html#T1" onmouseover="rs('euler_1/T1')" onmouseout="rh()">EULER_1:1</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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