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<b>let </b><font color="Maroon">c<sub>1</sub></font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>

<b>assume </b><a NAME="E1:46"/><i><font color="Green">E65</font></i>: 
 <a href="sin_cos.html#K23">cos</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K33">PI</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 1
 ;<br/>

<b>set </b><font color="Maroon">c<sub>2</sub></font> = <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="int_1.html#K1">/]</a></span>;<br/>
<b>assume </b><a NAME="E2:46"/>
not <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K4">INT</a> 
 ;<br/>

<a NAME="E3:46"/><b>then </b>
not <font color="Maroon">c<sub>1</sub></font> is <a href="int_1.html#V1">integer</a>
 
<b>by </b><i><a class="ref" href="int_1.html#D2">INT_1:def 2</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>3</sub></font> =  <a href="int_1.html#K4">frac</a> <font color="Maroon">c<sub>1</sub></font> as  <a href="xxreal_0.html#V2">positive</a> <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  <b>by </b><i><a class="ref" href="int_1.html#T72">INT_1:72</a></i>;<br/>
<a NAME="E5:46"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="int_1.html#K1">/]</a></span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="int_1.html#T68">INT_1:68</a></i>;<br/>
<i><font color="Green">E67</font></i>:  <a href="sin_cos.html#K23">cos</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K33">PI</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="int_1.html#K1">/]</a></span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> = 
 <a href="sin_cos.html#K23">cos</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K33">PI</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p3"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="int_1.html#K1">/]</a></span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K33">PI</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>

<br/>.= 
 <a href="sin_cos.html#K23">cos</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <a href="sin_cos.html#K33">PI</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="complex2.html#T10">COMPLEX2:10</a></i>
;<br/>
<b>reconsider </b><font color="Maroon">c<sub>4</sub></font> = 2 as  <a href="xxreal_0.html#V2">positive</a> <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  ;<br/>
<a NAME="E8:46"/>
<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;</a> 1
 
<b>by </b><i><a class="ref" href="int_1.html#T69">INT_1:69</a></i>;<br/>
<a NAME="E9:46"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="xcmplx_0.html#K3">*</a> <a href="sin_cos.html#K33">PI</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="xcmplx_0.html#K3">*</a> <a href="sin_cos.html#K33">PI</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/>
<b>hence </b><a NAME="E10:46"/>
contradiction
 <b>by </b><i><a class="txt" href="sin_cos6.html#E1:46"><i><font color="Green">E65</font></i></a>, <a class="txt" href="sin_cos6.html#E5:46"><i><font color="Green">E66</font></i></a>, <a class="txt" href="sin_cos6.html#E6:46"><i><font color="Green">E67</font></i></a>, <a class="ref" href="sin_cos6.html#T34" target="_self">Th34</a></i>;<br/>


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