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<b>let </b><font color="Maroon">x</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="E1:35"/>
0 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font>
 ;<br/>

<a NAME="E2:35"/><b>then </b><i><font color="Green">E21</font></i>: 
 <a href="sin_cos.html#K28">exp_R</a> <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1
 
<b>by </b><i><a class="ref" href="sin_cos7.html#T13" target="_self">Th13</a></i>;<br/>
<b>assume </b><a NAME="E3:35"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K28">exp_R</a> <font color="Maroon">x</font></span>)</span> <a href="square_1.html#K3">^2</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 <a href="hidden.html#R1">=</a> 0
 ;<br/>

<a NAME="E4:35"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K28">exp_R</a> <font color="Maroon">x</font></span>)</span> <a href="square_1.html#K3">^2</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> 1 <a href="hidden.html#R1">=</a> 0 <a href="binop_2.html#K23">+</a> 1
 
;<br/>
<a NAME="E5:35"/><b>then </b>
<span class="p1">(<span class="default"><a href="sin_cos.html#K28">exp_R</a> <font color="Maroon">x</font></span>)</span> <a href="square_1.html#K3">^2</a>  <a href="hidden.html#R1">=</a> 1
 
;<br/>
<a NAME="E6:35"/><b>then </b>
<span class="p1">(<span class="default"><a href="sin_cos.html#K28">exp_R</a> <font color="Maroon">x</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K28">exp_R</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a> 1
 
;<br/>
<a NAME="E7:35"/><b>then </b>
(  <a href="sin_cos.html#K28">exp_R</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 1 or  <a href="sin_cos.html#K28">exp_R</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="binop_2.html#K7">-</a> 1 )
 
<b>by </b><i><a class="ref" href="xcmplx_1.html#T183">XCMPLX_1:183</a></i>;<br/>
<b>hence </b><a NAME="E8:35"/>
contradiction
 <b>by </b><i><a class="txt" href="sin_cos7.html#E21"><i><font color="Green">Lemma42</font></i></a>, <a class="ref" href="sin_cos.html#T60">SIN_COS:60</a></i>;<br/>


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