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<span class="kw">let </span><font color="Maroon" title="c3">th</font> be   <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">th</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <a href="sin_cos.html#K25" title="SIN_COS:func.25">exp_R</a> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">th</font> &amp; <a href="sin_cos.html#K25" title="SIN_COS:func.25">exp_R</a> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">th</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:88"/>
<font color="Maroon" title="c3">th</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <a href="sin_cos.html#K25" title="SIN_COS:func.25">exp_R</a> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">th</font> &amp; <a href="sin_cos.html#K25" title="SIN_COS:func.25">exp_R</a> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">th</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 )</span><br/>

<a NAME="E2:88"/><span class="kw">then </span><span class="lab"><font color="Green" title="E70">A1</font></span>: 
<a href="sin_cos.html#K25" title="SIN_COS:func.25">exp_R</a> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c3">th</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="sin_cos.html#T52" target="_self" title="SIN_COS:th.52">Th57</a></span>;<br/>
<span class="lab"><font color="Green" title="E71">A2</font></span>: <span class="p1">(<span class="default"><a href="sin_cos.html#K25" title="SIN_COS:func.25">exp_R</a> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c3">th</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K25" title="SIN_COS:func.25">exp_R</a> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">th</font></span>)</span> = 
<a href="sin_cos.html#K25" title="SIN_COS:func.25">exp_R</a> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c3">th</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">th</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="sin_cos.html#E85"><span class="lab"><font color="Green" title="E66">Lm10</font></span></a></span>
<br/>.= 
1
<span class="kw">by</span> <span class="lab"><a class="txt" href="sin_cos.html#E87"><span class="lab"><font color="Green" title="E67">Lm11</font></span></a></span>
;<br/>
<a NAME="E4:88"/><span class="kw">then </span><span class="lab"><font color="Green" title="E72">A3</font></span>: 
<a href="sin_cos.html#K25" title="SIN_COS:func.25">exp_R</a> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">th</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K25" title="SIN_COS:func.25">exp_R</a> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c3">th</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xcmplx_1.html#T73" title="XCMPLX_1:th.73">XCMPLX_1:73</a></span>;<br/>
<span class="kw">thus </span><a NAME="E5:88"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <a href="sin_cos.html#K25" title="SIN_COS:func.25">exp_R</a> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">th</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:88"><span class="lab"><font color="Green" title="E70">A1</font></span></a>, <a class="txt" href="#E3:88"><span class="lab"><font color="Green" title="E71">A2</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <a href="sin_cos.html#K25" title="SIN_COS:func.25">exp_R</a> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">th</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1</span><br/>

<span class="kw">thus </span><a NAME="E6:88"/>
<a href="sin_cos.html#K25" title="SIN_COS:func.25">exp_R</a> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">th</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:88"><span class="lab"><font color="Green" title="E70">A1</font></span></a>, <a class="txt" href="#E4:88"><span class="lab"><font color="Green" title="E72">A3</font></span></a>, <a class="ref" href="xreal_1.html#T211" title="XREAL_1:th.211">XREAL_1:211</a></span>; <a class="txt" 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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