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<span class="kw">let </span><font color="Maroon" title="c1">z1</font>, <font color="Maroon" title="c2">z2</font> be   <a href="xcmplx_0.html#V1" title="XCMPLX_0:attr.1">complex</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="Red">::  thesis: </font></span></a><span class="hide">  <a href="sin_cos.html#K19" title="SIN_COS:func.19">exp</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">z1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">z2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K19" title="SIN_COS:func.19">exp</a> <font color="Maroon" title="c1">z1</font></span>)</span> <a href="sin_cos.html#K2" title="SIN_COS:func.2">*</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K19" title="SIN_COS:func.19">exp</a> <font color="Maroon" title="c2">z2</font></span>)</span></span><br/>

<span class="lab"><font color="Green" title="E38">A1</font></span>:  <a href="sin_cos.html#K19" title="SIN_COS:func.19">exp</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">z1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">z2</font></span>)</span> = 
 <a href="comseq_3.html#K7" title="COMSEQ_3:func.7">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">z1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">z2</font></span>)</span> <a href="sin_cos.html#K7" title="SIN_COS:func.7">ExpSeq</a> </span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="sin_cos.html#D18" target="_self" title="SIN_COS:def.18">Def18</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="comseq_3.html#K7" title="COMSEQ_3:func.7">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">z1</font> <a href="sin_cos.html#K7" title="SIN_COS:func.7">ExpSeq</a> </span>)</span></span>)</span> <a href="sin_cos.html#K2" title="SIN_COS:func.2">*</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K7" title="COMSEQ_3:func.7">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">z2</font> <a href="sin_cos.html#K7" title="SIN_COS:func.7">ExpSeq</a> </span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="sin_cos.html#E42"><span class="lab"><font color="Green" title="E36">Lm2</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="sin_cos.html#K19" title="SIN_COS:func.19">exp</a> <font color="Maroon" title="c1">z1</font></span>)</span> <a href="sin_cos.html#K2" title="SIN_COS:func.2">*</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K7" title="COMSEQ_3:func.7">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">z2</font> <a href="sin_cos.html#K7" title="SIN_COS:func.7">ExpSeq</a> </span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="sin_cos.html#D18" target="_self" title="SIN_COS:def.18">Def18</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="sin_cos.html#K19" title="SIN_COS:func.19">exp</a> <font color="Maroon" title="c1">z1</font></span>)</span> <a href="sin_cos.html#K2" title="SIN_COS:func.2">*</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K19" title="SIN_COS:func.19">exp</a> <font color="Maroon" title="c2">z2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="sin_cos.html#D18" target="_self" title="SIN_COS:def.18">Def18</a></span>
;<br/>
<span class="kw">thus </span><a NAME="E2:49"/>
 <a href="sin_cos.html#K19" title="SIN_COS:func.19">exp</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">z1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">z2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K19" title="SIN_COS:func.19">exp</a> <font color="Maroon" title="c1">z1</font></span>)</span> <a href="sin_cos.html#K2" title="SIN_COS:func.2">*</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K19" title="SIN_COS:func.19">exp</a> <font color="Maroon" title="c2">z2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:49"><span class="lab"><font color="Green" title="E38">A1</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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