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<span class="kw">let </span><font color="Maroon" title="c1">th</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K16" title="SIN_COS:func.16">exp</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">th</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a></span>)</span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; (  for <font color="Olive" title="b1">th</font> being   <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  holds <br/> (  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K18" title="SIN_COS:func.18">sin</a> <font color="Olive" title="b1">th</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 &amp;  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K21" title="SIN_COS:func.21">cos</a> <font color="Olive" title="b1">th</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 ) ) )</span><br/>

<span class="kw">thus </span><span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K16" title="SIN_COS:func.16">exp</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">th</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a></span>)</span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> = 
<span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><a href="comseq_3.html#K10" title="COMSEQ_3:func.10">Sum</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">th</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a></span>)</span> <a href="sin_cos.html#K4" title="SIN_COS:func.4">ExpSeq</a></span>)</span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="sin_cos.html#D14" target="_self" title="SIN_COS:def.14">Def18</a></span>
<br/>.= 
1
<span class="kw">by</span> <span class="lab"><a class="txt" href="sin_cos.html#E50"><span class="lab"><font color="Green" title="E42">Lm5</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">  for <font color="Olive" title="b1">th</font> being   <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  holds <br/> (  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K18" title="SIN_COS:func.18">sin</a> <font color="Olive" title="b1">th</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 &amp;  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K21" title="SIN_COS:func.21">cos</a> <font color="Olive" title="b1">th</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 )</span><br/>

<span class="kw">let </span><font color="Maroon" title="c2">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"> (  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K18" title="SIN_COS:func.18">sin</a> <font color="Maroon" title="c2">th</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 &amp;  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K21" title="SIN_COS:func.21">cos</a> <font color="Maroon" title="c2">th</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 )</span><br/>

<a NAME="E2:59"/><span class="lab"><font color="Green" title="E43">A1</font></span>: 
<font color="Maroon" title="c2">th</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:59"/>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K18" title="SIN_COS:func.18">sin</a> <font color="Maroon" title="c2">th</font></span>)</span> <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="sin_cos.html#E50"><span class="lab"><font color="Green" title="E42">Lm5</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="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K21" title="SIN_COS:func.21">cos</a> <font color="Maroon" title="c2">th</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1</span><br/>

<span class="kw">thus </span><a NAME="E4:59"/>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K21" title="SIN_COS:func.21">cos</a> <font color="Maroon" title="c2">th</font></span>)</span> <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="#E2:59"><span class="lab"><font color="Green" title="E43">A1</font></span></a>, <a class="txt" href="sin_cos.html#E50"><span class="lab"><font color="Green" title="E42">Lm5</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"> verum</span><br/>


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