<?xml version="1.0"?>
<div class="add">

<a NAME="E1:32"/>
for <font color="Olive" title="b1">x</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>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <a href="fdiff_9.html#K1" title="FDIFF_9:func.1">sec</a>  holds <br/>( <font color="Olive" title="b1">x</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K36" title="SIN_COS:func.36">PI</a> </span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <a href="fdiff_9.html#K1" title="FDIFF_9:func.1">sec</a>  &amp; <font color="Olive" title="b1">x</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K36" title="SIN_COS:func.36">PI</a> </span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <a href="fdiff_9.html#K1" title="FDIFF_9:func.1">sec</a>  &amp; <a href="fdiff_9.html#K1" title="FDIFF_9:func.1">sec</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="fdiff_9.html#K1" title="FDIFF_9:func.1">sec</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">x</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K36" title="SIN_COS:func.36">PI</a> </span>)</span></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">x</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="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <a href="fdiff_9.html#K1" title="FDIFF_9:func.1">sec</a>  implies ( <font color="Maroon" title="c1">x</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K36" title="SIN_COS:func.36">PI</a> </span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <a href="fdiff_9.html#K1" title="FDIFF_9:func.1">sec</a>  &amp; <font color="Maroon" title="c1">x</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K36" title="SIN_COS:func.36">PI</a> </span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <a href="fdiff_9.html#K1" title="FDIFF_9:func.1">sec</a>  &amp; <a href="fdiff_9.html#K1" title="FDIFF_9:func.1">sec</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="fdiff_9.html#K1" title="FDIFF_9:func.1">sec</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K36" title="SIN_COS:func.36">PI</a> </span>)</span></span>)</span> ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:32_1"/><span class="lab"><font color="Green" title="E15">A1</font></span>: 
<font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <a href="fdiff_9.html#K1" title="FDIFF_9:func.1">sec</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">x</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K36" title="SIN_COS:func.36">PI</a> </span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <a href="fdiff_9.html#K1" title="FDIFF_9:func.1">sec</a>  &amp; <font color="Maroon" title="c1">x</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K36" title="SIN_COS:func.36">PI</a> </span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <a href="fdiff_9.html#K1" title="FDIFF_9:func.1">sec</a>  &amp; <a href="fdiff_9.html#K1" title="FDIFF_9:func.1">sec</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="fdiff_9.html#K1" title="FDIFF_9:func.1">sec</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K36" title="SIN_COS:func.36">PI</a> </span>)</span></span>)</span> )</span><br/>

<a NAME="E2:32_1"/><span class="kw">then </span>
<font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <a href="sin_cos.html#K23" title="SIN_COS:func.23">cos</a> </span>)</span> <a href="xboole_0.html#K4" title="XBOOLE_0:func.4">\</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K23" title="SIN_COS:func.23">cos</a>  <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rfunct_1.html#D8" title="RFUNCT_1:def.8">RFUNCT_1:def 8</a></span>;<br/>
<a NAME="E3:32_1"/><span class="kw">then </span>
( <font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <a href="sin_cos.html#K23" title="SIN_COS:func.23">cos</a>  &amp; not <font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="sin_cos.html#K23" title="SIN_COS:func.23">cos</a>  <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/>
<a NAME="E4:32_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E16">A2</font></span>: 
not <a href="sin_cos.html#K23" title="SIN_COS:func.23">cos</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D13" title="FUNCT_1:def.13">FUNCT_1:def 13</a></span>;<br/>
<a NAME="E5:32_1"/><span class="kw">then </span>
<a href="sin_cos.html#K23" title="SIN_COS:func.23">cos</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<a NAME="E6:32_1"/><span class="kw">then </span>
<a href="sin_cos.html#K23" title="SIN_COS:func.23">cos</a>  <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K36" title="SIN_COS:func.36">PI</a> </span>)</span></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="sin_cos.html#T83" title="SIN_COS:th.83">SIN_COS:83</a></span>;<br/>
<a NAME="E7:32_1"/><span class="kw">then </span>
not <a href="sin_cos.html#K23" title="SIN_COS:func.23">cos</a>  <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K36" title="SIN_COS:func.36">PI</a> </span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<a NAME="E8:32_1"/><span class="kw">then </span>
( <font color="Maroon" title="c1">x</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K36" title="SIN_COS:func.36">PI</a> </span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <a href="sin_cos.html#K23" title="SIN_COS:func.23">cos</a>  &amp; not <font color="Maroon" title="c1">x</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K36" title="SIN_COS:func.36">PI</a> </span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="sin_cos.html#K23" title="SIN_COS:func.23">cos</a>  <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D13" title="FUNCT_1:def.13">FUNCT_1:def 13</a>, <a class="ref" href="sin_cos.html#T27" title="SIN_COS:th.27">SIN_COS:27</a></span>;<br/>
<a NAME="E9:32_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E17">A3</font></span>: 
<font color="Maroon" title="c1">x</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K36" title="SIN_COS:func.36">PI</a> </span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <a href="sin_cos.html#K23" title="SIN_COS:func.23">cos</a> </span>)</span> <a href="xboole_0.html#K4" title="XBOOLE_0:func.4">\</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K23" title="SIN_COS:func.23">cos</a>  <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/>
<a NAME="E10:32_1"/>
<a href="sin_cos.html#K23" title="SIN_COS:func.23">cos</a>  <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K36" title="SIN_COS:func.36">PI</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="sin_cos.html#K23" title="SIN_COS:func.23">cos</a>  <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p3">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K36" title="SIN_COS:func.36">PI</a> </span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K36" title="SIN_COS:func.36">PI</a> </span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="funct_9.html#E24"><span class="lab"><font color="Green" title="E12">L2</font></span></a>, <a class="ref" href="funct_9.html#T1" target="_self" title="FUNCT_9:th.1">Th1</a>, <a class="ref" href="sin_cos.html#T27" title="SIN_COS:th.27">SIN_COS:27</a></span>;<br/>
<a NAME="E11:32_1"/><span class="kw">then </span>
( <font color="Maroon" title="c1">x</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K36" title="SIN_COS:func.36">PI</a> </span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <a href="sin_cos.html#K23" title="SIN_COS:func.23">cos</a>  &amp; not <font color="Maroon" title="c1">x</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K36" title="SIN_COS:func.36">PI</a> </span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="sin_cos.html#K23" title="SIN_COS:func.23">cos</a>  <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:32_1"><span class="lab"><font color="Green" title="E16">A2</font></span></a>, <a class="ref" href="funct_1.html#D13" title="FUNCT_1:def.13">FUNCT_1:def 13</a>, <a class="ref" href="sin_cos.html#T27" title="SIN_COS:th.27">SIN_COS:27</a></span>;<br/>
<a NAME="E12:32_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E18">A4</font></span>: 
<font color="Maroon" title="c1">x</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K36" title="SIN_COS:func.36">PI</a> </span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <a href="sin_cos.html#K23" title="SIN_COS:func.23">cos</a> </span>)</span> <a href="xboole_0.html#K4" title="XBOOLE_0:func.4">\</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K23" title="SIN_COS:func.23">cos</a>  <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/>
<a NAME="E13:32_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E19">A5</font></span>: 
( <font color="Maroon" title="c1">x</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K36" title="SIN_COS:func.36">PI</a> </span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <a href="fdiff_9.html#K1" title="FDIFF_9:func.1">sec</a>  &amp; <font color="Maroon" title="c1">x</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K36" title="SIN_COS:func.36">PI</a> </span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <a href="fdiff_9.html#K1" title="FDIFF_9:func.1">sec</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:32_1"><span class="lab"><font color="Green" title="E17">A3</font></span></a>, <a class="ref" href="rfunct_1.html#D8" title="RFUNCT_1:def.8">RFUNCT_1:def 8</a></span>;<br/>
<a href="fdiff_9.html#K1" title="FDIFF_9:func.1">sec</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K36" title="SIN_COS:func.36">PI</a> </span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="sin_cos.html#K23" title="SIN_COS:func.23">cos</a>  <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p3">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K36" title="SIN_COS:func.36">PI</a> </span>)</span></span>)</span></span>)</span> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:32_1"><span class="lab"><font color="Green" title="E19">A5</font></span></a>, <a class="ref" href="rfunct_1.html#D8" title="RFUNCT_1:def.8">RFUNCT_1:def 8</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="sin_cos.html#K23" title="SIN_COS:func.23">cos</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="xcmplx_0.html#K5" title="XCMPLX_0:func.5">"</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="sin_cos.html#T83" title="SIN_COS:th.83">SIN_COS:83</a></span>
<br/>.= 
<a href="fdiff_9.html#K1" title="FDIFF_9:func.1">sec</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:32_1"><span class="lab"><font color="Green" title="E15">A1</font></span></a>, <a class="ref" href="rfunct_1.html#D8" title="RFUNCT_1:def.8">RFUNCT_1:def 8</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E15:32_1"/>
( <font color="Maroon" title="c1">x</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K36" title="SIN_COS:func.36">PI</a> </span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <a href="fdiff_9.html#K1" title="FDIFF_9:func.1">sec</a>  &amp; <font color="Maroon" title="c1">x</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K36" title="SIN_COS:func.36">PI</a> </span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <a href="fdiff_9.html#K1" title="FDIFF_9:func.1">sec</a>  &amp; <a href="fdiff_9.html#K1" title="FDIFF_9:func.1">sec</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="fdiff_9.html#K1" title="FDIFF_9:func.1">sec</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K36" title="SIN_COS:func.36">PI</a> </span>)</span></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:32_1"><span class="lab"><font color="Green" title="E17">A3</font></span></a>, <a class="txt" href="#E12:32_1"><span class="lab"><font color="Green" title="E18">A4</font></span></a>, <a class="ref" href="rfunct_1.html#D8" title="RFUNCT_1:def.8">RFUNCT_1:def 8</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/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E2:32"/>
 <a href="fdiff_9.html#K1" title="FDIFF_9:func.1">sec</a>  is 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K36" title="SIN_COS:func.36">PI</a>  <a href="funct_9.html#V1" title="FUNCT_9:attr.1">-periodic</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_9.html#T1" target="_self" title="FUNCT_9:th.1">Th1</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/>


</div>
