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<a NAME="E1:11"/><i><font color="Green">E4</font></i>: 
 <a href="sin_cos.html#K22">cos</a>  <a href="fdiff_1.html#R2">is_differentiable_on</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">3 <a href="binop_2.html#K12">/</a> 2</span>)</span> <a href="binop_2.html#K11">*</a> <a href="sin_cos.html#K33">PI</a> </span>)</span>,<span class="p2">(<span class="default">2 <a href="binop_2.html#K11">*</a> <a href="sin_cos.html#K33">PI</a> </span>)</span></span><a href="rcomp_1.html#K2">.[</a></span>
 
<b>by </b><i><a class="ref" href="fdiff_1.html#T34">FDIFF_1:34</a>, <a class="ref" href="sin_cos.html#T72">SIN_COS:72</a></i>;<br/>
<a NAME="E2:11"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="real_1.html#NM1">Real</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">3 <a href="binop_2.html#K12">/</a> 2</span>)</span> <a href="binop_2.html#K11">*</a> <a href="sin_cos.html#K33">PI</a> </span>)</span>,<span class="p2">(<span class="default">2 <a href="binop_2.html#K11">*</a> <a href="sin_cos.html#K33">PI</a> </span>)</span></span><a href="rcomp_1.html#K2">.[</a></span> holds <br/> <a href="fdiff_1.html#K1">diff</a> <a href="sin_cos.html#K22">cos</a> ,<font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="real_1.html#NM1">Real</a>;<br/>

<b>assume </b><a NAME="E1:11_1"/><i><font color="Green">E5</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">3 <a href="binop_2.html#K12">/</a> 2</span>)</span> <a href="binop_2.html#K11">*</a> <a href="sin_cos.html#K33">PI</a> </span>)</span>,<span class="p2">(<span class="default">2 <a href="binop_2.html#K11">*</a> <a href="sin_cos.html#K33">PI</a> </span>)</span></span><a href="rcomp_1.html#K2">.[</a></span>
 ;<br/>

<i><font color="Green">E6</font></i>:  <a href="fdiff_1.html#K1">diff</a> <a href="sin_cos.html#K22">cos</a> ,<font color="Maroon">c<sub>1</sub></font> = 
 <a href="binop_2.html#K7">-</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K19">sin</a>  <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K33">PI</a>  <a href="binop_2.html#K9">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sin_cos.html#K33">PI</a>  <a href="binop_2.html#K12">/</a> 2</span>)</span> <a href="binop_2.html#K9">+</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binop_2.html#K10">-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default">3 <a href="binop_2.html#K12">/</a> 2</span>)</span> <a href="binop_2.html#K11">*</a> <a href="sin_cos.html#K33">PI</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="sin_cos.html#T72">SIN_COS:72</a></i>
<br/>.= 
 <a href="binop_2.html#K7">-</a> <span class="p1">(<span class="default"><a href="binop_2.html#K7">-</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K19">sin</a>  <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sin_cos.html#K33">PI</a>  <a href="binop_2.html#K12">/</a> 2</span>)</span> <a href="binop_2.html#K9">+</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binop_2.html#K10">-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default">3 <a href="binop_2.html#K12">/</a> 2</span>)</span> <a href="binop_2.html#K11">*</a> <a href="sin_cos.html#K33">PI</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="sin_cos.html#T83">SIN_COS:83</a></i>
<br/>.= 
<a href="sin_cos.html#K22">cos</a>  <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binop_2.html#K10">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">3 <a href="binop_2.html#K12">/</a> 2</span>)</span> <a href="binop_2.html#K11">*</a> <a href="sin_cos.html#K33">PI</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="sin_cos.html#T83">SIN_COS:83</a></i>
;<br/>
<a NAME="E3:11_1"/>
( <span class="p1">(<span class="default">3 <a href="binop_2.html#K12">/</a> 2</span>)</span> <a href="binop_2.html#K11">*</a> <a href="sin_cos.html#K33">PI</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 2 <a href="binop_2.html#K11">*</a> <a href="sin_cos.html#K33">PI</a>  )
 
<b>by </b><i><a class="txt" href="sin_cos2.html#E1:11_1"><i><font color="Green">E5</font></i></a>, <a class="ref" href="rcomp_1.html#T47">RCOMP_1:47</a></i>;<br/>
<a NAME="E4:11_1"/><b>then </b>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default">3 <a href="binop_2.html#K12">/</a> 2</span>)</span> <a href="binop_2.html#K11">*</a> <a href="sin_cos.html#K33">PI</a> </span>)</span> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">3 <a href="binop_2.html#K12">/</a> 2</span>)</span> <a href="binop_2.html#K11">*</a> <a href="sin_cos.html#K33">PI</a> </span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>1</sub></font> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">3 <a href="binop_2.html#K12">/</a> 2</span>)</span> <a href="binop_2.html#K11">*</a> <a href="sin_cos.html#K33">PI</a> </span>)</span> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">3 <a href="binop_2.html#K12">/</a> 2</span>)</span> <a href="binop_2.html#K11">*</a> <a href="sin_cos.html#K33">PI</a> </span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K11">*</a> <a href="sin_cos.html#K33">PI</a> </span>)</span> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">3 <a href="binop_2.html#K12">/</a> 2</span>)</span> <a href="binop_2.html#K11">*</a> <a href="sin_cos.html#K33">PI</a> </span>)</span> )
 
<b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/>
<a NAME="E5:11_1"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">3 <a href="binop_2.html#K12">/</a> 2</span>)</span> <a href="binop_2.html#K11">*</a> <a href="sin_cos.html#K33">PI</a> </span>)</span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default">0,<span class="p2">(<span class="default"><a href="sin_cos.html#K33">PI</a>  <a href="binop_2.html#K12">/</a> 2</span>)</span></span><a href="rcomp_1.html#K2">.[</a></span>
 
<b>by </b><i><a class="ref" href="rcomp_1.html#T47">RCOMP_1:47</a></i>;<br/>
<b>hence </b><a NAME="E6:11_1"/>
 <a href="fdiff_1.html#K1">diff</a> <a href="sin_cos.html#K22">cos</a> ,<font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="txt" href="sin_cos2.html#E2:11_1"><i><font color="Green">E6</font></i></a>, <a class="ref" href="sin_cos.html#T85">SIN_COS:85</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:11"/>
 <a href="sin_cos.html#K22">cos</a>  <a href="rfunct_2.html#R1">is_increasing_on</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">3 <a href="binop_2.html#K12">/</a> 2</span>)</span> <a href="binop_2.html#K11">*</a> <a href="sin_cos.html#K33">PI</a> </span>)</span>,<span class="p2">(<span class="default">2 <a href="binop_2.html#K11">*</a> <a href="sin_cos.html#K33">PI</a> </span>)</span></span><a href="rcomp_1.html#K2">.[</a></span>
 <b>by </b><i><a class="txt" href="sin_cos2.html#E1:11"><i><font color="Green">E4</font></i></a>, <a class="txt" href="sin_cos2.html#E1"><i><font color="Green">Lemma1</font></i></a>, <a class="ref" href="rolle.html#T9">ROLLE:9</a></i>;<br/>


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