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

<b>let </b><font color="Maroon" title="c3">th1</font>, <font color="Maroon" title="c4">th2</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> ;<br/>



<b>assume </b><a NAME="E1:123"/><i><font color="Green" title="E103">A1</font></i>: 
( <font color="Maroon" title="c3">th1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default">0,1</span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> &amp; <font color="Maroon" title="c4">th2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default">0,1</span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> &amp; <a href="sin_cos.html#K32" title="SIN_COS:func.32">tan</a>  <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c3">th1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="sin_cos.html#K32" title="SIN_COS:func.32">tan</a>  <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">th2</font> )
 ;<br/>

<a NAME="E2:123"/><b>then </b><i><font color="Green" title="E104">A2</font></i>: 
( 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c3">th1</font> &amp; <font color="Maroon" title="c3">th1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 1 &amp; 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c4">th2</font> &amp; <font color="Maroon" title="c4">th2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 1 )
 
<b>by </b><i><a class="ref" href="rcomp_1.html#T47" title="RCOMP_1:th.47">RCOMP_1:47</a></i>;<br/>
<b>assume </b><a NAME="E3:123"/><i><font color="Green" title="E105">A3</font></i>: 
<font color="Maroon" title="c3">th1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">th2</font>
 ;<br/>

<a NAME="E4:123"/>
 ex <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>  st <br/>( <font color="Olive" title="b1">th</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default">0,1</span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> &amp;  <a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> <a href="sin_cos.html#K32" title="SIN_COS:func.32">tan</a> ,<font color="Olive" title="b1">th</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="123_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="123_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="123_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:123_1_1_1"/>
( <font color="Maroon" title="c3">th1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c4">th2</font> or <font color="Maroon" title="c4">th2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c3">th1</font> )
 </span><b>by </b><i><a class="txt" href="sin_cos.html#E3:123"><i><font color="Green" title="E105">A3</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="123_1_1_1_1"><b>suppose </b></a><a NAME="E1:123_1_1_1_1"/><i><font color="Green" title="E106">A4</font></i>: 
<font color="Maroon" title="c3">th1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c4">th2</font>
 ;<br/><div class="add"><a NAME="E2:123_1_1_1_1"/><i><font color="Green" title="E107">A5</font></i>: 
( <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c3">th1</font>,<font color="Maroon" title="c4">th2</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> is  <a href="rcomp_1.html#V3" title="RCOMP_1:attr.3">open</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  &amp; <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c3">th1</font>,<font color="Maroon" title="c4">th2</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default">0,1</span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> &amp; <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c3">th1</font>,<font color="Maroon" title="c4">th2</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="123_1_1_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:123_1_1_1_1_1"/><i><font color="Green" title="E108">A6</font></i>: 
for <font color="Olive" title="b1">th</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b1">th</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c3">th1</font>,<font color="Maroon" title="c4">th2</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> holds <br/><font color="Olive" title="b1">th</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default">0,1</span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="123_1_1_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">th</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/>

<b>assume </b><a NAME="E1:123_1_1_1_1_1_1"/>
<font color="Maroon" title="c5">th</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c3">th1</font>,<font color="Maroon" title="c4">th2</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
 ;<br/>

<a NAME="E2:123_1_1_1_1_1_1"/><b>then </b><i><font color="Green" title="E109">A7</font></i>: 
( <font color="Maroon" title="c3">th1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c5">th</font> &amp; <font color="Maroon" title="c5">th</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c4">th2</font> )
 
<b>by </b><i><a class="ref" href="rcomp_1.html#T47" title="RCOMP_1:th.47">RCOMP_1:47</a></i>;<br/>
<a NAME="E3:123_1_1_1_1_1_1"/><b>then </b><i><font color="Green" title="E110">A8</font></i>: 
0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c5">th</font>
 
<b>by </b><i><a class="txt" href="sin_cos.html#E2:123"><i><font color="Green" title="E104">A2</font></i></a></i>;<br/>
<a NAME="E4:123_1_1_1_1_1_1"/>
<font color="Maroon" title="c5">th</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 1
 
<b>by </b><i><a class="txt" href="sin_cos.html#E2:123"><i><font color="Green" title="E104">A2</font></i></a>, <a class="txt" href="sin_cos.html#E2:123_1_1_1_1_1_1"><i><font color="Green" title="E109">A7</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E5:123_1_1_1_1_1_1"/>
<font color="Maroon" title="c5">th</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default">0,1</span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
 <b>by </b><i><a class="txt" href="sin_cos.html#E3:123_1_1_1_1_1_1"><i><font color="Green" title="E110">A8</font></i></a>, <a class="ref" href="rcomp_1.html#T47" title="RCOMP_1:th.47">RCOMP_1:47</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:123_1_1_1_1_1"/>
for <font color="Olive" title="b1">th</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b1">th</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c3">th1</font>,<font color="Maroon" title="c4">th2</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> holds <br/><font color="Olive" title="b1">th</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="123_1_1_1_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">th</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/>

<b>assume </b><a NAME="E1:123_1_1_1_1_1_2"/>
<font color="Maroon" title="c5">th</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c3">th1</font>,<font color="Maroon" title="c4">th2</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 ;<br/>

<a NAME="E2:123_1_1_1_1_1_2"/><b>then </b><i><font color="Green" title="E109">A9</font></i>: 
( <font color="Maroon" title="c3">th1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">th</font> &amp; <font color="Maroon" title="c5">th</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">th2</font> )
 
<b>by </b><i><a class="ref" href="rcomp_1.html#T48" title="RCOMP_1:th.48">RCOMP_1:48</a></i>;<br/>
<a NAME="E3:123_1_1_1_1_1_2"/>
<font color="Maroon" title="c5">th</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 
<b>by </b><i><a class="txt" href="sin_cos.html#E2:123"><i><font color="Green" title="E104">A2</font></i></a>, <a class="txt" href="sin_cos.html#E2:123_1_1_1_1_1_2"><i><font color="Green" title="E109">A9</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E4:123_1_1_1_1_1_2"/>
<font color="Maroon" title="c5">th</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 <b>by </b><i><a class="txt" href="sin_cos.html#E2:123_1_1_1_1_1_2"><i><font color="Green" title="E109">A9</font></i></a>, <a class="txt" href="sin_cos.html#E2:123"><i><font color="Green" title="E104">A2</font></i></a>, <a class="ref" href="rcomp_1.html#T48" title="RCOMP_1:th.48">RCOMP_1:48</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:123_1_1_1_1_1"/>
( <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c3">th1</font>,<font color="Maroon" title="c4">th2</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> is  <a href="rcomp_1.html#V3" title="RCOMP_1:attr.3">open</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  &amp; <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c3">th1</font>,<font color="Maroon" title="c4">th2</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default">0,1</span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> &amp; <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c3">th1</font>,<font color="Maroon" title="c4">th2</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> )
 <b>by </b><i><a class="txt" href="sin_cos.html#E1:123_1_1_1_1_1"><i><font color="Green" title="E108">A6</font></i></a>, <a class="ref" href="subset_1.html#T7" title="SUBSET_1:th.7">SUBSET_1:7</a></i>;<br/>


</div><b>end;</b></div><a NAME="E3:123_1_1_1_1"/><b>then </b><i><font color="Green" title="E108">A10</font></i>: 
 <a href="sin_cos.html#K32" title="SIN_COS:func.32">tan</a>  <a href="fdiff_1.html#R2" title="FDIFF_1:pred.2">is_differentiable_on</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c3">th1</font>,<font color="Maroon" title="c4">th2</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
 <b>by </b><i><a class="txt" href="sin_cos.html#E130"><i><font color="Green" title="E100">Lm21</font></i></a>, <a class="ref" href="fdiff_1.html#T34" title="FDIFF_1:th.34">FDIFF_1:34</a></i>;<br/><a NAME="E4:123_1_1_1_1"/>
 <a href="sin_cos.html#K32" title="SIN_COS:func.32">tan</a>  <a href="fcont_1.html#R2" title="FCONT_1:pred.2">is_continuous_on</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c3">th1</font>,<font color="Maroon" title="c4">th2</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 <b>by </b><i><a class="txt" href="sin_cos.html#E2:123_1_1_1_1"><i><font color="Green" title="E107">A5</font></i></a>, <a class="ref" href="sin_cos.html#T76" target="_self" title="SIN_COS:th.76">Th76</a>, <a class="ref" href="fcont_1.html#T17" title="FCONT_1:th.17">FCONT_1:17</a></i>;<br/><b>then consider </b><font color="Maroon" title="c5">r</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E6:123_1_1_1_1"/><i><font color="Green" title="E109">A11</font></i>: 
( <font color="Maroon" title="c5">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c3">th1</font>,<font color="Maroon" title="c4">th2</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> &amp;  <a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> <a href="sin_cos.html#K32" title="SIN_COS:func.32">tan</a> ,<font color="Maroon" title="c5">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 )
 <b>by </b><i><a class="txt" href="sin_cos.html#E1:123"><i><font color="Green" title="E103">A1</font></i></a>, <a class="txt" href="sin_cos.html#E1:123_1_1_1_1"><i><font color="Green" title="E106">A4</font></i></a>, <a class="txt" href="sin_cos.html#E3:123_1_1_1_1"><i><font color="Green" title="E108">A10</font></i></a>, <a class="ref" href="rolle.html#T1" title="ROLLE:th.1">ROLLE:1</a></i>;<br/><b>take </b><font color="Maroon" title="c6">th</font> = <font color="Maroon" title="c5">r</font>;<br/><b>thus </b><a NAME="E7:123_1_1_1_1"/>
( <font color="Maroon" title="c6">th</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default">0,1</span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> &amp;  <a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> <a href="sin_cos.html#K32" title="SIN_COS:func.32">tan</a> ,<font color="Maroon" title="c6">th</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 )
 <b>by </b><i><a class="txt" href="sin_cos.html#E2:123_1_1_1_1"><i><font color="Green" title="E107">A5</font></i></a>, <a class="txt" href="sin_cos.html#E6:123_1_1_1_1"><i><font color="Green" title="E109">A11</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="123_1_1_1_2"><b>suppose </b></a><a NAME="E1:123_1_1_1_2"/><i><font color="Green" title="E106">A12</font></i>: 
<font color="Maroon" title="c4">th2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c3">th1</font>
 ;<br/><div class="add"><a NAME="E2:123_1_1_1_2"/><i><font color="Green" title="E107">A13</font></i>: 
( <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c4">th2</font>,<font color="Maroon" title="c3">th1</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> is  <a href="rcomp_1.html#V3" title="RCOMP_1:attr.3">open</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  &amp; <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c4">th2</font>,<font color="Maroon" title="c3">th1</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default">0,1</span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> &amp; <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c4">th2</font>,<font color="Maroon" title="c3">th1</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="123_1_1_1_2_1"><b>proof </b></a><div class="add">

<a NAME="E1:123_1_1_1_2_1"/><i><font color="Green" title="E108">A14</font></i>: 
for <font color="Olive" title="b1">th</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b1">th</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c4">th2</font>,<font color="Maroon" title="c3">th1</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> holds <br/><font color="Olive" title="b1">th</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default">0,1</span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="123_1_1_1_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">th</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/>

<b>assume </b><a NAME="E1:123_1_1_1_2_1_1"/>
<font color="Maroon" title="c5">th</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c4">th2</font>,<font color="Maroon" title="c3">th1</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
 ;<br/>

<a NAME="E2:123_1_1_1_2_1_1"/><b>then </b><i><font color="Green" title="E109">A15</font></i>: 
( <font color="Maroon" title="c4">th2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c5">th</font> &amp; <font color="Maroon" title="c5">th</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c3">th1</font> )
 
<b>by </b><i><a class="ref" href="rcomp_1.html#T47" title="RCOMP_1:th.47">RCOMP_1:47</a></i>;<br/>
<a NAME="E3:123_1_1_1_2_1_1"/>
<font color="Maroon" title="c5">th</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 1
 
<b>by </b><i><a class="txt" href="sin_cos.html#E2:123"><i><font color="Green" title="E104">A2</font></i></a>, <a class="txt" href="sin_cos.html#E2:123_1_1_1_2_1_1"><i><font color="Green" title="E109">A15</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E4:123_1_1_1_2_1_1"/>
<font color="Maroon" title="c5">th</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default">0,1</span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
 <b>by </b><i><a class="txt" href="sin_cos.html#E2:123_1_1_1_2_1_1"><i><font color="Green" title="E109">A15</font></i></a>, <a class="txt" href="sin_cos.html#E2:123"><i><font color="Green" title="E104">A2</font></i></a>, <a class="ref" href="rcomp_1.html#T47" title="RCOMP_1:th.47">RCOMP_1:47</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:123_1_1_1_2_1"/>
for <font color="Olive" title="b1">th</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b1">th</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c4">th2</font>,<font color="Maroon" title="c3">th1</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> holds <br/><font color="Olive" title="b1">th</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="123_1_1_1_2_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">th</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/>

<b>assume </b><a NAME="E1:123_1_1_1_2_1_2"/>
<font color="Maroon" title="c5">th</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c4">th2</font>,<font color="Maroon" title="c3">th1</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 ;<br/>

<a NAME="E2:123_1_1_1_2_1_2"/><b>then </b><i><font color="Green" title="E109">A16</font></i>: 
( <font color="Maroon" title="c4">th2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">th</font> &amp; <font color="Maroon" title="c5">th</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">th1</font> )
 
<b>by </b><i><a class="ref" href="rcomp_1.html#T48" title="RCOMP_1:th.48">RCOMP_1:48</a></i>;<br/>
<a NAME="E3:123_1_1_1_2_1_2"/>
<font color="Maroon" title="c5">th</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 
<b>by </b><i><a class="txt" href="sin_cos.html#E2:123"><i><font color="Green" title="E104">A2</font></i></a>, <a class="txt" href="sin_cos.html#E2:123_1_1_1_2_1_2"><i><font color="Green" title="E109">A16</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E4:123_1_1_1_2_1_2"/>
<font color="Maroon" title="c5">th</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 <b>by </b><i><a class="txt" href="sin_cos.html#E2:123_1_1_1_2_1_2"><i><font color="Green" title="E109">A16</font></i></a>, <a class="txt" href="sin_cos.html#E2:123"><i><font color="Green" title="E104">A2</font></i></a>, <a class="ref" href="rcomp_1.html#T48" title="RCOMP_1:th.48">RCOMP_1:48</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:123_1_1_1_2_1"/>
( <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c4">th2</font>,<font color="Maroon" title="c3">th1</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> is  <a href="rcomp_1.html#V3" title="RCOMP_1:attr.3">open</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  &amp; <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c4">th2</font>,<font color="Maroon" title="c3">th1</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default">0,1</span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> &amp; <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c4">th2</font>,<font color="Maroon" title="c3">th1</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> )
 <b>by </b><i><a class="txt" href="sin_cos.html#E1:123_1_1_1_2_1"><i><font color="Green" title="E108">A14</font></i></a>, <a class="ref" href="subset_1.html#T7" title="SUBSET_1:th.7">SUBSET_1:7</a></i>;<br/>


</div><b>end;</b></div><a NAME="E3:123_1_1_1_2"/><b>then </b><i><font color="Green" title="E108">A17</font></i>: 
 <a href="sin_cos.html#K32" title="SIN_COS:func.32">tan</a>  <a href="fdiff_1.html#R2" title="FDIFF_1:pred.2">is_differentiable_on</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c4">th2</font>,<font color="Maroon" title="c3">th1</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
 <b>by </b><i><a class="txt" href="sin_cos.html#E130"><i><font color="Green" title="E100">Lm21</font></i></a>, <a class="ref" href="fdiff_1.html#T34" title="FDIFF_1:th.34">FDIFF_1:34</a></i>;<br/><a NAME="E4:123_1_1_1_2"/>
 <a href="sin_cos.html#K32" title="SIN_COS:func.32">tan</a>  <a href="fcont_1.html#R2" title="FCONT_1:pred.2">is_continuous_on</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c4">th2</font>,<font color="Maroon" title="c3">th1</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 <b>by </b><i><a class="txt" href="sin_cos.html#E2:123_1_1_1_2"><i><font color="Green" title="E107">A13</font></i></a>, <a class="ref" href="sin_cos.html#T76" target="_self" title="SIN_COS:th.76">Th76</a>, <a class="ref" href="fcont_1.html#T17" title="FCONT_1:th.17">FCONT_1:17</a></i>;<br/><b>then consider </b><font color="Maroon" title="c5">r</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E6:123_1_1_1_2"/><i><font color="Green" title="E109">A18</font></i>: 
( <font color="Maroon" title="c5">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c4">th2</font>,<font color="Maroon" title="c3">th1</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> &amp;  <a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> <a href="sin_cos.html#K32" title="SIN_COS:func.32">tan</a> ,<font color="Maroon" title="c5">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 )
 <b>by </b><i><a class="txt" href="sin_cos.html#E1:123"><i><font color="Green" title="E103">A1</font></i></a>, <a class="txt" href="sin_cos.html#E1:123_1_1_1_2"><i><font color="Green" title="E106">A12</font></i></a>, <a class="txt" href="sin_cos.html#E3:123_1_1_1_2"><i><font color="Green" title="E108">A17</font></i></a>, <a class="ref" href="rolle.html#T1" title="ROLLE:th.1">ROLLE:1</a></i>;<br/><b>take </b><font color="Maroon" title="c6">th</font> = <font color="Maroon" title="c5">r</font>;<br/><b>thus </b><a NAME="E7:123_1_1_1_2"/>
( <font color="Maroon" title="c6">th</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default">0,1</span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> &amp;  <a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> <a href="sin_cos.html#K32" title="SIN_COS:func.32">tan</a> ,<font color="Maroon" title="c6">th</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 )
 <b>by </b><i><a class="txt" href="sin_cos.html#E2:123_1_1_1_2"><i><font color="Green" title="E107">A13</font></i></a>, <a class="txt" href="sin_cos.html#E6:123_1_1_1_2"><i><font color="Green" title="E109">A18</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E2:123_1"/>
 ex <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>  st <br/>( <font color="Olive" title="b1">th</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default">0,1</span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> &amp;  <a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> <a href="sin_cos.html#K32" title="SIN_COS:func.32">tan</a> ,<font color="Olive" title="b1">th</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 )
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:123"/>
contradiction
 <b>by </b><i><a class="txt" href="sin_cos.html#E130"><i><font color="Green" title="E100">Lm21</font></i></a></i>;<br/>


</div>
