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

<a NAME="E1:122"/>
for <font color="Olive">th</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st <font color="Olive">th</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span> holds <br/><a href="sin_cos.html#K32" target="_self">tan</a>  <a href="partfun1.html#K2">|</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span> <a href="fcont_1.html#R1">is_continuous_in</a> <font color="Olive">th</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="122_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">th</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>

<b>assume </b><a NAME="E1:122_1"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">th</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span>
 ;<br/>

<a NAME="E2:122_1"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">th</font> is   <a href="real_1.html#NM1">Real</a>
 
<b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/>
<a NAME="E3:122_1"/><i><font color="Green">E56</font></i>: 
 <a href="subset_1.html#K2">[#]</a> <a href="numbers.html#K1">REAL</a>  <a href="hidden.html#R1">=</a>  <a href="numbers.html#K1">REAL</a> 
 
<b>by </b><i><a class="ref" href="subset_1.html#D4">SUBSET_1:def 4</a></i>;<br/>
<a NAME="E4:122_1"/><b>then </b>
 <a href="sin_cos.html#K19" target="_self">sin</a>  <a href="fdiff_1.html#R1">is_differentiable_in</a> <font color="Maroon">th</font>
 
<b>by </b><i>, , <a class="ref" href="fcont_3.html#T4">FCONT_3:4</a>, <a class="ref" href="fdiff_1.html#T16">FDIFF_1:16</a></i>;<br/>
<a NAME="E5:122_1"/><b>then </b><i><font color="Green">E57</font></i>: 
 <a href="sin_cos.html#K19" target="_self">sin</a>  <a href="fcont_1.html#R1">is_continuous_in</a> <font color="Maroon">th</font>
 
<b>by </b><i><a class="ref" href="fdiff_1.html#T32">FDIFF_1:32</a></i>;<br/>
<a NAME="E6:122_1"/>
 <a href="sin_cos.html#K22" target="_self">cos</a>  <a href="fdiff_1.html#R1">is_differentiable_in</a> <font color="Maroon">th</font>
 
<b>by </b><i>, , , <a class="ref" href="fcont_3.html#T4">FCONT_3:4</a>, <a class="ref" href="fdiff_1.html#T16">FDIFF_1:16</a></i>;<br/>
<a NAME="E7:122_1"/><b>then </b><i><font color="Green">E59</font></i>: 
 <a href="sin_cos.html#K22" target="_self">cos</a>  <a href="fcont_1.html#R1">is_continuous_in</a> <font color="Maroon">th</font>
 
<b>by </b><i><a class="ref" href="fdiff_1.html#T32">FDIFF_1:32</a></i>;<br/>
<a NAME="E8:122_1"/>
<a href="sin_cos.html#K22" target="_self">cos</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">th</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i>, </i>;<br/>
<a NAME="E9:122_1"/><b>then </b><i><font color="Green">E61</font></i>: 
 <a href="sin_cos.html#K32" target="_self">tan</a>  <a href="fcont_1.html#R1">is_continuous_in</a> <font color="Maroon">th</font>
 
<b>by </b><i>, , <a class="ref" href="fcont_1.html#T11">FCONT_1:11</a></i>;<br/>
<a NAME="E10:122_1"/>
( <font color="Maroon">th</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K32" target="_self">tan</a>  <a href="partfun1.html#K2">|</a> <span class="p2"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span></span>)</span> &amp; ( for <font color="Olive">rseq</font> being  <a href="seq_1.html#NM1">Real_Sequence</a>  st  <a href="relset_1.html#K5">rng</a> <font color="Olive">rseq</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K32" target="_self">tan</a>  <a href="partfun1.html#K2">|</a> <span class="p2"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span></span>)</span> &amp; <font color="Olive">rseq</font> is <a href="seq_2.html#V4">convergent</a> &amp;  <a href="seq_2.html#K2">lim</a> <font color="Olive">rseq</font> <a href="hidden.html#R1">=</a> <font color="Maroon">th</font> holds <br/>( <span class="p1">(<span class="default"><a href="sin_cos.html#K32" target="_self">tan</a>  <a href="partfun1.html#K2">|</a> <span class="p2"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span></span>)</span> <a href="rfunct_2.html#K1">*</a> <font color="Olive">rseq</font> is <a href="seq_2.html#V4">convergent</a> &amp; <span class="p1">(<span class="default"><a href="sin_cos.html#K32" target="_self">tan</a>  <a href="partfun1.html#K2">|</a> <span class="p2"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">th</font> <a href="hidden.html#R1">=</a>  <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K32" target="_self">tan</a>  <a href="partfun1.html#K2">|</a> <span class="p3"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span></span>)</span> <a href="rfunct_2.html#K1">*</a> <font color="Olive">rseq</font></span>)</span> ) ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="122_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="122_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">rseq</font> be   <a href="seq_1.html#NM1">Real_Sequence</a>;<br/><b>assume </b><a NAME="E1:122_1_1_1"/><i><font color="Green">E62</font></i>: 
(  <a href="relset_1.html#K5">rng</a> <font color="Maroon">rseq</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K32" target="_self">tan</a>  <a href="partfun1.html#K2">|</a> <span class="p2"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span></span>)</span> &amp; <font color="Maroon">rseq</font> is <a href="seq_2.html#V4">convergent</a> &amp;  <a href="seq_2.html#K2">lim</a> <font color="Maroon">rseq</font> <a href="hidden.html#R1">=</a> <font color="Maroon">th</font> )
 ;<br/><a NAME="E2:122_1_1_1"/><b>then </b><i><font color="Green">E91</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">rseq</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <a href="sin_cos.html#K32" target="_self">tan</a> 
 <b>by </b><i>, , <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E3:122_1_1_1"/><b>then </b><i><font color="Green">E92</font></i>: 
( <a href="sin_cos.html#K32" target="_self">tan</a>  <a href="rfunct_2.html#K1">*</a> <font color="Maroon">rseq</font> is <a href="seq_2.html#V4">convergent</a> &amp; <a href="sin_cos.html#K32" target="_self">tan</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">th</font> <a href="hidden.html#R1">=</a>  <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K32" target="_self">tan</a>  <a href="rfunct_2.html#K1">*</a> <font color="Maroon">rseq</font></span>)</span> )
 <b>by </b><i>, , <a class="ref" href="fcont_1.html#D1">FCONT_1:def 1</a></i>;<br/><a NAME="E4:122_1_1_1"/>
<span class="p1">(<span class="default"><a href="sin_cos.html#K32" target="_self">tan</a>  <a href="partfun1.html#K2">|</a> <span class="p2"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span></span>)</span> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">rseq</font> <a href="funct_2.html#R2">=</a> <a href="sin_cos.html#K32" target="_self">tan</a>  <a href="rfunct_2.html#K1">*</a> <font color="Maroon">rseq</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="122_1_1_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="122_1_1_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">k1</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><a NAME="E1:122_1_1_1_1_1"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">rseq</font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K5">NAT</a> 
 <b>by </b><i><a class="ref" href="seq_1.html#T3">SEQ_1:3</a></i>;<br/><a NAME="E2:122_1_1_1_1_1"/><b>then </b>
<font color="Maroon">rseq</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">k1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">rseq</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E3:122_1_1_1_1_1"/><b>then </b><i><font color="Green">E93</font></i>: 
<span class="p1">(<span class="default"><a href="sin_cos.html#K32" target="_self">tan</a>  <a href="partfun1.html#K2">|</a> <span class="p2"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">rseq</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">k1</font></span>)</span> <a href="hidden.html#R1">=</a> <a href="sin_cos.html#K32" target="_self">tan</a>  <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">rseq</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">k1</font></span>)</span>
 <b>by </b><i>, </i>;<br/><a NAME="E4:122_1_1_1_1_1"/>
<span class="p1">(<span class="default"><a href="sin_cos.html#K32" target="_self">tan</a>  <a href="partfun1.html#K2">|</a> <span class="p2"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">rseq</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">k1</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K32" target="_self">tan</a>  <a href="partfun1.html#K2">|</a> <span class="p3"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span></span>)</span> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">rseq</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">k1</font>
 <b>by </b><i>, <a class="ref" href="rfunct_2.html#T21">RFUNCT_2:21</a></i>;<br/><b>hence </b><a NAME="E5:122_1_1_1_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K32" target="_self">tan</a>  <a href="partfun1.html#K2">|</a> <span class="p3"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span></span>)</span> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">rseq</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">k1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K32" target="_self">tan</a>  <a href="rfunct_2.html#K1">*</a> <font color="Maroon">rseq</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">k1</font>
 <b>by </b><i>, , <a class="ref" href="rfunct_2.html#T21">RFUNCT_2:21</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:122_1_1_1_1"/>
<span class="p1">(<span class="default"><a href="sin_cos.html#K32" target="_self">tan</a>  <a href="partfun1.html#K2">|</a> <span class="p2"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span></span>)</span> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">rseq</font> <a href="funct_2.html#R2">=</a> <a href="sin_cos.html#K32" target="_self">tan</a>  <a href="rfunct_2.html#K1">*</a> <font color="Maroon">rseq</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T113">FUNCT_2:113</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E5:122_1_1_1"/>
( <span class="p1">(<span class="default"><a href="sin_cos.html#K32" target="_self">tan</a>  <a href="partfun1.html#K2">|</a> <span class="p2"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span></span>)</span> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">rseq</font> is <a href="seq_2.html#V4">convergent</a> &amp; <span class="p1">(<span class="default"><a href="sin_cos.html#K32" target="_self">tan</a>  <a href="partfun1.html#K2">|</a> <span class="p2"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">th</font> <a href="hidden.html#R1">=</a>  <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K32" target="_self">tan</a>  <a href="partfun1.html#K2">|</a> <span class="p3"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span></span>)</span> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">rseq</font></span>)</span> )
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:122_1_1"/>
( <font color="Maroon">th</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K32" target="_self">tan</a>  <a href="partfun1.html#K2">|</a> <span class="p2"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span></span>)</span> &amp; ( for <font color="Olive">rseq</font> being  <a href="seq_1.html#NM1">Real_Sequence</a>  st  <a href="relset_1.html#K5">rng</a> <font color="Olive">rseq</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K32" target="_self">tan</a>  <a href="partfun1.html#K2">|</a> <span class="p2"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span></span>)</span> &amp; <font color="Olive">rseq</font> is <a href="seq_2.html#V4">convergent</a> &amp;  <a href="seq_2.html#K2">lim</a> <font color="Olive">rseq</font> <a href="hidden.html#R1">=</a> <font color="Maroon">th</font> holds <br/>( <span class="p1">(<span class="default"><a href="sin_cos.html#K32" target="_self">tan</a>  <a href="partfun1.html#K2">|</a> <span class="p2"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span></span>)</span> <a href="rfunct_2.html#K1">*</a> <font color="Olive">rseq</font> is <a href="seq_2.html#V4">convergent</a> &amp; <span class="p1">(<span class="default"><a href="sin_cos.html#K32" target="_self">tan</a>  <a href="partfun1.html#K2">|</a> <span class="p2"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">th</font> <a href="hidden.html#R1">=</a>  <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K32" target="_self">tan</a>  <a href="partfun1.html#K2">|</a> <span class="p3"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span></span>)</span> <a href="rfunct_2.html#K1">*</a> <font color="Olive">rseq</font></span>)</span> ) ) )
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E11:122_1"/>
<a href="sin_cos.html#K32" target="_self">tan</a>  <a href="partfun1.html#K2">|</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span> <a href="fcont_1.html#R1">is_continuous_in</a> <font color="Maroon">th</font>
 <b>by </b><i><a class="ref" href="fcont_1.html#D1">FCONT_1:def 1</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:122"/>
 <a href="sin_cos.html#K32" target="_self">tan</a>  <a href="fcont_1.html#R2">is_continuous_on</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i>, <a class="ref" href="fcont_1.html#D2">FCONT_1:def 2</a></i>;<br/>


</div>
