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<b>let </b><font color="Maroon">c<sub>3</sub></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="E1:130"/><i><font color="Green">E110</font></i>: 
<font color="Maroon">c<sub>3</sub></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="E2:130"/><b>then </b><i><font color="Green">E111</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 
<b>by </b><i><a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/>
<a NAME="E3:130"/>
<a href="sin_cos.html#K19" target="_self">sin</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <a href="sin_cos.html#K19" target="_self">sin</a>  <a href="seq_1.html#K2">.</a> 0
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="130_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="130_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="130_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:130_1_1_1"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 0 or 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>3</sub></font> )
 </span><b>by </b><i><a class="txt" href="sin_cos.html#E1:130"><i><font color="Green">E110</font></i></a>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="130_1_1_1_1"><b>suppose </b></a><a NAME="E1:130_1_1_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>hence </b><a NAME="E2:130_1_1_1_1"/>
<a href="sin_cos.html#K19" target="_self">sin</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <a href="sin_cos.html#K19" target="_self">sin</a>  <a href="seq_1.html#K2">.</a> 0
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="130_1_1_1_2"><b>suppose </b></a><a NAME="E1:130_1_1_1_2"/><i><font color="Green">E112</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><div class="add"><a NAME="E2:130_1_1_1_2"/><i><font color="Green">E113</font></i>: 
 <a href="sin_cos.html#K19" target="_self">sin</a>  <a href="fdiff_1.html#R2">is_differentiable_on</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default">0,<font color="Maroon">c<sub>3</sub></font></span><a href="rcomp_1.html#K2">.[</a></span>
 <b>by </b><i><a class="ref" href="sin_cos.html#T73" target="_self">Th73</a>, <a class="ref" href="fdiff_1.html#T34">FDIFF_1:34</a></i>;<br/><a NAME="E3:130_1_1_1_2"/><i><font color="Green">E114</font></i>: 
<font color="Maroon">c<sub>3</sub></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="E4:130_1_1_1_2"/>
 <a href="sin_cos.html#K19" target="_self">sin</a>  <a href="fcont_1.html#R2">is_continuous_on</a>  <a href="numbers.html#K1">REAL</a> 
 <b>by </b><i><a class="ref" href="sin_cos.html#T73" target="_self">Th73</a>, <a class="ref" href="fdiff_1.html#T33">FDIFF_1:33</a></i>;<br/><a NAME="E5:130_1_1_1_2"/><b>then </b>
 <a href="sin_cos.html#K19" target="_self">sin</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,<font color="Maroon">c<sub>3</sub></font></span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i><a class="ref" href="fcont_1.html#T17">FCONT_1:17</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>4</sub></font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E7:130_1_1_1_2"/><i><font color="Green">E115</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default">0,<font color="Maroon">c<sub>3</sub></font></span><a href="rcomp_1.html#K2">.[</a></span> &amp;  <a href="fdiff_1.html#K1">diff</a> <a href="sin_cos.html#K19" target="_self">sin</a> ,<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K19" target="_self">sin</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K19" target="_self">sin</a>  <a href="seq_1.html#K2">.</a> 0</span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="xcmplx_0.html#K6">-</a> 0</span>)</span> )
 <b>by </b><i><a class="txt" href="sin_cos.html#E1:130_1_1_1_2"><i><font color="Green">E112</font></i></a>, <a class="txt" href="sin_cos.html#E2:130_1_1_1_2"><i><font color="Green">E113</font></i></a>, <a class="txt" href="sin_cos.html#E3:130_1_1_1_2"><i><font color="Green">E114</font></i></a>, <a class="ref" href="rolle.html#T3">ROLLE:3</a></i>;<br/><a NAME="E8:130_1_1_1_2"/><i><font color="Green">E116</font></i>: 
 <a href="fdiff_1.html#K1">diff</a> <a href="sin_cos.html#K19" target="_self">sin</a> ,<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <a href="sin_cos.html#K22" target="_self">cos</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="sin_cos.html#T73" target="_self">Th73</a></i>;<br/><a NAME="E9:130_1_1_1_2"/>
( 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="sin_cos.html#E7:130_1_1_1_2"><i><font color="Green">E115</font></i></a>, <a class="ref" href="rcomp_1.html#T47">RCOMP_1:47</a></i>;<br/><a NAME="E10:130_1_1_1_2"/><b>then </b>
( 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;</a> 1 )
 <b>by </b><i><a class="txt" href="sin_cos.html#E2:130"><i><font color="Green">E111</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E11:130_1_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>4</sub></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>
 <b>by </b><i><a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/><a NAME="E12:130_1_1_1_2"/><b>then </b><i><font color="Green">E117</font></i>: 
<a href="sin_cos.html#K22" target="_self">cos</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="ref" href="sin_cos.html#T74" target="_self">Th74</a></i>;<br/><a NAME="E13:130_1_1_1_2"/>
<span class="p1">(<span class="default"><a href="sin_cos.html#K19" target="_self">sin</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K19" target="_self">sin</a>  <a href="seq_1.html#K2">.</a> 0</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K22" target="_self">cos</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="sin_cos.html#E1:130_1_1_1_2"><i><font color="Green">E112</font></i></a>, <a class="txt" href="sin_cos.html#E7:130_1_1_1_2"><i><font color="Green">E115</font></i></a>, <a class="txt" href="sin_cos.html#E8:130_1_1_1_2"><i><font color="Green">E116</font></i></a>, <a class="ref" href="xcmplx_1.html#T88">XCMPLX_1:88</a></i>;<br/><a NAME="E14:130_1_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="sin_cos.html#K19" target="_self">sin</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K19" target="_self">sin</a>  <a href="seq_1.html#K2">.</a> 0</span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="txt" href="sin_cos.html#E1:130_1_1_1_2"><i><font color="Green">E112</font></i></a>, <a class="txt" href="sin_cos.html#E12:130_1_1_1_2"><i><font color="Green">E117</font></i></a>, <a class="ref" href="real_2.html#T122">REAL_2:122</a></i>;<br/><b>hence </b><a NAME="E15:130_1_1_1_2"/>
<a href="sin_cos.html#K19" target="_self">sin</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <a href="sin_cos.html#K19" target="_self">sin</a>  <a href="seq_1.html#K2">.</a> 0
 <b>by </b><i><a class="ref" href="real_2.html#T106">REAL_2:106</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E2:130_1"/>
<a href="sin_cos.html#K19" target="_self">sin</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <a href="sin_cos.html#K19" target="_self">sin</a>  <a href="seq_1.html#K2">.</a> 0
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:130"/>
<a href="sin_cos.html#K19" target="_self">sin</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="ref" href="sin_cos.html#T33" target="_self">Th33</a></i>;<br/>


</div>
