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<a NAME="E1:5"/><i><font color="Green" title="E4">A1</font></i>: 
 <a href="sin_cos.html#K19" title="SIN_COS:func.19">sin</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"><span class="p2">(<span class="default"><a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a>  <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> 2</span>)</span>,<a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
 
<b>by </b><i><a class="ref" href="fdiff_1.html#T34" title="FDIFF_1:th.34">FDIFF_1:34</a>, <a class="ref" href="sin_cos.html#T73" title="SIN_COS:th.73">SIN_COS:73</a></i>;<br/>
<a NAME="E2:5"/>
for <font color="Olive" title="b1">th1</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b1">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"><span class="p2">(<span class="default"><a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a>  <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> 2</span>)</span>,<a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> holds <br/> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> <a href="sin_cos.html#K19" title="SIN_COS:func.19">sin</a> ,<font color="Olive" title="b1">th1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c1">th1</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">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"><span class="p2">(<span class="default"><a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a>  <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> 2</span>)</span>,<a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> implies  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> <a href="sin_cos.html#K19" title="SIN_COS:func.19">sin</a> ,<font color="Maroon" title="c1">th1</font> )</span><br/>

<b>assume </b><a NAME="E1:5_1"/>
<font color="Maroon" title="c1">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"><span class="p2">(<span class="default"><a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a>  <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> 2</span>)</span>,<a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> <a href="sin_cos.html#K19" title="SIN_COS:func.19">sin</a> ,<font color="Maroon" title="c1">th1</font></span><br/>

<a NAME="E2:5_1"/><b>then </b>
( <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a>  <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> 2 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">th1</font> &amp; <font color="Maroon" title="c1">th1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a>  )
 
<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="E3:5_1"/><b>then </b>
( <span class="p1">(<span class="default"><a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a>  <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> 2</span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a>  <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> 2</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">th1</font> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a>  <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> 2</span>)</span> &amp; <font color="Maroon" title="c1">th1</font> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a>  <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> 2</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a>  <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a>  <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> 2</span>)</span> )
 
<b>by </b><i><a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/>
<a NAME="E4:5_1"/><b>then </b><i><font color="Green" title="E5">A2</font></i>: 
<font color="Maroon" title="c1">th1</font> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a>  <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> 2</span>)</span> <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"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p2">(<span class="default"><a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a>  <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> 2</span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
 
<b>by </b><i><a class="ref" href="rcomp_1.html#T47" title="RCOMP_1:th.47">RCOMP_1:47</a></i>;<br/>
<i><font color="Green" title="E6">A3</font></i>:  <a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> <a href="sin_cos.html#K19" title="SIN_COS:func.19">sin</a> ,<font color="Maroon" title="c1">th1</font> = 
<a href="sin_cos.html#K22" title="SIN_COS:func.22">cos</a>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a>  <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> 2</span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">th1</font> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a>  <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> 2</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="sin_cos.html#T73" title="SIN_COS:th.73">SIN_COS:73</a></i>
<br/>.= 
 <a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K19" title="SIN_COS:func.19">sin</a>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">th1</font> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a>  <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> 2</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="sin_cos.html#T83" title="SIN_COS:th.83">SIN_COS:83</a></i>
;<br/>
<a NAME="E6:5_1"/>
<a href="sin_cos.html#K19" title="SIN_COS:func.19">sin</a>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">th1</font> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a>  <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> 2</span>)</span></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="sin_cos2.html#E4:5_1"><i><font color="Green" title="E5">A2</font></i></a>, <a class="txt" href="sin_cos2.html#E4"><i><font color="Green" title="E3">Lm2</font></i></a></i>;<br/>
<a NAME="E7:5_1"/><b>then </b>
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K19" title="SIN_COS:func.19">sin</a>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">th1</font> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a>  <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="ref" href="real_2.html#T105" title="REAL_2:th.105">REAL_2:105</a></i>;<br/>
<b>hence </b><a NAME="E8:5_1"/>
 <a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> <a href="sin_cos.html#K19" title="SIN_COS:func.19">sin</a> ,<font color="Maroon" title="c1">th1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="sin_cos2.html#E5:5_1"><i><font color="Green" title="E6">A3</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:5"/>
 <a href="sin_cos.html#K19" title="SIN_COS:func.19">sin</a>  <a href="rfunct_2.html#R2" title="RFUNCT_2:pred.2">is_decreasing_on</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a>  <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> 2</span>)</span>,<a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
 <b>by </b><i><a class="txt" href="sin_cos2.html#E1:5"><i><font color="Green" title="E4">A1</font></i></a>, <a class="txt" href="sin_cos2.html#E1"><i><font color="Green" title="E1">Lm1</font></i></a>, <a class="ref" href="rolle.html#T10" title="ROLLE:th.10">ROLLE:10</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
