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

<span class="kw">let </span><font color="Maroon" title="c2">A</font> be   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="measure5.html#V2" title="MEASURE5:attr.2">closed_interval</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>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">Z</font> being   <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>  st <font color="Maroon" title="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">Z</font> holds <br/> <a href="integra5.html#K2" title="INTEGRA5:func.2">integral</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="partfun2.html#K1" title="PARTFUN2:func.1">id</a> <font color="Olive" title="b1">Z</font></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <a href="sin_cos.html#K18" title="SIN_COS:func.18">cos</a></span>)</span>,<font color="Maroon" title="c2">A</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="partfun2.html#K1" title="PARTFUN2:func.1">id</a> <font color="Olive" title="b1">Z</font></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <a href="sin_cos.html#K16" title="SIN_COS:func.16">sin</a></span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <a href="sin_cos.html#K18" title="SIN_COS:func.18">cos</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="seq_4.html#K2" title="SEQ_4:func.2">upper_bound</a> <font color="Maroon" title="c2">A</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="partfun2.html#K1" title="PARTFUN2:func.1">id</a> <font color="Olive" title="b1">Z</font></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <a href="sin_cos.html#K16" title="SIN_COS:func.16">sin</a></span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <a href="sin_cos.html#K18" title="SIN_COS:func.18">cos</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <font color="Maroon" title="c2">A</font></span>)</span></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c3">Z</font> be   <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>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">Z</font> implies  <a href="integra5.html#K2" title="INTEGRA5:func.2">integral</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="partfun2.html#K1" title="PARTFUN2:func.1">id</a> <font color="Maroon" title="c3">Z</font></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <a href="sin_cos.html#K18" title="SIN_COS:func.18">cos</a></span>)</span>,<font color="Maroon" title="c2">A</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="partfun2.html#K1" title="PARTFUN2:func.1">id</a> <font color="Maroon" title="c3">Z</font></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <a href="sin_cos.html#K16" title="SIN_COS:func.16">sin</a></span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <a href="sin_cos.html#K18" title="SIN_COS:func.18">cos</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="seq_4.html#K2" title="SEQ_4:func.2">upper_bound</a> <font color="Maroon" title="c2">A</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="partfun2.html#K1" title="PARTFUN2:func.1">id</a> <font color="Maroon" title="c3">Z</font></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <a href="sin_cos.html#K16" title="SIN_COS:func.16">sin</a></span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <a href="sin_cos.html#K18" title="SIN_COS:func.18">cos</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <font color="Maroon" title="c2">A</font></span>)</span></span>)</span> )</span><br/>



<span class="kw">assume </span><a NAME="E1:9"/><span class="lab"><font color="Green" title="E5">A1</font></span>: 
<font color="Maroon" title="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">Z</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="integra5.html#K2" title="INTEGRA5:func.2">integral</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="partfun2.html#K1" title="PARTFUN2:func.1">id</a> <font color="Maroon" title="c3">Z</font></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <a href="sin_cos.html#K18" title="SIN_COS:func.18">cos</a></span>)</span>,<font color="Maroon" title="c2">A</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="partfun2.html#K1" title="PARTFUN2:func.1">id</a> <font color="Maroon" title="c3">Z</font></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <a href="sin_cos.html#K16" title="SIN_COS:func.16">sin</a></span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <a href="sin_cos.html#K18" title="SIN_COS:func.18">cos</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="seq_4.html#K2" title="SEQ_4:func.2">upper_bound</a> <font color="Maroon" title="c2">A</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="partfun2.html#K1" title="PARTFUN2:func.1">id</a> <font color="Maroon" title="c3">Z</font></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <a href="sin_cos.html#K16" title="SIN_COS:func.16">sin</a></span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <a href="sin_cos.html#K18" title="SIN_COS:func.18">cos</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <font color="Maroon" title="c2">A</font></span>)</span></span>)</span></span><br/>

 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="partfun2.html#K1" title="PARTFUN2:func.1">id</a> <font color="Maroon" title="c3">Z</font></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <a href="sin_cos.html#K16" title="SIN_COS:func.16">sin</a></span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <a href="sin_cos.html#K18" title="SIN_COS:func.18">cos</a></span>)</span> = 
<span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="partfun2.html#K1" title="PARTFUN2:func.1">id</a> <font color="Maroon" title="c3">Z</font></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <a href="sin_cos.html#K16" title="SIN_COS:func.16">sin</a></span>)</span></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>
<span class="kw">by</span> <span class="lab"><a class="ref" href="sin_cos.html#T24" onmouseover="rs('sin_cos/T24')" onmouseout="rh()">SIN_COS:24</a>, <a class="ref" href="valued_1.html#D1" onmouseover="rs('valued_1/D1')" onmouseout="rh()">VALUED_1:def 1</a></span>
<br/>.= 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="partfun2.html#K1" title="PARTFUN2:func.1">id</a> <font color="Maroon" title="c3">Z</font></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <a href="sin_cos.html#K16" title="SIN_COS:func.16">sin</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T28" onmouseover="rs('xboole_1/T28')" onmouseout="rh()">XBOOLE_1:28</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><a href="partfun2.html#K1" title="PARTFUN2:func.1">id</a> <font color="Maroon" title="c3">Z</font></span>)</span></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>
<span class="kw">by</span> <span class="lab"><a class="ref" href="sin_cos.html#T24" onmouseover="rs('sin_cos/T24')" onmouseout="rh()">SIN_COS:24</a>, <a class="ref" href="valued_1.html#D4" onmouseover="rs('valued_1/D4')" onmouseout="rh()">VALUED_1:def 4</a></span>
<br/>.= 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="partfun2.html#K1" title="PARTFUN2:func.1">id</a> <font color="Maroon" title="c3">Z</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T28" onmouseover="rs('xboole_1/T28')" onmouseout="rh()">XBOOLE_1:28</a></span>
;<br/>
<a NAME="E3:9"/><span class="kw">then </span><span class="lab"><font color="Green" title="E6">A2</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="partfun2.html#K1" title="PARTFUN2:func.1">id</a> <font color="Maroon" title="c3">Z</font></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <a href="sin_cos.html#K16" title="SIN_COS:func.16">sin</a></span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <a href="sin_cos.html#K18" title="SIN_COS:func.18">cos</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">Z</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T45" onmouseover="rs('relat_1/T45')" onmouseout="rh()">RELAT_1:45</a></span>;<br/>
<a NAME="E4:9"/><span class="kw">then </span><span class="lab"><font color="Green" title="E7">A3</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="partfun2.html#K1" title="PARTFUN2:func.1">id</a> <font color="Maroon" title="c3">Z</font></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <a href="sin_cos.html#K16" title="SIN_COS:func.16">sin</a></span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <a href="sin_cos.html#K18" title="SIN_COS:func.18">cos</a> <a href="fdiff_1.html#R2" title="FDIFF_1:pred.2">is_differentiable_on</a> <font color="Maroon" title="c3">Z</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="fdiff_4.html#T47" onmouseover="rs('fdiff_4/T47')" onmouseout="rh()">FDIFF_4:47</a></span>;<br/>
<a NAME="E5:9"/><span class="lab"><font color="Green" title="E8">A4</font></span>: 
 for <font color="Olive" title="b1">x</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Z</font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="partfun2.html#K1" title="PARTFUN2:func.1">id</a> <font color="Maroon" title="c3">Z</font></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <a href="sin_cos.html#K18" title="SIN_COS:func.18">cos</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K18" title="SIN_COS:func.18">cos</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font></span>)</span>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="9_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">x</font> be   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Z</font> implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="partfun2.html#K1" title="PARTFUN2:func.1">id</a> <font color="Maroon" title="c3">Z</font></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <a href="sin_cos.html#K18" title="SIN_COS:func.18">cos</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K18" title="SIN_COS:func.18">cos</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:9_2"/><span class="lab"><font color="Green" title="E9">A5</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Z</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="partfun2.html#K1" title="PARTFUN2:func.1">id</a> <font color="Maroon" title="c3">Z</font></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <a href="sin_cos.html#K18" title="SIN_COS:func.18">cos</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K18" title="SIN_COS:func.18">cos</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font></span>)</span></span><br/>

<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="partfun2.html#K1" title="PARTFUN2:func.1">id</a> <font color="Maroon" title="c3">Z</font></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <a href="sin_cos.html#K18" title="SIN_COS:func.18">cos</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="partfun2.html#K1" title="PARTFUN2:func.1">id</a> <font color="Maroon" title="c3">Z</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K18" title="SIN_COS:func.18">cos</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#T5" onmouseover="rs('valued_1/T5')" onmouseout="rh()">VALUED_1:5</a></span>
<br/>.= 
<font color="Maroon" title="c4">x</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K18" title="SIN_COS:func.18">cos</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:9_2"><span class="lab"><font color="Green" title="E9">A5</font></span></a>, <a class="ref" href="funct_1.html#T18" onmouseover="rs('funct_1/T18')" onmouseout="rh()">FUNCT_1:18</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E3:9_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="partfun2.html#K1" title="PARTFUN2:func.1">id</a> <font color="Maroon" title="c3">Z</font></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <a href="sin_cos.html#K18" title="SIN_COS:func.18">cos</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K18" title="SIN_COS:func.18">cos</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font></span>)</span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E6:9"/><span class="lab"><font color="Green" title="E9">A6</font></span>: 
 for <font color="Olive" title="b1">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>   st <font color="Olive" title="b1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="partfun2.html#K1" title="PARTFUN2:func.1">id</a> <font color="Maroon" title="c3">Z</font></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <a href="sin_cos.html#K16" title="SIN_COS:func.16">sin</a></span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <a href="sin_cos.html#K18" title="SIN_COS:func.18">cos</a></span>)</span> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c3">Z</font></span>)</span> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="partfun2.html#K1" title="PARTFUN2:func.1">id</a> <font color="Maroon" title="c3">Z</font></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <a href="sin_cos.html#K16" title="SIN_COS:func.16">sin</a></span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <a href="sin_cos.html#K18" title="SIN_COS:func.18">cos</a></span>)</span> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c3">Z</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="partfun2.html#K1" title="PARTFUN2:func.1">id</a> <font color="Maroon" title="c3">Z</font></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <a href="sin_cos.html#K18" title="SIN_COS:func.18">cos</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="9_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="partfun2.html#K1" title="PARTFUN2:func.1">id</a> <font color="Maroon" title="c3">Z</font></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <a href="sin_cos.html#K16" title="SIN_COS:func.16">sin</a></span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <a href="sin_cos.html#K18" title="SIN_COS:func.18">cos</a></span>)</span> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c3">Z</font></span>)</span> implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="partfun2.html#K1" title="PARTFUN2:func.1">id</a> <font color="Maroon" title="c3">Z</font></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <a href="sin_cos.html#K16" title="SIN_COS:func.16">sin</a></span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <a href="sin_cos.html#K18" title="SIN_COS:func.18">cos</a></span>)</span> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c3">Z</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="partfun2.html#K1" title="PARTFUN2:func.1">id</a> <font color="Maroon" title="c3">Z</font></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <a href="sin_cos.html#K18" title="SIN_COS:func.18">cos</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:9_3"/>
<font color="Maroon" title="c4">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="partfun2.html#K1" title="PARTFUN2:func.1">id</a> <font color="Maroon" title="c3">Z</font></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <a href="sin_cos.html#K16" title="SIN_COS:func.16">sin</a></span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <a href="sin_cos.html#K18" title="SIN_COS:func.18">cos</a></span>)</span> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c3">Z</font></span>)</span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="partfun2.html#K1" title="PARTFUN2:func.1">id</a> <font color="Maroon" title="c3">Z</font></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <a href="sin_cos.html#K16" title="SIN_COS:func.16">sin</a></span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <a href="sin_cos.html#K18" title="SIN_COS:func.18">cos</a></span>)</span> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c3">Z</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="partfun2.html#K1" title="PARTFUN2:func.1">id</a> <font color="Maroon" title="c3">Z</font></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <a href="sin_cos.html#K18" title="SIN_COS:func.18">cos</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font></span><br/>

<a NAME="E2:9_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E10">A7</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">Z</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:9"><span class="lab"><font color="Green" title="E7">A3</font></span></a>, <a class="ref" href="fdiff_1.html#D7" onmouseover="rs('fdiff_1/D7')" onmouseout="rh()">FDIFF_1:def 7</a></span>;<br/>
<span class="kw">then </span><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="partfun2.html#K1" title="PARTFUN2:func.1">id</a> <font color="Maroon" title="c3">Z</font></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <a href="sin_cos.html#K16" title="SIN_COS:func.16">sin</a></span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <a href="sin_cos.html#K18" title="SIN_COS:func.18">cos</a></span>)</span> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c3">Z</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font> = 
<font color="Maroon" title="c4">x</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K18" title="SIN_COS:func.18">cos</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:9"><span class="lab"><font color="Green" title="E6">A2</font></span></a>, <a class="ref" href="fdiff_4.html#T47" onmouseover="rs('fdiff_4/T47')" onmouseout="rh()">FDIFF_4:47</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="partfun2.html#K1" title="PARTFUN2:func.1">id</a> <font color="Maroon" title="c3">Z</font></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <a href="sin_cos.html#K18" title="SIN_COS:func.18">cos</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:9"><span class="lab"><font color="Green" title="E8">A4</font></span></a>, <a class="txt" href="#E2:9_3"><span class="lab"><font color="Green" title="E10">A7</font></span></a></span>
;<br/>
<span class="kw">hence </span><a NAME="E4:9_3"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="partfun2.html#K1" title="PARTFUN2:func.1">id</a> <font color="Maroon" title="c3">Z</font></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <a href="sin_cos.html#K16" title="SIN_COS:func.16">sin</a></span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <a href="sin_cos.html#K18" title="SIN_COS:func.18">cos</a></span>)</span> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c3">Z</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="partfun2.html#K1" title="PARTFUN2:func.1">id</a> <font color="Maroon" title="c3">Z</font></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <a href="sin_cos.html#K18" title="SIN_COS:func.18">cos</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="lab"><font color="Green" title="E10">A8</font></span>:  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="partfun2.html#K1" title="PARTFUN2:func.1">id</a> <font color="Maroon" title="c3">Z</font></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <a href="sin_cos.html#K18" title="SIN_COS:func.18">cos</a></span>)</span> = 
<span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><a href="partfun2.html#K1" title="PARTFUN2:func.1">id</a> <font color="Maroon" title="c3">Z</font></span>)</span></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>
<span class="kw">by</span> <span class="lab"><a class="ref" href="sin_cos.html#T24" onmouseover="rs('sin_cos/T24')" onmouseout="rh()">SIN_COS:24</a>, <a class="ref" href="valued_1.html#D4" onmouseover="rs('valued_1/D4')" onmouseout="rh()">VALUED_1:def 4</a></span>
<br/>.= 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="partfun2.html#K1" title="PARTFUN2:func.1">id</a> <font color="Maroon" title="c3">Z</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T28" onmouseover="rs('xboole_1/T28')" onmouseout="rh()">XBOOLE_1:28</a></span>
<br/>.= 
<font color="Maroon" title="c3">Z</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T45" onmouseover="rs('relat_1/T45')" onmouseout="rh()">RELAT_1:45</a></span>
;<br/>
<a NAME="E8:9"/><span class="kw">then </span>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="partfun2.html#K1" title="PARTFUN2:func.1">id</a> <font color="Maroon" title="c3">Z</font></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <a href="sin_cos.html#K16" title="SIN_COS:func.16">sin</a></span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <a href="sin_cos.html#K18" title="SIN_COS:func.18">cos</a></span>)</span> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c3">Z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="partfun2.html#K1" title="PARTFUN2:func.1">id</a> <font color="Maroon" title="c3">Z</font></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <a href="sin_cos.html#K18" title="SIN_COS:func.18">cos</a></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:9"><span class="lab"><font color="Green" title="E7">A3</font></span></a>, <a class="ref" href="fdiff_1.html#D7" onmouseover="rs('fdiff_1/D7')" onmouseout="rh()">FDIFF_1:def 7</a></span>;<br/>
<a NAME="E9:9"/><span class="kw">then </span><span class="lab"><font color="Green" title="E11">A9</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="partfun2.html#K1" title="PARTFUN2:func.1">id</a> <font color="Maroon" title="c3">Z</font></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <a href="sin_cos.html#K16" title="SIN_COS:func.16">sin</a></span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <a href="sin_cos.html#K18" title="SIN_COS:func.18">cos</a></span>)</span> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Maroon" title="c3">Z</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><a href="partfun2.html#K1" title="PARTFUN2:func.1">id</a> <font color="Maroon" title="c3">Z</font></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <a href="sin_cos.html#K18" title="SIN_COS:func.18">cos</a>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:9"><span class="lab"><font color="Green" title="E9">A6</font></span></a>, <a class="ref" href="partfun1.html#T5" onmouseover="rs('partfun1/T5')" onmouseout="rh()">PARTFUN1:5</a></span>;<br/>
<a NAME="E10:9"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="partfun2.html#K1" title="PARTFUN2:func.1">id</a> <font color="Maroon" title="c3">Z</font></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <a href="sin_cos.html#K18" title="SIN_COS:func.18">cos</a></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c2">A</font> is  <a href="fcont_1.html#V1" title="FCONT_1:attr.1">continuous</a> 
 
;<br/>
<a NAME="E11:9"/><span class="kw">then </span><span class="lab"><font color="Green" title="E12">A10</font></span>: 
<span class="p1">(<span class="default"><a href="partfun2.html#K1" title="PARTFUN2:func.1">id</a> <font color="Maroon" title="c3">Z</font></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <a href="sin_cos.html#K18" title="SIN_COS:func.18">cos</a> <a href="integra5.html#R1" title="INTEGRA5:pred.1">is_integrable_on</a> <font color="Maroon" title="c2">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:9"><span class="lab"><font color="Green" title="E5">A1</font></span></a>, <a class="txt" href="#E7:9"><span class="lab"><font color="Green" title="E10">A8</font></span></a>, <a class="ref" href="integra5.html#T11" onmouseover="rs('integra5/T11')" onmouseout="rh()">INTEGRA5:11</a></span>;<br/>
<a NAME="E12:9"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="partfun2.html#K1" title="PARTFUN2:func.1">id</a> <font color="Maroon" title="c3">Z</font></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <a href="sin_cos.html#K18" title="SIN_COS:func.18">cos</a></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c2">A</font> is  <a href="comseq_2.html#V1" title="COMSEQ_2:attr.1">bounded</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:9"><span class="lab"><font color="Green" title="E5">A1</font></span></a>, <a class="txt" href="#E7:9"><span class="lab"><font color="Green" title="E10">A8</font></span></a>, <a class="ref" href="integra5.html#T10" onmouseover="rs('integra5/T10')" onmouseout="rh()">INTEGRA5:10</a></span>;<br/>
<span class="kw">hence </span><a NAME="E13:9"/>
 <a href="integra5.html#K2" title="INTEGRA5:func.2">integral</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="partfun2.html#K1" title="PARTFUN2:func.1">id</a> <font color="Maroon" title="c3">Z</font></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <a href="sin_cos.html#K18" title="SIN_COS:func.18">cos</a></span>)</span>,<font color="Maroon" title="c2">A</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="partfun2.html#K1" title="PARTFUN2:func.1">id</a> <font color="Maroon" title="c3">Z</font></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <a href="sin_cos.html#K16" title="SIN_COS:func.16">sin</a></span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <a href="sin_cos.html#K18" title="SIN_COS:func.18">cos</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="seq_4.html#K2" title="SEQ_4:func.2">upper_bound</a> <font color="Maroon" title="c2">A</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="partfun2.html#K1" title="PARTFUN2:func.1">id</a> <font color="Maroon" title="c3">Z</font></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <a href="sin_cos.html#K16" title="SIN_COS:func.16">sin</a></span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <a href="sin_cos.html#K18" title="SIN_COS:func.18">cos</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="seq_4.html#K3" title="SEQ_4:func.3">lower_bound</a> <font color="Maroon" title="c2">A</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:9"><span class="lab"><font color="Green" title="E5">A1</font></span></a>, <a class="txt" href="#E3:9"><span class="lab"><font color="Green" title="E6">A2</font></span></a>, <a class="txt" href="#E11:9"><span class="lab"><font color="Green" title="E12">A10</font></span></a>, <a class="txt" href="#E9:9"><span class="lab"><font color="Green" title="E11">A9</font></span></a>, <a class="ref" href="fdiff_4.html#T47" onmouseover="rs('fdiff_4/T47')" onmouseout="rh()">FDIFF_4:47</a>, <a class="ref" href="integra5.html#T13" onmouseover="rs('integra5/T13')" onmouseout="rh()">INTEGRA5:13</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
