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

<span class="kw">let </span><font color="Maroon" title="c1">b</font>, <font color="Maroon" title="c2">a</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" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="sin_cos.html#K25" title="SIN_COS:func.25">exp_R</a> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c1">b</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K25" title="SIN_COS:func.25">exp_R</a> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integra5.html#K4" title="INTEGRA5:func.4">integral</a> (<a href="sin_cos.html#K25" title="SIN_COS:func.25">exp_R</a>,<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c1">b</font>)</span><br/>



<a NAME="E1:3"/><span class="lab"><font color="Green" title="E3">A1</font></span>: 
 <a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c1">b</font>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T17" title="XXREAL_0:th.17">XXREAL_0:17</a></span>;<br/>
<a NAME="E2:3"/><span class="lab"><font color="Green" title="E4">A2</font></span>: 
<span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c1">b</font>)</span>)</span>,<span class="p2">(<span class="default"><a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> (<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c1">b</font>)</span>)</span></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>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 
;<br/>
<a NAME="E3:3"/>
( <a href="sin_cos.html#K25" title="SIN_COS:func.25">exp_R</a> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> is  <a href="fcont_1.html#V1" title="FCONT_1:attr.1">continuous</a>  &amp; <font color="Maroon" title="c2">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> (<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c1">b</font>) )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T25" title="XXREAL_0:th.25">XXREAL_0:25</a></span>;<br/>
<a NAME="E4:3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E5">A3</font></span>: 
<a href="sin_cos.html#K25" title="SIN_COS:func.25">exp_R</a> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> (<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c1">b</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="integra5.html#K4" title="INTEGRA5:func.4">integral</a> (<a href="sin_cos.html#K25" title="SIN_COS:func.25">exp_R</a>,<span class="p2">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c1">b</font>)</span>)</span>,<span class="p2">(<span class="default"><a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> (<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c1">b</font>)</span>)</span>)</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K25" title="SIN_COS:func.25">exp_R</a> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c1">b</font>)</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:3"><span class="lab"><font color="Green" title="E3">A1</font></span></a>, <a class="txt" href="#E2:3"><span class="lab"><font color="Green" title="E4">A2</font></span></a>, <a class="ref" href="integra7.html#T20" title="INTEGRA7:th.20">INTEGRA7:20</a>, <a class="ref" href="integra7.html#T27" title="INTEGRA7:th.27">INTEGRA7:27</a>, <a class="ref" href="sin_cos.html#T47" title="SIN_COS:th.47">SIN_COS:47</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E5:3"/><span class="lab"><font color="Green" title="E6">A4</font></span>: 
(  <a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c1">b</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> implies <span class="p1">(<span class="default"><a href="sin_cos.html#K25" title="SIN_COS:func.25">exp_R</a> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c1">b</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K25" title="SIN_COS:func.25">exp_R</a> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integra5.html#K4" title="INTEGRA5:func.4">integral</a> (<a href="sin_cos.html#K25" title="SIN_COS:func.25">exp_R</a>,<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c1">b</font>) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:3_1"/><span class="lab"><font color="Green" title="E7">A5</font></span>: 
 <a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c1">b</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="sin_cos.html#K25" title="SIN_COS:func.25">exp_R</a> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c1">b</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K25" title="SIN_COS:func.25">exp_R</a> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integra5.html#K4" title="INTEGRA5:func.4">integral</a> (<a href="sin_cos.html#K25" title="SIN_COS:func.25">exp_R</a>,<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c1">b</font>)</span><br/>

<a NAME="E2:3_1"/><span class="kw">then </span>
 <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> (<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c1">b</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T36" title="XXREAL_0:th.36">XXREAL_0:36</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:3_1"/>
<span class="p1">(<span class="default"><a href="sin_cos.html#K25" title="SIN_COS:func.25">exp_R</a> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c1">b</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K25" title="SIN_COS:func.25">exp_R</a> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integra5.html#K4" title="INTEGRA5:func.4">integral</a> (<a href="sin_cos.html#K25" title="SIN_COS:func.25">exp_R</a>,<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c1">b</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:3"><span class="lab"><font color="Green" title="E5">A3</font></span></a>, <a class="txt" href="#E1:3_1"><span class="lab"><font color="Green" title="E7">A5</font></span></a></span>; <a class="txt" 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="E6:3"/>
(  <a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c1">b</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">b</font> implies <span class="p1">(<span class="default"><a href="sin_cos.html#K25" title="SIN_COS:func.25">exp_R</a> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c1">b</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K25" title="SIN_COS:func.25">exp_R</a> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integra5.html#K4" title="INTEGRA5:func.4">integral</a> (<a href="sin_cos.html#K25" title="SIN_COS:func.25">exp_R</a>,<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c1">b</font>) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:3_2"/><span class="lab"><font color="Green" title="E7">A6</font></span>: 
 <a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c1">b</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">b</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="sin_cos.html#K25" title="SIN_COS:func.25">exp_R</a> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c1">b</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K25" title="SIN_COS:func.25">exp_R</a> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integra5.html#K4" title="INTEGRA5:func.4">integral</a> (<a href="sin_cos.html#K25" title="SIN_COS:func.25">exp_R</a>,<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c1">b</font>)</span><br/>

<a NAME="E2:3_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E8">A7</font></span>: 
 <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> (<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c1">b</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T36" title="XXREAL_0:th.36">XXREAL_0:36</a></span>;<br/>
<a NAME="E3:3_2"/>
( <font color="Maroon" title="c1">b</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">a</font> implies <span class="p1">(<span class="default"><a href="sin_cos.html#K25" title="SIN_COS:func.25">exp_R</a> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c1">b</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K25" title="SIN_COS:func.25">exp_R</a> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integra5.html#K4" title="INTEGRA5:func.4">integral</a> (<a href="sin_cos.html#K25" title="SIN_COS:func.25">exp_R</a>,<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c1">b</font>) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:3_2_1"/>
<font color="Maroon" title="c1">b</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">a</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="sin_cos.html#K25" title="SIN_COS:func.25">exp_R</a> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c1">b</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K25" title="SIN_COS:func.25">exp_R</a> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integra5.html#K4" title="INTEGRA5:func.4">integral</a> (<a href="sin_cos.html#K25" title="SIN_COS:func.25">exp_R</a>,<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c1">b</font>)</span><br/>

<a NAME="E2:3_2_1"/><span class="kw">then </span>
 <a href="integra5.html#K4" title="INTEGRA5:func.4">integral</a> (<a href="sin_cos.html#K25" title="SIN_COS:func.25">exp_R</a>,<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c1">b</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default"><a href="integra5.html#K2" title="INTEGRA5:func.2">integral</a> (<a href="sin_cos.html#K25" title="SIN_COS:func.25">exp_R</a>,<span class="p2"><a href="integra5.html#K3" title="INTEGRA5:func.3">['</a><span class="default"><font color="Maroon" title="c1">b</font>,<font color="Maroon" title="c2">a</font></span><a href="integra5.html#K3" title="INTEGRA5:func.3">']</a></span>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="integra5.html#D4" title="INTEGRA5:def.4">INTEGRA5:def 4</a></span>;<br/>
<a NAME="E3:3_2_1"/><span class="kw">then </span>
<a href="sin_cos.html#K25" title="SIN_COS:func.25">exp_R</a> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p2">(<span class="default"><a href="integra5.html#K4" title="INTEGRA5:func.4">integral</a> (<a href="sin_cos.html#K25" title="SIN_COS:func.25">exp_R</a>,<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c1">b</font>)</span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K25" title="SIN_COS:func.25">exp_R</a> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c1">b</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:3"><span class="lab"><font color="Green" title="E3">A1</font></span></a>, <a class="txt" href="#E4:3"><span class="lab"><font color="Green" title="E5">A3</font></span></a>, <a class="txt" href="#E1:3_2"><span class="lab"><font color="Green" title="E7">A6</font></span></a>, <a class="txt" href="#E2:3_2"><span class="lab"><font color="Green" title="E8">A7</font></span></a>, <a class="ref" href="integra5.html#D4" title="INTEGRA5:def.4">INTEGRA5:def 4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:3_2_1"/>
<span class="p1">(<span class="default"><a href="sin_cos.html#K25" title="SIN_COS:func.25">exp_R</a> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c1">b</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K25" title="SIN_COS:func.25">exp_R</a> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integra5.html#K4" title="INTEGRA5:func.4">integral</a> (<a href="sin_cos.html#K25" title="SIN_COS:func.25">exp_R</a>,<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c1">b</font>)
 ; <a class="txt" 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="kw">hence </span><a NAME="E4:3_2"/>
<span class="p1">(<span class="default"><a href="sin_cos.html#K25" title="SIN_COS:func.25">exp_R</a> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c1">b</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K25" title="SIN_COS:func.25">exp_R</a> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integra5.html#K4" title="INTEGRA5:func.4">integral</a> (<a href="sin_cos.html#K25" title="SIN_COS:func.25">exp_R</a>,<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c1">b</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:3"><span class="lab"><font color="Green" title="E3">A1</font></span></a>, <a class="txt" href="#E5:3"><span class="lab"><font color="Green" title="E6">A4</font></span></a>, <a class="txt" href="#E1:3_2"><span class="lab"><font color="Green" title="E7">A6</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>; <a class="txt" 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="kw">hence </span><a NAME="E7:3"/>
<span class="p1">(<span class="default"><a href="sin_cos.html#K25" title="SIN_COS:func.25">exp_R</a> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c1">b</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K25" title="SIN_COS:func.25">exp_R</a> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integra5.html#K4" title="INTEGRA5:func.4">integral</a> (<a href="sin_cos.html#K25" title="SIN_COS:func.25">exp_R</a>,<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c1">b</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:3"><span class="lab"><font color="Green" title="E6">A4</font></span></a>, <a class="ref" href="xxreal_0.html#T15" title="XXREAL_0:th.15">XXREAL_0:15</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
