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

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c1">x1</font>, <font color="Maroon" title="c2">x2</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><b>that </b><br/><a NAME="E1:20_1"/><i><font color="Green" title="E19">A1</font></i>: 
<font color="Maroon" title="c1">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <a href="sin_cos.html#K27" title="SIN_COS:func.27">exp_R</a> 
 <b>and </b><br/><a NAME="E2:20_1"/><i><font color="Green" title="E20">A2</font></i>: 
<font color="Maroon" title="c2">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <a href="sin_cos.html#K27" title="SIN_COS:func.27">exp_R</a> 
 <b>and </b><br/><a NAME="E3:20_1"/><i><font color="Green" title="E21">A3</font></i>: 
<a href="sin_cos.html#K27" title="SIN_COS:func.27">exp_R</a>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c1">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="sin_cos.html#K27" title="SIN_COS:func.27">exp_R</a>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c2">x2</font>
 ;<br/><b>reconsider </b><font color="Maroon" title="c3">p1</font> = <font color="Maroon" title="c1">x1</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <b>by </b><i><a class="txt" href="taylor_1.html#E1:20_1"><i><font color="Green" title="E19">A1</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c4">p2</font> = <font color="Maroon" title="c2">x2</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <b>by </b><i><a class="txt" href="taylor_1.html#E2:20_1"><i><font color="Green" title="E20">A2</font></i></a></i>;<br/><b>thus </b><font color="Maroon" title="c1">x1</font> = 
 <a href="power.html#K6" title="POWER:func.6">log</a> <a href="power.html#K8" title="POWER:func.8">number_e</a> ,<span class="p1">(<span class="default"><a href="sin_cos.html#K27" title="SIN_COS:func.27">exp_R</a>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">p1</font></span>)</span>
<b>by </b><i><a class="ref" href="taylor_1.html#T13" target="_self" title="TAYLOR_1:th.13">Th13</a></i>
<br/>.= 
 <a href="power.html#K6" title="POWER:func.6">log</a> <a href="power.html#K8" title="POWER:func.8">number_e</a> ,<span class="p1">(<span class="default"><a href="sin_cos.html#K27" title="SIN_COS:func.27">exp_R</a>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">p2</font></span>)</span>
<b>by </b><i><a class="txt" href="taylor_1.html#E3:20_1"><i><font color="Green" title="E21">A3</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c2">x2</font>
<b>by </b><i><a class="ref" href="taylor_1.html#T13" target="_self" title="TAYLOR_1:th.13">Th13</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:20"/>
 <a href="sin_cos.html#K27" title="SIN_COS:func.27">exp_R</a>  is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 <b>by </b><i><a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></i>;<br/>

<b>thus </b><a NAME="E3:20"/>
(  <a href="sin_cos.html#K27" title="SIN_COS:func.27">exp_R</a>  <a href="fdiff_1.html#R2" title="FDIFF_1:pred.2">is_differentiable_on</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  &amp;  <a href="sin_cos.html#K27" title="SIN_COS:func.27">exp_R</a>  <a href="fdiff_1.html#R2" title="FDIFF_1:pred.2">is_differentiable_on</a>  <a href="subset_1.html#K2" title="SUBSET_1:func.2">[#]</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  )
 <b>by </b><i><a class="ref" href="sin_cos.html#T71" title="SIN_COS:th.71">SIN_COS:71</a></i>;<br/>

<b>thus </b><a NAME="E4:20"/>
for <font color="Olive" title="b1">x</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> holds   <a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> <a href="sin_cos.html#K27" title="SIN_COS:func.27">exp_R</a> ,<font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="sin_cos.html#K27" title="SIN_COS:func.27">exp_R</a>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">x</font>
 <b>by </b><i><a class="ref" href="sin_cos.html#T71" title="SIN_COS:th.71">SIN_COS:71</a></i>;<br/>

<div><b>hereby </b>
<div class="add"><b>let </b><font color="Maroon" title="c1">x</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/><a NAME="E1:20_2"/>
 <a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> <a href="sin_cos.html#K27" title="SIN_COS:func.27">exp_R</a> ,<font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="sin_cos.html#K27" title="SIN_COS:func.27">exp_R</a>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c1">x</font>
 <b>by </b><i><a class="ref" href="sin_cos.html#T71" title="SIN_COS:th.71">SIN_COS:71</a></i>;<br/><b>hence </b><a NAME="E2:20_2"/>
 <a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> <a href="sin_cos.html#K27" title="SIN_COS:func.27">exp_R</a> ,<font color="Maroon" title="c1">x</font> <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="ref" href="sin_cos.html#T59" title="SIN_COS:th.59">SIN_COS:59</a></i>;<br/></div>
<b>end;</b></div>

<b>thus </b><a NAME="E6:20"/>
(  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <a href="sin_cos.html#K27" title="SIN_COS:func.27">exp_R</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <a href="sin_cos.html#K27" title="SIN_COS:func.27">exp_R</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="subset_1.html#K2" title="SUBSET_1:func.2">[#]</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  )
 <b>by </b><i><a class="ref" href="sin_cos.html#T51" title="SIN_COS:th.51">SIN_COS:51</a></i>;<br/>

<b>thus </b><a NAME="E7:20"/>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <a href="sin_cos.html#K27" title="SIN_COS:func.27">exp_R</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="limfunc1.html#K4" title="LIMFUNC1:func.4">right_open_halfline</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_3"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_3_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c1">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:20_3_1"/><i><font color="Green" title="E19">A4</font></i>: 
<font color="Maroon" title="c1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <a href="sin_cos.html#K27" title="SIN_COS:func.27">exp_R</a> 
 ;<br/><b>consider </b><font color="Maroon" title="c2">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:20_3_1"/><i><font color="Green" title="E20">A5</font></i>: 
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <a href="sin_cos.html#K27" title="SIN_COS:func.27">exp_R</a> 
 <b>and </b><br/><a NAME="E4:20_3_1"/><i><font color="Green" title="E21">A6</font></i>: 
<font color="Maroon" title="c1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="sin_cos.html#K27" title="SIN_COS:func.27">exp_R</a>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c2">x</font>
 <b>by </b><i><a class="txt" href="taylor_1.html#E1:20_3_1"><i><font color="Green" title="E19">A4</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c3">x</font> = <font color="Maroon" title="c2">x</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <b>by </b><i><a class="txt" href="taylor_1.html#E3:20_3_1"><i><font color="Green" title="E20">A5</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c4">y1</font> = <font color="Maroon" title="c1">y</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <b>by </b><i><a class="txt" href="taylor_1.html#E4:20_3_1"><i><font color="Green" title="E21">A6</font></i></a></i>;<br/><a NAME="E7:20_3_1"/>
<a href="sin_cos.html#K27" title="SIN_COS:func.27">exp_R</a>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">x</font> <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="ref" href="sin_cos.html#T59" title="SIN_COS:th.59">SIN_COS:59</a></i>;<br/><a NAME="E8:20_3_1"/><b>then </b>
<font color="Maroon" title="c4">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">g</font> where <font color="Olive" title="b1">g</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> :  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b1">g</font> </span>}</span> 
 <b>by </b><i><a class="txt" href="taylor_1.html#E4:20_3_1"><i><font color="Green" title="E21">A6</font></i></a></i>;<br/><b>hence </b><a NAME="E9:20_3_1"/>
<font color="Maroon" title="c1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="limfunc1.html#K4" title="LIMFUNC1:func.4">right_open_halfline</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="ref" href="limfunc1.html#T152" title="LIMFUNC1:th.152">LIMFUNC1:152</a></i>;<br/></div><b>end;</b></div>
<a NAME="E2:20_3"/><b>then </b><i><font color="Green" title="E19">A7</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <a href="sin_cos.html#K27" title="SIN_COS:func.27">exp_R</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="limfunc1.html#K4" title="LIMFUNC1:func.4">right_open_halfline</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_3_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c1">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:20_3_2"/><i><font color="Green" title="E20">A8</font></i>: 
<font color="Maroon" title="c1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="limfunc1.html#K4" title="LIMFUNC1:func.4">right_open_halfline</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><a NAME="E2:20_3_2"/>
<font color="Maroon" title="c1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">g</font> where <font color="Olive" title="b1">g</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> :  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b1">g</font> </span>}</span> 
 <b>by </b><i><a class="txt" href="taylor_1.html#E1:20_3_2"><i><font color="Green" title="E20">A8</font></i></a>, <a class="ref" href="limfunc1.html#T152" title="LIMFUNC1:th.152">LIMFUNC1:152</a></i>;<br/><b>then consider </b><font color="Maroon" title="c2">g</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E4:20_3_2"/><i><font color="Green" title="E21">A9</font></i>: 
( <font color="Maroon" title="c1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">g</font> &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c2">g</font> )
 ;<br/><b>reconsider </b><font color="Maroon" title="c3">y1</font> = <font color="Maroon" title="c1">y</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <b>by </b><i><a class="txt" href="taylor_1.html#E4:20_3_2"><i><font color="Green" title="E21">A9</font></i></a></i>;<br/><a NAME="E6:20_3_2"/>
<font color="Maroon" title="c3">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="sin_cos.html#K27" title="SIN_COS:func.27">exp_R</a>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="power.html#K6" title="POWER:func.6">log</a> <a href="power.html#K8" title="POWER:func.8">number_e</a> ,<font color="Maroon" title="c3">y1</font></span>)</span>
 <b>by </b><i><a class="txt" href="taylor_1.html#E4:20_3_2"><i><font color="Green" title="E21">A9</font></i></a>, <a class="ref" href="taylor_1.html#T15" target="_self" title="TAYLOR_1:th.15">Th15</a></i>;<br/><b>hence </b><a NAME="E7:20_3_2"/>
<font color="Maroon" title="c1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <a href="sin_cos.html#K27" title="SIN_COS:func.27">exp_R</a> 
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a>, <a class="ref" href="sin_cos.html#T51" title="SIN_COS:th.51">SIN_COS:51</a></i>;<br/></div><b>end;</b></div>
<a NAME="E4:20_3"/><b>then </b>
 <a href="limfunc1.html#K4" title="LIMFUNC1:func.4">right_open_halfline</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <a href="sin_cos.html#K27" title="SIN_COS:func.27">exp_R</a> 
 
<b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<b>hence </b><a NAME="E5:20_3"/>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <a href="sin_cos.html#K27" title="SIN_COS:func.27">exp_R</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="limfunc1.html#K4" title="LIMFUNC1:func.4">right_open_halfline</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="taylor_1.html#E2:20_3"><i><font color="Green" title="E19">A7</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/>


</div><b>end;</b></div>


</div>
