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

<a NAME="E1:26"/><i><font color="Green" title="E22">A1</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K27" title="SIN_COS:func.27">exp_R</a>  <a href="partfun2.html#K2" title="PARTFUN2:func.2">"</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="limfunc1.html#K4" title="LIMFUNC1:func.4">right_open_halfline</a> 0
 
<b>by </b><i><a class="ref" href="taylor_1.html#T16" target="_self" title="TAYLOR_1:th.16">Th16</a>, <a class="ref" href="funct_1.html#T55" title="FUNCT_1:th.55">FUNCT_1:55</a></i>;<br/>
<a NAME="E2:26"/><b>then </b><i><font color="Green" title="E23">A2</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K27" title="SIN_COS:func.27">exp_R</a>  <a href="partfun2.html#K2" title="PARTFUN2:func.2">"</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <a href="taylor_1.html#K3" title="TAYLOR_1:func.3">ln</a> 
 
<b>by </b><i><a class="ref" href="taylor_1.html#D2" target="_self" title="TAYLOR_1:def.2">Def2</a></i>;<br/>
<a NAME="E3:26"/>
for <font color="Olive" title="b1">d</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">d</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> 0 holds <br/><span class="p1">(<span class="default"><a href="sin_cos.html#K27" title="SIN_COS:func.27">exp_R</a>  <a href="partfun2.html#K2" title="PARTFUN2:func.2">"</a> </span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="taylor_1.html#K3" title="TAYLOR_1:func.3">ln</a>  <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">d</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c1">d</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> ;<br/>

<b>assume </b><a NAME="E1:26_1"/><i><font color="Green" title="E24">A3</font></i>: 
<font color="Maroon" title="c1">d</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> 0
 ;<br/>

<a NAME="E2:26_1"/>
<span class="p1">(<span class="default"><a href="sin_cos.html#K27" title="SIN_COS:func.27">exp_R</a>  <a href="partfun2.html#K2" title="PARTFUN2:func.2">"</a> </span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c1">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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="c1">d</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:26_1_1"/>
<font color="Maroon" title="c1">d</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> : 0 <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#E1:26_1"><i><font color="Green" title="E24">A3</font></i></a>, <a class="ref" href="limfunc1.html#D3" title="LIMFUNC1:def.3">LIMFUNC1:def 3</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="E3:26_1_1"/><i><font color="Green" title="E25">A4</font></i>: 
( <font color="Maroon" title="c2">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">d</font> &amp; <font color="Maroon" title="c2">g</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 )
 ;<br/>
<a NAME="E4:26_1_1"/>
<font color="Maroon" title="c1">d</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#K2" title="SEQ_1:func.2">.</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="c1">d</font></span>)</span>
 
<b>by </b><i><a class="txt" href="taylor_1.html#E3:26_1_1"><i><font color="Green" title="E25">A4</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="E5:26_1_1"/>
<span class="p1">(<span class="default"><a href="sin_cos.html#K27" title="SIN_COS:func.27">exp_R</a>  <a href="partfun2.html#K2" title="PARTFUN2:func.2">"</a> </span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c1">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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="c1">d</font>
 <b>by </b><i><a class="ref" href="taylor_1.html#T16" target="_self" title="TAYLOR_1:th.16">Th16</a>, <a class="ref" href="funct_1.html#T54" title="FUNCT_1:th.54">FUNCT_1:54</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:26_1"/>
<span class="p1">(<span class="default"><a href="sin_cos.html#K27" title="SIN_COS:func.27">exp_R</a>  <a href="partfun2.html#K2" title="PARTFUN2:func.2">"</a> </span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c1">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="taylor_1.html#K3" title="TAYLOR_1:func.3">ln</a>  <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c1">d</font>
 <b>by </b><i><a class="txt" href="taylor_1.html#E1:26_1"><i><font color="Green" title="E24">A3</font></i></a>, <a class="ref" href="taylor_1.html#D2" target="_self" title="TAYLOR_1:def.2">Def2</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:26"/><i><font color="Green" title="E24">A5</font></i>: 
 <a href="taylor_1.html#K3" title="TAYLOR_1:func.3">ln</a>  <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="partfun2.html#K2" title="PARTFUN2:func.2">"</a> 
 <b>by </b><i><a class="txt" href="taylor_1.html#E1:26"><i><font color="Green" title="E22">A1</font></i></a>, <a class="txt" href="taylor_1.html#E2:26"><i><font color="Green" title="E23">A2</font></i></a>, <a class="ref" href="partfun1.html#T34" title="PARTFUN1:th.34">PARTFUN1:34</a></i>;<br/>

<b>hence </b><a NAME="E5:26"/>
 <a href="taylor_1.html#K3" title="TAYLOR_1:func.3">ln</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#T62" title="FUNCT_1:th.62">FUNCT_1:62</a></i>;<br/>

<a NAME="E6:26"/><i><font color="Green" title="E25">A6</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <a href="taylor_1.html#K3" title="TAYLOR_1:func.3">ln</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="limfunc1.html#K4" title="LIMFUNC1:func.4">right_open_halfline</a> 0
 
<b>by </b><i><a class="ref" href="taylor_1.html#D2" target="_self" title="TAYLOR_1:def.2">Def2</a></i>;<br/>
<a NAME="E7:26"/><i><font color="Green" title="E26">A7</font></i>: 
for <font color="Olive" title="b1">x</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b1">x</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 holds <br/> <a href="taylor_1.html#K3" title="TAYLOR_1:func.3">ln</a>  <a href="fdiff_1.html#R1" title="FDIFF_1:pred.1">is_differentiable_in</a> <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_2"><b>proof </b></a><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/>

<b>assume </b><a NAME="E1:26_2"/><i><font color="Green" title="E27">A8</font></i>: 
<font color="Maroon" title="c1">x</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 ;<br/>

<a NAME="E2:26_2"/><i><font color="Green" title="E28">A9</font></i>: 
<font color="Maroon" title="c1">x</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> 0
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_2_1"><b>proof </b></a><div class="add">

<a NAME="E1:26_2_1"/>
<font color="Maroon" title="c1">x</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> : 0 <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#E1:26_2"><i><font color="Green" title="E27">A8</font></i></a></i>;<br/>
<b>hence </b><a NAME="E2:26_2_1"/>
<font color="Maroon" title="c1">x</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> 0
 <b>by </b><i><a class="ref" href="limfunc1.html#D3" title="LIMFUNC1:def.3">LIMFUNC1:def 3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:26_2"/>
(  <a href="limfunc1.html#K4" title="LIMFUNC1:func.4">right_open_halfline</a> 0 is  <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>  &amp;  <a href="limfunc1.html#K4" title="LIMFUNC1:func.4">right_open_halfline</a> 0 <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <a href="taylor_1.html#K3" title="TAYLOR_1:func.3">ln</a>  )
 
<b>by </b><i><a class="txt" href="taylor_1.html#E1:26"><i><font color="Green" title="E22">A1</font></i></a>, <a class="ref" href="taylor_1.html#D2" target="_self" title="TAYLOR_1:def.2">Def2</a>, <a class="ref" href="taylor_1.html#T17" target="_self" title="TAYLOR_1:th.17">Th17</a>, <a class="ref" href="fdiff_1.html#T17" title="FDIFF_1:th.17">FDIFF_1:17</a></i>;<br/>
<b>hence </b><a NAME="E4:26_2"/>
 <a href="taylor_1.html#K3" title="TAYLOR_1:func.3">ln</a>  <a href="fdiff_1.html#R1" title="FDIFF_1:pred.1">is_differentiable_in</a> <font color="Maroon" title="c1">x</font>
 <b>by </b><i><a class="txt" href="taylor_1.html#E1:26"><i><font color="Green" title="E22">A1</font></i></a>, <a class="txt" href="taylor_1.html#E4:26"><i><font color="Green" title="E24">A5</font></i></a>, <a class="txt" href="taylor_1.html#E2:26_2"><i><font color="Green" title="E28">A9</font></i></a>, <a class="ref" href="taylor_1.html#T17" target="_self" title="TAYLOR_1:th.17">Th17</a>, <a class="ref" href="fdiff_1.html#T16" title="FDIFF_1:th.16">FDIFF_1:16</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:26"/>
for <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="limfunc1.html#K4" title="LIMFUNC1:func.4">right_open_halfline</a> 0  holds 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> <a href="taylor_1.html#K3" title="TAYLOR_1:func.3">ln</a> ,<font color="Olive" title="b1">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c1">x</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="limfunc1.html#K4" title="LIMFUNC1:func.4">right_open_halfline</a> 0;<br/>

<a NAME="E1:26_3"/>
<font color="Maroon" title="c1">x</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> 0
 
;<br/>
<a NAME="E2:26_3"/><b>then </b>
<font color="Maroon" title="c1">x</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> : 0 <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="ref" href="limfunc1.html#D3" title="LIMFUNC1:def.3">LIMFUNC1:def 3</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="E4:26_3"/><i><font color="Green" title="E27">A10</font></i>: 
( <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">g</font> &amp; 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c2">g</font> )
 ;<br/>
<a NAME="E5:26_3"/><i><font color="Green" title="E28">A11</font></i>: 
<font color="Maroon" title="c1">x</font> <a href="real_1.html#K2" title="REAL_1:func.2">"</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 
<b>by </b><i><a class="txt" href="taylor_1.html#E4:26_3"><i><font color="Green" title="E27">A10</font></i></a>, <a class="ref" href="xreal_1.html#T124" title="XREAL_1:th.124">XREAL_1:124</a></i>;<br/>
<a NAME="E6:26_3"/>
1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> 
 
<b>by </b><i><a class="ref" href="xcmplx_1.html#T217" title="XCMPLX_1:th.217">XCMPLX_1:217</a></i>;<br/>
<b>hence </b><a NAME="E7:26_3"/>
0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> <a href="taylor_1.html#K3" title="TAYLOR_1:func.3">ln</a> ,<font color="Maroon" title="c1">x</font>
 <b>by </b><i><a class="txt" href="taylor_1.html#E1:26"><i><font color="Green" title="E22">A1</font></i></a>, <a class="txt" href="taylor_1.html#E4:26"><i><font color="Green" title="E24">A5</font></i></a>, <a class="txt" href="taylor_1.html#E5:26_3"><i><font color="Green" title="E28">A11</font></i></a>, <a class="ref" href="taylor_1.html#T17" target="_self" title="TAYLOR_1:th.17">Th17</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E9:26"/>
(  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <a href="taylor_1.html#K3" title="TAYLOR_1:func.3">ln</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="limfunc1.html#K4" title="LIMFUNC1:func.4">right_open_halfline</a> 0 &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <a href="taylor_1.html#K3" title="TAYLOR_1:func.3">ln</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="taylor_1.html#K3" title="TAYLOR_1:func.3">ln</a>  <a href="fdiff_1.html#R2" title="FDIFF_1:pred.2">is_differentiable_on</a>  <a href="limfunc1.html#K4" title="LIMFUNC1:func.4">right_open_halfline</a> 0 &amp; ( for <font color="Olive" title="b1">x</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b1">x</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 holds <br/> <a href="taylor_1.html#K3" title="TAYLOR_1:func.3">ln</a>  <a href="fdiff_1.html#R1" title="FDIFF_1:pred.1">is_differentiable_in</a> <font color="Olive" title="b1">x</font> ) &amp; ( for <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="limfunc1.html#K4" title="LIMFUNC1:func.4">right_open_halfline</a> 0 holds   <a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> <a href="taylor_1.html#K3" title="TAYLOR_1:func.3">ln</a> ,<font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <font color="Olive" title="b1">x</font> ) &amp; ( for <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="limfunc1.html#K4" title="LIMFUNC1:func.4">right_open_halfline</a> 0  holds 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> <a href="taylor_1.html#K3" title="TAYLOR_1:func.3">ln</a> ,<font color="Olive" title="b1">x</font> ) )
 <b>by </b><i><a class="txt" href="taylor_1.html#E4:26"><i><font color="Green" title="E24">A5</font></i></a>, <a class="txt" href="taylor_1.html#E6:26"><i><font color="Green" title="E25">A6</font></i></a>, <a class="txt" href="taylor_1.html#E7:26"><i><font color="Green" title="E26">A7</font></i></a>, <a class="ref" href="taylor_1.html#T16" target="_self" title="TAYLOR_1:th.16">Th16</a>, <a class="ref" href="taylor_1.html#T17" target="_self" title="TAYLOR_1:th.17">Th17</a>, <a class="ref" href="funct_1.html#T55" title="FUNCT_1:th.55">FUNCT_1:55</a></i>;<br/>


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